Embed Size (px)
Citation preview
OT58 Operator Theory: Advances and Applications Vol. 58
Editor: I. Gobberg Tel Aviv University Ramat Aviv, Israel
Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel
Editorial Board:
A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) L. de Branges (West Lafayette) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P. A. Fillmore (Halifax) C. Foias (Bloomington) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) J. A. Helton (La Jolla)
Honorary and Advisory Editorial Board:
P. R. Halmos (Santa Clara) T. Kato (Berkeley) P. D. Lax (New York)
Springer Basel AG
M. A. Kaashoek (Amsterdam) T. Kailath (Stanford) H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) J. D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville)
M. S. Livsic (Beer Sheva) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)
Continuous and Discrete Fourier Dransforlns, Extension Problems and Wiener-Hopf Equations
Edited by
I. Gobberg
Springer Basel AG
Editor's address:
Prof. 1. Gohberg Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel Aviv University 69978 Tel Aviv, Israel
A CIP catalogue record for this book is available from the Library of Congress. Washington D.C., USA
Deutsche Bibiiothek Cataioging-in-Publication Data
Continuous and discrete Fourier transforms, extension problems, and Wiener Hopf equations / ed. by 1. Gohberg. -Basel ; Boston ; Berlin : Birkhăuser, 1992
(Operator theory ; Vo1.58) ISBN 978-3-0348-9695-5 ISBN 978-3-0348-8596-6 (eBook) DOI 10.1007/978-3-0348-8596-6
NE: Gochberg, Izrai!' C. [Hrsg.]; GT
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law, where copies are made for other than private use a fee is payable to )>Verwertungsgesellschaft Wort«, Munich.
© 1992 Springer Basel AG Originally published by Birkhăuser Verlag Basel in 1992 Softcover reprint of the hardcover 1st edition 1992
Table of Contents
Editorial Introduction
J. Benedetto, C. Heil, D. Walnut Uncertainty principles for time-frequency operators
1. Introduction . . . . . . . . . . . . . . . . . . 2. Sampling results for time-frequency transformations 3. Uncertainty principles for exact Gabor and wavelet frames References . . . . . . . . . . . . . . . . . . . . . .
R.L. Ellis, 1. Gohberg, D. C. Lay Distribution of zeros of matrix-valued continuous analogues of orthogonal polynomials
Table of contents . . Introduction . . . . . . 1. Preliminary results . .
1.1. Matrix-valued Krein functions of the first and second kinds . 1.2. Partitioned integral operators
2. Orthogonal operator-valued polynomials 2.1. Stein equations for operators . . . 2.2. Zeros of orthogonal polynomials 2.3. On Toeplitz matrices with operator entries
3. Zeros of matrix-valued Krein functions 3.1 On Wiener-Hopf operators 3.2. Proof of the main theorem
References . . . . . .
I. Gohberg, M.A. Kaashoek The band extension of the real line as a limit of discrete band extensions, II. The entropy principle
o. Introduction I. Preliminaries II. Main results References . .
v
. Vll
1 1 6
15 24
26 26 26 30 30 35 43 43 46 52 56 56 60 68
71 71 72 80 90
VI
M. Coilar, C. Sadosky Weakly positive matrix measures, generalized Toeplitz forms, and their applications to Hankel and Hilbert transform operators 93
Introduction . . . . . . . . . . . . . . . . . . . . . . 93 1. Lifting properties of generalized Toeplitz forms and weakly
positive matrix measures. . . . . . . . . . . . . . . 95 2. The GBT and the theorems of Helson-Szego and Nehari . . 101 3. GNS construction, Wold decomposition and abstract lifting theorems 109 4. Multiparameter and n-conditionallifting theorems,
the A-A-K theorem and applications in several variables 111 References . . . . . . . . . . . . . . . . . . . . . . 11 7
J.A. Ball, 1. Gohberg, M.A. Kaashoek Reduction of the abstract four block problem to a Nehari problem 121
O. Introduction . . . . . . . 121 1. Main theorems . . . . . . 123 2. Proofs of the main theorems 126 References
A.B. Kuijper The state space method for integro-differential equations of Wiener-Hopf type with rational matrix symbols
1. Introduction and main theorems . . . . . . 2. Preliminaries on matrix pencils. . . . . . . 3. Singular differential equations on the full-line 4. Singular differential equations on the half-line 5. Preliminaries on realizations . 6. Proof of theorem 1.1 7. Proofs of theorems 1.2 and 1.3 8. An example References . . . . . . . . . .
H. Widom Symbols and asymptotic expansions
O. Introduction . . . . . . . . . I. Smooth symbols on Rn II. Piecewise smooth symbols on T III. Piecewise smooth symbols on Rn IV. Symbols discontinuous across a hyperplane in Rn x Rn References . . . . .
Program of Workshop
140
142 142 151 154 158 160 164 165 182 186
189 189 195 196 198 206 209
211
VII
EDITORIAL INTRODUCTION
The present book is a selection of papers in modern analysis of Fourier transforms and
their applications. It consists of seven papers in which the continuous and discrete Fourier
transforms appear as subjects of research, or are used as tools in solving other problems.
In some papers the interplay between the continuous and discrete Fourier transforms is
very important.
The first four papers are based on talks presented at a workshop held in the Depart
ment of Mathematics, Maryland University, College Park, September 25 and 26, 1991.
(The program of the workshop is included at the end of this volume.) This conference was
dedicated to the continuous and discrete Fourier transforms and extension problems.
The last three papers were added later. They fit the material described above and are
concerned with Wiener-Hopf equations and extension problems.
The first paper, J.Benedetto, C.Heil, D.Walnut, "Uncertainty principles for time
frequency operators" is dedicated to sampling results for frequency transformations and
uncertainty principles for exact Gabor and wavelets frames.
In the second paper, R.E.Ellis, I.Gohberg, D.C.Lay, "Distribution of zeros of matrix
valued continuous analogues of orthogonal polynomials" continuous analogues of matrix
valued orthogonal polynomials are analyzed. The number of zeros in the upper half plane
of such polynomials is computed in terms of the number of negative eigenvalues of an
associated integral operator. Both continuous and discrete Fourier transforms are used
here.
In the third paper I.Gohberg, M.A.Kaashoek, "The band extension on the real line
as a limit of discrete band extensions, II. The entropy principle" the continuous analogue
of the entropy principles is deduced from the maximum entropy principle in the discrete
case. The interplay between continuous and discrete Fourier transforms is essential.
The fourth paper, M.Cotlar, C.Sadosky, "Weakly positive matrix measures, general
ized Toeplitz forms, and their applications to Hankel and Hilbert transform operators" is
VIII
a review paper in which a number of important recent results of the same authors, are
reviewed and commented upon.
The fifth paper, J.A.Ball, I.Gohberg, M.A.Kaashoek, "Reduction of the abstract four
block problem to a Nehari problem" is dedicated to the four-block extension problem. Here
this problem in an abstract setting is reduced to a Nehari problem. The band method serves
as a tool.
The sixth paper, A.B.Kuijper, "The state space method for integro-differential equa
tions of Wiener-Hopf type with rational matrix symbols" is dedicated to the theory of
Wiener-Hopf integro-differential systems of equations. In it are analyzed and explicitly
solved such systems, where the symbol is a rational matrix function. The method is based
on the realization of the symbol.
In the seventh paper, H.Widom, "Symbols and asymptotic expansions" is proposed a
general principle which enables the computation of asymptotic expansions for the trace of
functions of Wiener-Hopf or pseudodifferential operators. This principle is based on the
use of the appropriate symbol in the continuous and discontinuous cases.
I. Gohberg
Tel-Aviv, June 1992
Operator Theory: Advances and Applications, Vol. 58 © 1992 Birkhiiuser Verlag Basel
UNCERTAINTY PRINCIPLES FOR TIME-FREQUENCY OPERATORS
John Benedetto*, Christopher Reilt, and David Walnutt
1
Abstract: This paper explores some of the connections between classical Fourier analysis and time-frequency operators, as related to the role of the uncertainty principle in Gabor and wavelet basis expansions.
1. INTRODUCTION.
This paper will explore some of the connections between classical Fourier analysis
and recent results in the area of time-frequency analysis, specifically Gabor and wavelet
theory. Our starting point is the Poisson Summation Formula, which connects Fourier
series on the circle with Fourier transforms on the line. This extraordinarily useful formula
can be used, for example, to derive sampling formulas for band-limited functions and
estimates for aliasing error when an arbitrary function is sampled. Following Section lA,
which contains a review of basic Fourier analysis, we present in Section lB a proof of this
sampling theorem and, as a corollary, the sampling theorem of Shannon and Whittaker.
Next we consider the Gabor transform, which separates the time and frequency content of
a function by using a windowed Fourier transform. The sampling theory of this transform
can be cast in the language of decompositions of arbitrary functions in sets of basis
like functions and leads us to a discussion of frames and unconditional bases for Hilbert
spaces. The wavelet transform also separates time and frequency content in a function
* Also Professor of Mathematics, University of Maryland, College Park; acknowledges partial support from NSF Grant DMS-9002420. tAlso Pure Mathematics Instructor, Massachusetts Institute of Technology; acknowledges partial support from NSF Grant DMS-9007212. tAlso Assistant Professor of Mathematics, George Mason University, Fairfax, Virginia.
2 Benedetto et al.
but substitutes scale for frequency and does not explicitly use a Fourier transform. The
sampling theory here leads to the notion of multiresolution analysis and the construction
of wavelet orthonormal bases. Sections 2A, B, and C contain the statements and proofs of
theorems related to these two time-frequency operators.
Underlying our entire discussion is the classical uncertainty principle, which
limits the degree to which a function can be localized both in time and frequency. The
sampling and decomposition formulas outlined in the first two sections of the paper can be
seen as the realization of an arbitrary function as combinations of functions well-localized
in time and frequency. The ~ncertainty principle leads us to ask how well-localized those
basis-like functions can be. For band-limited functions and the Fourier transform, the
sparsest set of sampling which will permit exact reconstruction requires functions that are
not well-localized in time, namely sinl;x) and its translates. A denser sampling set leads
to better localized basis functions. For the Gabor transform, the Balian-Low theorem tells
us how well we can do for bases of Gabor functions, namely, that simultaneous quadratic
localization is not possible in both time and frequency. In Section 3A we present an
operator-theoretic proof of this theorem due to Battle. In this case, as with the previous
one, allowing denser sets of sampling and overdetermined systems (frames) enables us to
get better localization. For the wavelet transform, we present a generalization of a theorem
due to Battle which states that any wavelet basis cannot have simultaneous exponential
localization in time and frequency.
A. Basic Fourier analysis. Let L2[-T /2, T /2) denote the Hilbert space of T-periodic,
square-integrable functions. Given J E L2[-T /2, T /2), the Fourier coefficients of J are
1 jT/2 cn(f) = - J(t) e-27rint/T dt. n -T/2
Then we have
(Ll)
where the sum converges in L2[-T /2, T /2). The Plancherel formula is
Benedetto et al.
and the Parseval formula is
JT / 2
J(t)g(t) dt -T/2
for every J, 9 E L2 [-T/2, T/2).
3
n
Let L2(R) denote the Hilbert space of functions on the line which are square
integrable. Given J E L2(R), the Fourier transform of J is
where the integral is defined as an L2-limit in the usual way if it does not converge abso
lutely. Then we have
where, as before, the integral is defined as an L2-limit if it does not converge absolutely.
The Plancherel formula is
and the Parseval formula is
for every J, 9 E L 2(R).
B. The Poisson Summation Formula. We define PWn to be the space of functions J E
L2(R) such that supp(!) c [-f2, f2]. We call this the space of f2-band-limited functions.
The Poisson Summation Formula (PSF) is:
THEOREM 1.1. Let J E L2(R) be such that
IJ( t)1 :::; C(1 + ltD-a and (1.2)
for some C > 0 and a> 1. Then J and j are continuous functions, and, for any P > 0,
L J(t + nP) = ~ L j(n/ P) e21rint/P (1.3) n n
4 Benedetto et al.
and
L::i(-y+nP) = ~ L::J(n/P)e-27riR"Y/P. (1.4) n n
PROOF: By (1.2), the function F(t) = 'LJ(t + nP), the P-periodization of J, converges
absolutely and uniformly on compact sets and is in L2[-P/2, P/2). Moreover,
jP/2 L::J(t + nP)e-27rint/P dt = jJ(t)e-21fint/p dt = f(n/P). -P/2 n
Therefore, by (1.1),
so (1.3) holds. Equation (1.4) follows similarly. I
If J E L2(R) is compactly supported then the P-periodization of J is an element
of L2[-P/2, P/2). In this case, (1.3) holds as written with the sums on each side converging
in L2[-P/2, P/2), and the assumption (1.2) is not required. The right side of (1.3) is the
Fourier series of the function defined by the sum on the left. Similarly, if J E PWn then
(1.4) holds as written with the sums converging in L2[-P/2, P/2).
We now state and prove as a corollary to the PSF the sampling formula of
Shannon and Whittaker.
THEOREM 1.2. Let J E PWn. Then given T ~ 2f2,
J(t) = ~"J(n/T) sin7r(Tt - n). T L..J 7r(Tt - n)
n
Moreover,
j IJ(tW dt = ~ L:: IJ(n/TW· n
PROOF: By (1.3) and the remarks following Theorem 1.1,
L::i(-y+nT) = ~L::J(n/T)e-21fin'Y/T. n n
We can write
jT/2 L f( -y + nT) e21fit 'Y d-y = -T/2 n
1jT/2 - E J(n/T) e-27ri'Y(n/T-t) d-y. T -T/2 n
(1.5)
(1.6)
Benedetto et al. 5
Because the left hand side of (1.6) converges in Ll[-T/2,T/2) we have
Also, since the right hand side of (1.6) is the Fourier series of a function in L2[-T/2, T/2)
we can integrate term-by-term, obtaining
L f(n/T)'!" (T/2 e27ri-y(t-n/T) d"{ = L f(n/T) sin 7I"(Tt - n). T LT/2 7I"(Tt - n)
n n
. Finally, (1.5) follows since {v'T;j(;.~~)-n)} is an orthonormal set in L2[-T/2, T/2). I
We conclude this section with a variant of the sampling formula which is in
keeping with the spirit of this paper. S(R) denotes the Schwartz class of infinitely differ
entiable, rapidly decreasing functions on the line.
THEOREM 1.3. Let f E PWn and let T > 20 be given. Then there exists a function
s E S(R) such that 1
f(t) = fjLf(n/T)s(t-n/T). n
PROOF: Let s be be any function such that S E COO(R), s = 1 on [-0, OJ, and s = 0
outside [-T/2,T/2]. Then clearly s E S(R) and l("{)s("{) = l("{). The PSF implies that
J L l("{ + nT) s("{) e27rit -y d"{ = ~ J L f(n/T) e-27ri-y(n/T-t) s("{) d"{. (1.7) n n
Using an argument similar to one used in the proof of Theorem 1.2, the left hand side of
(1. 7) is
f l("{) s("{) e27rit-y d"{ = J 1("{)e27rit-y d"{ = f(t),
and the right hand side of (1.7) is
Lf(n/T) ~ f s("{)e27ri-y(t-n/T) d"{ = ~ Lf(n/T)s(t - niT). I n n
6 Benedetto et al.
2. SAMPLING RESULTS FOR TIME-FREQUENCY TRANSFORMATIONS.
A. Time-frequency operators and frames. In this section we investigate some results
concerning two time-frequency transformations, the continuous Gabor transform and the
continuous wavelet transform. These operators are closely related to the Fourier transform
and display simultaneously the time and frequency content of a signal.
DEFINITION 2.1. Let 9 E L2(R) be fixed. The continuous Gabor transform (GGT) with
analyzing function g, denoted W g, is defined by
wgf(t,w) = i: f(x)e-21fiwx g(x - t)dx,
where f E L2(R).
In the above definition, t is the time variable and w the frequency variable. The
transform wgf(t,w) is formed by shifting the window function 9 so that it is centered at t,
then taking the Fourier transform. In this way, wgf(t,w) displays the frequency content
of f near time t.
The definition of the continuous wavelet transform uses the same idea of a sliding
window but replaces the frequency variable by a scale variable.
DEFINITION 2.2. Let 1j; E L2(R) be fixed. The continuous wavelet transform (GWT) with
analyzing function 1j;, denoted ell "', is defined by
1 100 (X - b) elI",f(a,b) = ~ -00 f(x)1j; -a- dx,
where f E L2(R).
Note that ell", is a time-frequency localization operator where a is the scale
(frequency) variable and b the time variable.
We now present some elementary properties of the CGT and CWT. Proofs of
these results may be found in numerous places, e.g., [Dl; HW]. We point out here that
the results are immediate consequences of the Plancherel and Parseval formulas for the
Fourier transform.
Benedetto et al.
THEOREM 2.3. Let 9 E L2(R). Then for all f E L2(R),
i: i: Iwgf(t,wW dtcU.J = i: Ig(xW dx i: If(xW dx.
THEOREM 2.4. Let t/J E L2(R) and suppose that
Then
JOO l~b)12 -1-1- d"l < 00.
-00 "I
[00 [00 I~",f(a, bW db da 10 Loo a
= [00 li(w)12 dw [00 l~b)12 d"l 10 10 1"11
+ [0 li(wW cU.J [0 l~bW d"l. Loo Loo 1"11
7
The Poisson Summation Formula can be seen as a sampling result which char
acterizes the recovery of a function from regular samples of its Fourier transform. In some
cases the recovery is exact, for example when f is supported in the interval [0,1]. Such a
sampling result can be cast in the language of Fourier series, i.e., in the representation of
a function as a superposition of complex exponentials. In the case of the time-frequency
transformations considered here, there is a well-developed theory of the recovery of a func
tion from samples of its CGT or CWT. Here recovery is exact in all of L2(R), and, as in
the Fourier series case, sampling results can be cast in the language of the representation of
a function as a superposition of a fixed collection of basic functions. The most convenient
setting for such representations is that of a frame in a Hilbert space, first described by
Duffin and Schaeffer [DS) in relation to non-harmonic Fourier series.
DEFINITION 2.5. Let H be a separable Hilbert space. A collection {xn} CHis a frame
if there exist constants A, B > 0 such that for all x E H,
n
A frame is said to be tight if A = B and exact if {x n } ceases to be a frame upon the
removal of any element.
We say {x n } is a Riesz basis if there exist constants A, B > 0 such that A ~
IIxnll ~ B for all n and if every x E H can be written x = 2: cn(x) Xn for a unique choice of
8 Benedetto et al.
scalars {cn ( x)} and this sum converges unconditionally (every rearrangement of the sum
also converges, and to the same value).
A frame is a generalization of an orthonormal basis or a Riesz basis. With a
frame, one has representations of elements in the Hilbert space in terms of the frame, and
one can recover elements in a constructive way from the frame coefficients {(x, X n) }. The
difference is that frames may be overcomplete in the sense that each element of a frame is
in the closed linear span of the remaining elements. The following theorem collects some
standard results about frames, proofs of which can be found in, e.g., [DSj HWj DIJ.
THEOREM 2.6. Let H be a separable Hilbert space and {xn} a collection of vectors in H.
Then
(a) {xn} is a frame for H if and only if the operator S defined by Sx = 2:(x, xn) Xn
converges in H for each x and is a topological isomorphism of H onto itself. In this
case, we have the identities x = 2:(x, xn) S-1 Xn = 2:(x, S-1 Xn) Xn, and {S-1Xn}
is also a frame for H, called the dual frame of {x n }.
(b) {xn} is an exact frame if and only if {xn} is a Riesz basis for H.
(c) H {Xn} is such that Ilxnll = 1 for all n, and if {xn} forms a tight frame with
A = B = 1, then {xn} is an orthonormal basis for H.
PROOF: For a proof of (a), see, e.g., [HW, Theorem 2.1.3J. For (b), see [HW, Theorem
2.2.2J. To see (c), observe that for n fixed,
II Xnl1 2 = I: I(Xn,XkW = IIxnll4 + I: I(Xn,XkW· k k#n
Since IIxnll = 1, (Xn,Xk) = 0 for all n -# k. Thus {xn} is an orthonormal system. Because
of (a), the span of {Xn} is dense in H, i.e., {xn} is complete in H. A complete orthonormal
sequence is an orthonormal basis. I
B. Frames and sampling for the CGT. In this section we investigate the connections
between sampling results for the CGT and the existence of frames in the Hilbert space
£2(R).
DEFINITION 2.7. Let a, b > 0 and 9 E £2(R) be fixed. Then ifthe collection {e211"imbxg(x_
na)} is a frame for £2(R), it is called a Gabor frame. We will denote the elements of such
Benedetto et aI. 9
a frame by {EmbTnag}, where
In his fundamental paper [GJ, Gabor investigated the case when a = b = 1 and
g(x) = e-lI'X2 (d., Section 3A). In this case, the collection {EmTng} is complete in L2(R)
but does not form a frame. The idea of applying the notion of a frame to sets of the form
{EmbTnag} was first described in [DGM].
The following result can be viewed as a sampling theorem for the CGT. Its proof
is immediate from the definition of a Gabor frame.
THEOREM 2.8. The collection {EmbTnag} is a Gabor frame if and only if there exist
constants 0 < A < B such that for all J E L2(R),
n m
We now introduce the Zak transform, which we will use to partially characterize
Gabor frames (see [AT; D; HW; J]). We let Q denote the unit square, i.e., Q = [0,1)2.
DEFINITION 2.9. Given a > 0, the Zak transform, or Weil-Brezin map, denoted Za,
is the unitary mapping from L2(R) onto L2(Q) given by
ZaJ(t,w) = a1/ 2 L e2l1'ikw J(a(t - k)). k
ZaJ satisfies the following quasi-periodicity relations:
ZaJ(t + 1,w)
ZaJ(t,w + 1) ZaJ(t,w).
LEMMA 2.10. Suppose that ab = liN for some integer N :::: 1. Then
THEOREM 2.11. Let 9 E L2(R), and suppose that ab = liN for some a, b > 0, NEZ.
Then in order that {EmbTnag} be a Gabor frame, it is necessary and sufficient that there
exist constants A, B > 0 such that N-1
A < L IZag(t,w-jINW < B. (2.1) j=O
10 Benedetto et al.
PROOF: Given 1 E L2(R), we have by the unitarity of Za, Lemma 2.10, and Plancherel's
formula for Fourier series,
L: ~ l(f, EmbTnagW n m
= LLll111 Zal(t,W)Zag(t,w-m/N)e-21rimt/N e21rinWdtdwl2 n mOO
= EL:L:ll111 Zal(t,W)Zag(t,w-j/N)e-21rijt/N e-21rikte21rinWdtdwI2 j=O n k 0 0
1 1 N-l
= llIZal(t,wW ~ IZag(t,W - j/N)ldtdw.
Therefore, if (2.1) holds then
1 1 N-l
A IIZa/lli2(Q) ~ llIZal(t,wW L IZag(t,w - j/N)I dtdw ~ B IIZa/lli2(Q)' o 0 j=O
which by the unitarity of Za is
n m
Conversely, suppose (2.1) does not hold, say that N-l
ess inf L IZag(t,w-j/NW = o. (t,w)EQ j=O
Then given f > 0, there is a set E C Q of positive measure such that N-l L: IZag(t,w - j/NW < f
j=O
for (t,w) E E. Let 1 E L2(R) be such that
1 Zal(t,w) = IEII/2 XE(t,W)
for (t,w) E Q. Then 11/112 = IIZa/llL2(Q) = 1, but
1 1 N-I
llIZal(t,wW L: IZag(t,w-j/N)ldtdw = L:L:I(f,EmbTnagW < f.
o 0 j=O n m
That is, {EmbTnag} fails to have a lower frame bound. A similar argument shows that
{EmbTnag} fails to have an upper frame bound if N-l
ess sup L IZag(t,w - j /N)1 2 = +00. I (t,w)EQ j=O
Benedetto et aI. 11
C. Frames and sampling for the CWT. In this section we investigate the connections
between sampling results for the CWT and the existence of orthonormal bases of wavelets
in L2(R).
DEFINITION 2.12. Let t/J E L2(R) be fixed. Then ifthe collection {2i/2 t/J(2ix - k)} is an
orthonormal basis for L2(R), it is called a wavelet orthonormal basis.
For simplicity, we will write t/Jjk(X) = 2j/2 t/J(2 jx - k). The existence of wavelet
orthonormal bases has been known since Haar, who showed that if t/J = X[O,1/2) - X[1/2,1)
then {t/Jjd is a wavelet orthonormal basis, known as the Haar system. More recently,
Meyer has shown that there exists t/J, infinitely differentiable and rapidly decreasing such
that {tPjd is a wavelet orthonormal basis (see Section 3B) and Daubechies has shown that
for any r > 0, there exist t/J, compactly supported and r-times continuously differentiable
such that {t/Jjd is a wavelet orthonormal basis. Wavelet frames, especially those which
are exact, are discussed in Section 3B.
The following result can be viewed as a sampling theorem for the CWT. Its
proof is immediate from the definition of a wavelet orthonormal basis.
THEOREM 2.13. Ift/J E L2(R) is such that 1It/J1I2 = 1, then {tPjd is a wavelet orthonormal
basis if and only if
IIfll~ = LLI IP",f(2-i ,2-i k)1 2 •
j k
In order to partially characterize wavelet orthonormal bases, we introduce the
notion of a multiresolution analysis due to S. Mallat (see [Mj Maj D2]).
DEFINITION 2.14. A multiresolution analysis (MRA) of L2(R) is a collection of closed
subspaces {Vj} jEZ such that the following conditions hold.
(a) Vj C Vi+l for all j.
(b) UVj is dense in L2(R).
(c) nVj = {o}.
(d) f(x) E Vj if and only if f(2x) E Vj+! for all j.
(e) There exists a function r.p E L2(R) such that the collection {r.p(x - k)hEZ is an
orthonormal basis for Vo.
12 Benedetto et al.
Given a MRA, one can construct a wavelet orthonormal basis.
LEMMA 2.15. Let 9 E L2(R). Then {g(x - k)hEZ is an orthonormal set if and only if
EnEZ 19b + n W == 1.
PROOF: {g(x - k)hEZ is an orthonormal set if and only if g(x) is orthogonal to g(x - k)
for all k -::j:. 0 and IIgll2 = 1. That is, if and only if
j g(X)9(X-k)dX = jlgbWe-27rik'Yd'Y = 11'Llgb+nWe-27rik'Yd'Y = 6k. o nEZ
This is true if and only if EnEZ 19b + n W == 1. I
The following result is due to Y. Meyer.
THEOREM 2.16. Let {Vi} be a MRA. Then there exists"p E L2(R) such that {'IjJjd is a
wavelet orthonormal basis.
PROOF: Since Vi C Vj+1, we may define Wj as the orthogonal complement of Vj in Vj+1,
so that Vi+1 = Vi EB Wj for all j. Then {Wj} is a collection of mutually orthogonal
closed subspaces of L2(R). Let Pj and Qj be the orthogonal projectors onto Vi and Wj,
respectively. Then Pj+1 = Pj + Q j for all j.
Now suppose that "p E L2(R) is given so that {'IjJok} is an orthonormal basis
for Woo Then we claim that {'IjJjd is a wavelet orthonormal basis for L2(R). To see this,
note that I(x) E Wo if and only if 1(2jx) E Wj. Therefore for each fixed j, {'IjJjkhEZ is
an orthonormal basis for Wj. In particular, {"pjk}j,kEZ is an orthonormal system. To see
that {'IjJjk} is complete, note that for each I E L2(R), Pjl ---+ I and P_jl ---+ 0 as j ---+ 00.
Thus, III - (Pj - P-j)11I2 ---+ 0 as j ---+ 00. However,
j-1 j-1 Pj - P_j = 'L (Pk+1 - Pk) = 'L Qk.
k=-j k=-j
Since Qjl lies within the closed span of {"pjkhEZ for each j, we conclude that the span
of {'IjJjk}j,kEZ is dense in all of L2(R) and that {'IjJjdj,kEZ is a wavelet orthonormal basis.
In order to construct such a "p, note that since cP E Vo C V1 and since {cpa hEZ
is an orthonormal basis for V1 , cp satisfies the dilation equation
cp(x) = LhkCP(2x-k), k
Benedetto et al.
where hk = 21/ 2 ('P,'Plk). Now define
t/J(x) = 1)-1)kh1_k'P(2x-k). k
We claim that {t/JOk} is an orthonormal basis for Wo. In order to see this, define
mo(,) = ~ Lhke-211"ik, k
and m1{-r) = ~ L (_l)k h1- k e-211"ih.
Then
0{-r) = mo{-r/2)0{-r/2),
~(-r) = m1{-r/2)0{-r/2),
k
m1 (,) = e211"ih+1/2) mo{-r + 1/2).
By Lemma 2.15, since {'POn} is an orthonormal set,
n
n
n even n odd
Imo(,/2W L 10{-r/2 + j)12 + Imo{-r/2 + 1/2W L 10{-r/2 + j + 1/2)12 j j
= Imo(,/2W + Imo{-r/2 + 1/2W·
Therefore, by Lemma 2.15,
Imo(,/2W + Imo{-r/2 + 1/2W == 1.
And, since Im1{-r)1 = Imo{-r + 1/2)1,
Then, as above,
n
13
(2.2)
14 Benedetto et al.
so that {tPOIe} is an orthonormal set by Lemma 2.15.
It remains to show that Wo = span{tPole}. Now, for any I E L2(R),
Pol = L Ck <POle Ie
where Cle = (I, <POle), and
(Pof)"(-y) = c('Y)<j?(-y) = C('Y) mo(-y/2) <j?(-y/2),
(PI!)"('Y) = a(-y/2) <j?(-y/2) ,
where a('Y) = 2-1/2 L;ale e-211'ile,,( E L2[0, 1] and ale = (I,<Plk)'
Consider the system of equations
(2.3)
(2.4)
( mo(-y) m1(-Y») (c(-y») ( a(-y) ) mo(-y + 1/2) m1 (-y + 1/2) d(-y) - a(-y + 1/2) , (2.5)
where a( 'Y) E L2 [0, 1] is given. The determinant of the matrix on the left reduces to
which is non-zero for all 'Y. The solution of the system is therefore
C(-y) = e-27ri("(+l/2) (m1(-Y + 1/2)a(-y) - ml('Y)a(-y + 1/2»,
d('Y) = e-211'i(,,(+1/2) (mo(-y)a('Y + 1/2) - mo(-y + 1/2)a(-y».
Since mo and ml are bounded, c, dE L2[0, 1].
Now, for any I E L2(R), Qol = PI! - Pol, and so by (2.3) and (2.4),
(Qof)"(-y) = (a('Y/2) - c(-y)mo(-y/2» <j?('Y/2).
However, by (2.5), a(-y/2) - c('Y)mo('Y/2) = d(-y)ml(-y/2), so that
(Qof)"('Y) = d(-y)ml(-y/2)<j?(-y/2) = d(-y)~(-y)
for some d(-y) = L;dlee-27rile"( E L2[0, 1]. Thus, Qol = L;dletPole, which implies that the
span of {tPOIe} is dense in Wo, and hence that {tPi,.} is a wavelet orthonormal basis for
L2(R). I
In some cases it is possible to define a MRA starting with the auxiliary function
mo defined by (2.2). The following result of Cohen [e], which we state without proof,
characterizes when this is possible.
Benedetto et al.
THEOREM 2.17. Let hk be a real-valued sequence and suppose that
(a) L: Ihkl kE < 00 for some f > 0,
(b) mo(O) = 1,
(c) ImoC!W + ImoC! + 1/2W = 1 for all,.
Let 'P be defined by 00
<PC!) = II moC!/2i). i=l
15
Then {'P( x - k)} is an orthonormal set if and only if there exists a compact set K such
that
(d) K is a finite union of closed intervals of total measure 1.
(e) for every x E [-1/2, 1/2J there is an integer k such that x + k E K,
(f) K contains a neighborhood of zero,
(g) there exists f > 0 such that Imo(2-k,)1 ~ f for all k ~ 1 and, E K.
In this case, taking Vo = span{'P(x - k)} and defining Vi = U(2- i x) : f E Vo}, the {Vi}
are a MRA, and therefore a wavelet orthonormal basis exists for L 2(R).
In the special case obtained by putting K = [-1/2, 1/2]' conditions (d )-(g) are
satisfied if and only if ImoC!)1 > 0 for hi < 1/4. This fact was proved by Mallat [Maj.
3. UNCERTAINTY PRINCIPLES FOR EXACT GABOR AND WAVELET FRAMES.
The previous sections described several time-frequency analysis techniques, in
cluding Gabor and wavelet frames. In applications, exact frames (especially those which
are actually orthonormal bases) are often appealing, as they can often be implemented
with fast discrete algorithms. We focus on such exact systems in this section, establishing
some limitations on the joint time-frequency concentration of the elements of Gabor and
wavelet systems which form exact frames. These results may thus be considered "uncer
tainty principles" for such nonredundant systems.
First, however, we review some general results on exact frames and the classical
uncertainty principle in Hilbert spaces. By Theorem 2.6(a), any exact frame {x n } for
a Hilbert space H is a Riesz basis for H. Moreover, cn(x) = (x, xn) where {Xn} is the
16 Benedetto et al.
dual frame to {xn}, i.e., xn = S-lXn. Therefore, {xn} and {xn} are biorthonormal, i.e.,
(xm,xn) = Dmn.
The following inequality is the classical uncertainty principle. Its proof, which
is quite elementary, was recorded by N. Wiener and H. Weyl.
Given a Hilbert space H and given (not necessarily continuous) operators A,
B mapping domains D(A), D(B) c H into H, respectively, define the commutator of
A, B to be the operator [A, B] = AB - BA. If A is self-adjoint then the expectation of
A at I E D(A) is Ef(A) = (AI,!), and the variance of A at I E D(A2) is aJ(A) =
Ef (A2) - Ef(A)2. An uncertainty principle inequality can now be formulated on H. Its
statement (Theorem 3.2) and simple computational proof are part of the folklore in the
Hilbert space community.
THEOREM 3.2. Given self-adjoint operators A, B on a Hilbert space H. If IE D(A2) n
D(B2) n D(i[A,B]) and 11/11 = 1 then Ef(i[A,B])2 :::; 4aJ(A)aJ(B).
The classical uncertainty principle (Theorem 3.1) follows as a corollary. Define
the position operator P (operating on functions f) by
PI(t) = tl(t),
and the momentum operator M by
MI = (Pit = (-y i(-YW·
The domains of P, M include the Schwartz class S(R) c L2(R), and both P, Mare
self-adjoint. If IE S(R) then
(f'Y' = 27ri pi, I' 27ri MI,
[P,M]I -2~J. (3.1)
From Theorem 3.2 we therefore have
Ef( -2~I)2 < aJ(P)aJ(M),
Benedetto et a1. 17
where I is the identity operator on S(R). However, Ef(I) = IIfll~, C1}(P) = IIPfll~ -(Pf,f)2 ~ IIPfll~, and C1}(M) = IIMfll~ - (Mf,J)2 ~ IIMfll~, from which Theorem 3.1
follows immediately for f E S(R). A standard closure argument extends the inequality to
all f E L2(R).
A. An uncertainty principle for exact Gabor frames. If ab = 1 and {EmbTnag}
is a Gabor frame for L2(R) then it must be exact. Conversely, if {EmbTnag} is an exact
Gabor frame then necessarily ab = 1. By dilating 9 if necessary we therefore assume
a = b = 1 in this section without loss of generality, and for simplicity write gmn(x) = EmTng(x) = e21rimx g(x - n), and let Z denote the Zak transform Z = Za = Zt. Note
that the dual frame to {gmn} is another exact Gabor frame {gmn}, where g = s-tg is a
uniquely determined function in L2(R).
The Balian-Low Theorem (BLT) is an uncertainty principle-like result for exact
Gabor frames. It imposes severe restictions on the time-frequency localization of any
function 9 which generates an exact Gabor frame:
THEOREM 3.3. Given 9 E L2(R). If {gmn} is an exact Gabor frame then
In particular, note that if 9 is the Gaussian function g(x) = e-1rX2 then {gmn}
is not an exact Gabor frame.
The BLT was first stated by Balian [B], and later (independently) by Low [L],
for the special case of Gabor systems which are orthonormal bases. Their proofs contained
a technical gap which was filled by Coifman, Semmes, and Daubechies [Dl]; this group
also extended the result to all exact Gabor frames. The resulting proof required the use
of the Zak transform and, implicitly, both classical and distributional differentiation. Ex
plicit justification for this proof, along with a non-distributional version, was presented in
[BHW]. As all versions of this proof of the BLT are quite technical, we shall instead dis
cuss a proof inspired by an elegant, natural, and elementary argument due to Battle [Bat],
originally stated for the case of orthonormal bases only, extended formally to exact frames
by Daubechies and Janssen [DJ], and justified in [BHW]. This argument depends criti
cally on (3.1), from which the classical uncertainty principle can also be derived. Precisely,
18 Benedetto et al.
(3.1) is used in the following form:
LEMMA 3.4. Given i, g E L2(R). If Pi, Pg, P j, Pg E L2(R) tben
(Pi,Mg) - (Mi,Pg) = 2~i (f,g). (3.2)
Note that (3.2) follows immediately from (3.1) for i, g E S(R), and can be
extended to all i, g E P(R) satisfying the hypotheses of Lemma 3.4 by a closure argument.
Lemma 3.4 suffices to prove a weak version of the BLT:
THEOREM 3.5. Given g E L2(R). If {gmn} is an exact Gabor frame tben
PROOF: Assume Pg, Pg, Pg, Pg E L2(R). An easy computation yields the formulas
Since {gmn} and {gmn} are biorthonormal, and recalling that g = goo and 9 = goo, it
follows that
Since e211'imn = 1 for all m, n, we therefore have
(Pg,Mg) = L(Pg,gmn) (gmn,Mg)
L((Pg)mn,g) (g, (Mg)mn)
= L(gmn,pg) (Mg,gmn)
(Mg, Pg).
Therefore, (g,g) = 0 by (3.2). However, we also have (g,g) = 1 by biorthonormality, a
contradiction. I
Note that Theorems 3.3 and 3.5 are equivalent if {gmn} is an orthonormal basis,
for then g = g. We demonstrate now that Theorems 3.3 and 3.5 are equivalent in general,
for which it suffices to prove:
Benedetto et al. 19
PROPOSITION 3.6. Given g E £2(R). If {gmn} is an exact Gabor frame then
(3.3)
and
(3.4)
PROOF: We give only a formal argument, due to Daubechies and Janssen [DJj for (3.3);
the proof of (3.4) is entirely symmetricaL
Given! E £2(R), we formally compute
ZP!(t,w) = t L!(t+k)e211'ik", + L!(t+k)ke211'ik",
t Z!(t,w) + 2~JhZ!(t,w). (3.5)
Note that from Lemma 2.10, Zgmn = E(m,n)Zg and ZYmn = E(m,n)Zg, where
E(m,n)(t,w) = e211';mt e211'in",. It therefore follows from the biorthonormality of {gmn} and
{Ymn} that Zg = 1/Zg. Therefore, using (3.5) we formally compute
ZPg(t,w) = Z / ) - -21 . ~(1/Zg)(t,w) g t,w 11"
= t Zg(t,w) + ~ ~Zg(t,w)
Zg(t,w)2
ZPg(t,w) = Zg(t,w)2 . (3.6)
From Lemma 2.10, IZgl is an essentially constant function (i.e., is bounded below from
zero and above from infinity). It therefore follows from (3.6) that ZPg E L2( Q) if and
only if ZPy E £2(Q), from which (3.3) follows by the unitarity of Z. I
The formal arguments in the proof of Proposition 3.6 above are justified in
[BHW] through a mixture of classical and distributional differentiation. We therefore
pose the following problem which, if answered positively, would establish Proposition 3.6
without the use of differentiation.
PROBLEM 3.7. Given g E L2(R) such that {gmn} is an exact Gabor frame. Assume
! E £foc(R) is such that!· gmn E Ll(R) and (f,gmn) = 0 for all m, n E Z. Does it then
follow that ! = O?
20 Benedetto et al.
That Proposition 3.6 follows from a positive answer to Problem 3.7 is established
as follows. Assume {gmn} is an exact Gabor frame and that Pg E L2(R). Note that
Pg E Lloc(R) and Pg . gmn = 9 . Pgmn E Ll(R) for all m, n. Moreover,
(Pg,gmn) = (g,Pgmn )
= (Zg,ZPgmn )
(1/ Zg, E(m,n)ZPg)
= (ZPg/(Zg)2, E(m,n)Zg)
= (ZPg/(Zg)2, Zgmn). (3.7)
Note that ZPg E L2(Q) since Pg E L2(R). As IZgl is essentially constant, it follows
that ZPg/(Zg)2 E L2(Q). Therefore h = Z-1(ZPg/(Zg)2) E L2(R). From (3.7) and the
unitarity of Z, we therefore have
for all m, n. Hence, if the answer to Problem 3.7 is positive then Pfj = h E L 2 (R).
We note that the analogue of Problem 3.7 with the exact Gabor frame {gmn}
replaced by an arbitrary (exact) frame is false. For example, if {-!Pjd is the Haar system
(cf., Section 2C), then t/J E Ll(R)nL2(R) and Jt/J(x)dx = O. Therefore, taking f == 1 we
have f E Lloc(R) yet (j,t/Jjk) = 0 for all j, k. Other examples relevant to Problem 3.7 are
given in [BHW].
B. An uncertainty principle for exact wavelet frames. The previous section estab
lished that all exact Gabor frames suffer a severe time-frequency localization constraint.
In this section we will consider whether exact wavelet frames suffer any similar constraints.
We will use the notation t/Jjk(X) = 2i/2 t/J(2ix - k), cf., Section 3C.
First, note that the classical example of a wavelet orthonormal basis {1/Jjd, i.e.,
the Haar system t/J = X[O,1/2) - X[1/2,1), satisfies lit t/J(t) 112 II, ~(-y )112 = +00. However,
the "wavelet BLT" is false in general, as shown by the first modern example of a wavelet
orthonormal basis. The Meyer wavelet is the function t/J E L2(R) defined by ~(-y) =
Benedetto et al. 21
ei 'Y/2 w(hl), where
and v E COO(R) is such that v(r) = 0 for, ::; 0, v(,) = 1 for, ~ 1,0::; v(r) ::; 1 for, E
[0,1], and v(r) + v(1-,) = 1 for, E [0,1]. That {1Pjd is an orthonormal basis for L2(R)
is shown in [M]. As .,jy E C~(R), it follows that tP E S(R). Thus tP has better than any
polynomial localization in both time and frequency, i.e., IIp(t)tP(t)1121Iq(r).,jy(r)112 < +00 for all polynomials p, q. However, tP does not possess exponential localization in both time
and frequency, and it is therefore natural to ask whether there exist wavelet orthonormal
bases, or, more generally, exact wavelet frames, which do. Battle [Ba2] answered this in
the negative for the case of wavelet orthonormal bases, and we now extend this result in a
weak manner to exact wavelet frames.
First, however, note that while the dual frame of any Gabor frame is itself a
Gabor frame, it is not always the case that the dual frame of any wavelet frame is itself a
wavelet frame [Dl]. However, this is true for exact wavelet frames. For, if {tPjd is an exact
wavelet frame then there exists a unique function If; E L2(R) such that (tPjk, If;) = bj bk,
namely If; = S-ltP. It follows then that (tPjk,lf;jlk l) = bjjl bUI, whence {If;jd is the dual
frame to {tPjd. We have then:
THEOREM 3.8. Given tP E L2(R). If {1Pjd is an exact wavelet frame then
The proof of Theorem 3.8 depends on the following two lemmas.
LEMMA 3.9. Given tP E L2(R) such that {1Pjd is an exact wavelet frame. If.,jy E C(R) n
Ll(R) n LOO(R) and ~ E C(R) n Ll(R) n LOO(R) then .,jy(0) = 0 = ~(O).
PROOF: Note that If; E Co(R); therefore If;(xo) i= 0 for some dyadic point Xo = 2-jo ko
where jo, ko E Z are not both zero. Given j > jo, set kj = 2j - jo ko. Then
o (tPjkj' If;)
(¢jkj,lf;)
22 Benedetto et al.
Now,
pointwise as j -t 00. Also,
= Ti/2 j e-21ri2 -;-yk; "j;(Ti,)~(,)d,
= Ti/2 j e-21rixO'Y"j;(Ti,)~({)d,.
and -¢ E Ll(R), so by (3.8) and the Lebesgue Dominated Convergence Theorem,
= "j;(0) j e21rix O'Y ~(() d,
= "j;(O)-¢(xo).
As -¢(xo) i= 0 we conclude that "j;(0) = O. By symmetry, -¢(O) = 0 as well. I
The following result extends Lemma 3.9 to general moments.
(3.8)
LEMMA 3.10. Given 1jJ E L2(R) such that {1jJid is an exact wavelet frame. If"j;, -¢ E
C N +1(R) n LOO(R) and
(1 + 1,J)N+l "j;({), (1 + 1,J)N+l ~(() E Ll(R)
then
for m = 0, ... , N.
PROOF: That J 1jJ(x)dx = "j;(0) = 0 follows from Lemma 3.9. Assume now that
iffi(Dffi"j;)(O) = jXffi 1jJ(X)dX = 0
for m = 0, ... , k -1 for some k :::; N, and assume Dk"j;(O) i= o. Then
Benedetto et a1. 23
where Rk is the Taylor remainder
for some e between 0 and 'Y. As Dk~ is continuous, there exists some dyadic point Xo =
2-io ko such that Dk~(xo) =1= O. Given i > io, set kj = 2j - jo ko. Then, as in (3.8),
Now,
Thus
o = J e-21riXo-r~(2-j'Y)~("'()d'Y = J e-21rixo-r 'Yk~:~tO) ~("'()d'Y + J e-21rixo-r Rk(Tj'Y)~("'()d'Y
= II (j) + I2(j).
= Cl 1'Ylk+1 < Cl (1 + l'YI)N+l 2i(k+l) 2i(k+l)
I2(j) ~ 2ig~1) J (1 + l'YI)N+1I~(",()1 d'Y = 2ig:1)'
so I 2(j) E O(2-i (k+l»). Also,
C ·k --..".----
= 23i~ ((i'Y)k~('Y)r(xo)
C ·k_-=--_ 2:~ Dk~(xo).
Thus Il(j) =1= 0 for all j and it(j) E O(2-ik ) \ O(2-i(k+l»). As Il(j) = -I2(j), this is a
contradiction. I
The proof of Theorem 3.8 now follows immediately.
PROOF OF THEOREM 3.8: Assume eltl 1/1(t), el-rl ~('Y), eltl ~(t), ehl ~("'() E L2(R). It then
follows that both 1/1 and ~ E S(R). Therefore, from Lemma 3.10, all moments of 1/1 and ~
vanish, which implies 'if; = -If = 0, a contradiction .•
24 Benedetto et al.
Theorem 3.8 is closer in analogy to the weak BLT, Theorem 3.5, than to the
BLT, Theorem 3.3. In the Gabor case, the weak BLT and the BLT have been shown to
be equivalent; we leave as an open problem whether Theorem 3.8 can be improved to a
strong form which states that if {1jJjk} is an exact wavelet frame then neither t/J nor ~ can
have exponential localization both in time and frequency.
REFERENCES
[AT] L. Auslander and R. Tolimieri, "Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold," Lecture Notes in Mathematics, No. 436, Springer-Verlag, New York, 1975.
[B] R. Balian, Un principe d'incertitude fort en theorie du signal ou en mecanique quantique, C. R. Acad. Sci. Paris 292 (1981), 1357-1362.
[Ba1] G. Battle, Heisenberg proof of the Balian-Low theorem, Lett. Math. Phys. 15 (1988),175-177.
[Ba2] , Phase space localization theorem for ondelettes, J. Math. Phys. 30 (1989), 2195-2196.
[BHW] J. Benedetto, C. Heil, and D. Walnut, Remarks on the proof of the Balian-Low theorem, preprint.
[C] A. Cohen, Ondelettes, analysis multiresolutions et filters mirTOirs en quadrature, Annales de l'institute Henri Poincare 7 (1990), 439-459.
[D1] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Th. 39 (1990), 961-1005.
[D2] , Orthonormal bases of compactly suported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996.
[DGM] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283.
[DJ] I. Daubechies and A.J.E.M. Janssen, Two theorems on lattice expansions, IEEE Trans. Inform. Th., to appear.
[DS] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier "erie", Trans. Amer. Math. Soc. 72 (1952), 341-366.
[G] D. Gabor, Theory of communications, J. lost. Elec. Eng. (London) 93 (1946), 429-457.
Benedetto et al. 25
[HW) C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628-666.
P) A.J.E.M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982), 720-731.
[L) F. Low, Complete sets of wave packets, "A Passion for Physics-Essays in Honor of Geoffrey Chew," C. DeTar, et al., ed., World Scientific, Singapore (1985), 17-22.
[Ma) S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc. 315 (1989), 69-87.
[M) Y. Meyer, Principe d'incertitude, bases hilbertiennes et algebres d'operateurs, Seminaire Bourbaki 662 (1985-86).
The MITRE Corporation McLean, Virginia 22102
MSC 1991: Primary 42A99 Secondary 42A65, 46B15
26 Operator Theory: Advances and Applications, Vol. 58 © 1992 Birkhiiuser Verlag Basel
DISTRIBUTION OF ZEROS OF MATRIX-VALUED CONTINUOUS ANALOGUES OF ORTHOGONAL POLYNOMIALS
Robert L. Ellis, Israel Gohberg, and David C. Lay
The number of zeros in the upper half-plane, counting multiplicities, of a matrix-valued continuous analogue of an orthogonal polynomial is shown to equal the number of negative eigenvalues of the associated integral operator with a matrix-valued kernel. This result generalizes a theorem of Krein and Langer on scalar-valued orthogonal functions to a noncommutative case. The proof relies on properties of orthogonal operator polynomials, Toeplitz operators, and Wiener-Hopf equations.
TABLE OF CONTENTS
Introduction 1. Preliminary Results
1.1. Matrix-Valued Krein Functions of the First and Second Kinds 1.2. Partitioned Integral Operators
2. Orthogonal Operator-Valued Polynomials 2.1. Stein Equations for Operators 2.2. Zeros of Orthogonal Polynomials 2.3. On Toeplitz Matrices with Operator Entries
3. Zeros of Matrix-Valued Krein Functions 3.1. On Wiener-Hopf Operators 3.2. Proof of the Matrix Version of the Krein-Langer Theorem.
INTRODUCTION
In a series of papers beginning at least as early as 1955, M. G. Krein introduced
and studied a class of entire functions of the form
(0.1)
Here a is a real number and 9 is a solution in L1 (0, a) of the equation
g(t) -la k(t - s)g(s)ds = k(t) (0::::; t ::::; a), (0.2)
Ellis et al. 27
where k is in L1(-a,a) and k(-t) = k(t). Krein viewed the functions q,a as continuous
analogues of the classical Szego orthogonal polynomials. To justify this, we recall that the
classical Szego polynomials determined by a positive function h on the unit circle l' with
an absolutely convergent Fourier series 2::;-00 hjzj may be written in the form
n
Pn(z) = bozn + b1zn- 1 + ... + bn-1z + bn = L bkzn- k, k=O
where the coefficients satisfy the equation
(0.3)
(0.4)
The polynomials pn (n ~ 0) are orthogonal with respect to the scalar product on L 2(1')
determined by h:
The Toeplitz matrix H that appears in (0.4) is positive definite. In the continuous case,
the role of H is played by 1- K, where K is a compact integral operator on Ll(O,a) with
a selfadjoint kernel k(t - s) as in (0.2). The formal analogue of (0.4) is
f(t) -la k(t - s)f(s)ds = b(t), (0 ~ t ~ a). (0.5)
To avoid using the delta function we substitute g(t) = f(t) - b(t), so that (0.5) may be
rewritten as (0.2).
To obtain a continuous analogue of the Szego polynomials Pn (n ~ 0) in (0.4),
we replace the integer n by a positive real number a, and pn by a function q,a with support
in [O,a]. We write z = eiA, replace zk (0 ~ k ~ n) by eiAt (0 ~ t ~ a), and integrate from
o to a rather than sum from 0 to n. Thus the continuous analogue of Pn is given by
28 Ellis et aI.
Substituting get) = J(t) - 6(t) as before, we have
Setting get) = 0 for t outside [0, a] and introducing the Fourier transform
g().) = 100 g(t)ei),.t dt = r g(t)ei),.t dt, -00 Jo
we may also write
(0.6)
The classical theorem on the zeros of Szego orthogonal polynomials has been generalized
in several ways. In [K3], M. G. Krein considered the case of (0.4) in which the weight h
is not strictly positive. He showed that if Pn is the orthogonal polynomial determined by
(0.3) and (0.4), then Pn has no zeros on the unit circle, and the number of zeros of Pn in the
exterior of the unit disk equals the number of neg.dive eigenvalues of the Toeplitz matrix
H in (0.4). For another approach to the proof of Krein's theorem, see [EGL]. Recently,
attention has been given to the case in which the weight h is an indefinite matrix- or
operator-valued function. See [G] for six papers that contain several new methods and
substantial generalizations: [AG], [A], [BG], [D], [GL], and [L].
For the case of a scalar weight, M. G. Krein [Kl] introduced continuous ana
logues of orthogonal polynomials for a definite inner product. In [KLl] Krein and H. Langer
considered the indefinite scalar case and stated that the entire function 4ia in (0.1) does
not vanish on the real line and has as many zeros in the lower half-plane as the operator
1- K determined by the left side of (0.2) has negative eigenvalues. A full proof, along
with other important results and manifold applications, appears in [KL2].
The aim of the present paper is to generalize the result of Krein and Langer to
the case of matrix-valued orthogonal functions. In the following theorem and elsewhere,
Li x m ( a, b) denotes the space of all m x m matrix functions whose entries are in L1 ( a, b).
Ellis et al. 29
The norm of each k in L;nxm(a, b) is given by
Ilkll = 16 Ilk(t)1I dt,
where IIk(t)1I is the norm of the matrix k(t) as an operator on en with its Euclidean norm.
MAIN THEOREM. Let k E L;nxm( -a, a) with k( -t) = k(t)* for 0 :5 t :5 a.
Assume that there are 9 and h in L;nxm(O,a) that satisfy the integral equations
and
respectively. Let
g(t) -10. k(t - s)g(s)ds = k(t) (O:5t:5a)
h(t) -10. h(s)k(t - s)ds = k(t) (0 :5 t :5 a),
Wo.('\) = eio.A (I + 10. e-iAB 9(S)dS)
Wo.('\) = eio.A (I + 10. e- iAB h(S)dS) .
Then ~o. and WOo are invertible on the real line, and the number of zeros of det(~o.) in
the lower half plane and the number of zeros of det(W a ) in the lower half plane, counting
multiplicities, both equal the number of negative eigenvalues of the operator 1- K, where
K is defined on L;nxm(O,a) by
(K<p)(t) = 10. k(t - s)<p(s)ds (O:5t:5a).
Our approach to the problem is different from that of Krein and Langer. The main idea is
to change the problem into one analogous to the discrete case but involving operator-valued
polynomials. To do this, we partition the interval [0, a] into suitably small subintervals of
length T, and "pull back" all the integration onto the single interval [0, r]. We show that the
operator K above is intertwined with a "partitioned integral operator" that is represented
by a Toeplitz matrix whose entries are integral operators on [0, r]. Our treatment of this
30 Ellis et aI.
partitioned integral operator is modeled after the block-matrix case studied in [GL]. Once
this is done, we translate our results back to the continuous case. This procedure is not
accomplished easily and involves several steps.
The paper consists of three chapters. Chapter 1 contains general definitions
and preliminary results. Theorem 1.5 connects the number of negative eigenvalues of the
integral operator 1- K with the number of zeros, in the exterior of the unit disk, of an
operator polynomial whose coefficients arise from an operator equation that is analogous to
(0.4). The Toeplitz matrix that appears in this operator equation represents the partitioned
integral operator for I - K.
Chapter 2 contains general facts about operator polynomials, which are used
to prove Theorem 1.5. Chapter 3 contains the proof of the matrix generalization of the
Krein-Langer theorem. An outline of its proof appears in Section 3.2.
CHAPTER 1. PRELIMINARY RESULTS
1.1. Krein Functions of the First and Second Kind
Let a be a positive real number and let K be a function In
L;"xm( -a, a). Then k determines an operator K on L;"xm(O, a) defined by
(Kcp)(t) = loa k(t - s)cp(s)ds
Suppose that there is a solution 9 E L;"xm(o, a) of the equation (I - K)g = k, that is,
g(t) -la k(t - s)g(s)ds = k(t) (0:5 t :5 a)
and let g(t) = 0 for t outside [O,a]. Let
I!(t) = -2k(t) + g(t) - 21t k(t - s)g(s)ds (0:5 t :5 a).
and set I!(t) = 0 outside [0, a]. Using (1.1) to substitute for -2k(t), we find that
I!(t) = -g(t) + 21B k(t - s)g(s) ds
(1.1)
(1.2)
(1.3)
Ellis et al.
We define the Krein functions of the first and second kind associated with k by
4>a('x) = eiaA (I + la e-iAt 9(t)dt)
Wa('x) = eiaA (I + l a e-iAt .e(t)dt)
31
The entire functions 4> a and Wa are matrix generalizations of the orthogonal functions of
the first and second kind introduced in [Kl]. It turns out to be easier to deal with the
function 1+ g(,X) rather than t.he Krein function 4>a('x) itself. So we define
(1.4)
and
(1.5)
where i is the Fourier transform of the function .e in (1.2). Also, for any matrix-valued
function f of a complex variable, we let
f*(,X) = [1(,\)]* (1.6)
where the asterisk on the right denotes the adjoint of a matrix.
The goal of this section is to show that on the real line the values of the Krein
functions of the first and second kind are invertible matrices. This will follow from an iden
tity satisfied by <i> a and ~ a defined above. The following theorem generalizes Theorem 6.1
in [KL2] for the scalar case. The proof for the noncommutative case is a slight modification
of the proof of the theorem in [KL2].
THEOREM 1.1. Let k E Li"xm(-a,a) satisfy k(-t) = k(t)* for 0:::; t:::; a,
and assume that 9 is a solution of the equation
g(t) -la k(t - s)g(s) ds = k(t) (O:::;t:::;a)
32 Ellis et al.
Then for any complex nu.mber A,
(1.7)
where 41a and ~a are defined as in (1.4) and (1.5), respectively.
PROOF. From (1.4), (1.5), and (1.6) we have
~:(A)41a(A) + 41:(A)~a(A) = 21 + 1a eiAt g(t)dt + 1a
e-iAtf(t)* dt
+ 1a 1a eiA{s-t)f(t)*g(s)dsdt + 1a eiAtf(t)dt
+ 1a e-iAtg(t)* dt
+ 1a 1a eiA{s-t)g(t)*f(s) ds dt. (1.8)
By reversing the order of integration and substituting u for t - s, we find that
1a 1a 1afa--eiA{B-t)f(t)*g(s) ds dt = e-iAUf(u -I- s)*g(s)duds o 0 0 -8
= rfO e-iAUf(u+s)*g(s)duds Jo -8
Reversing the order of integration in both double integrals on the right, we can rewrite the
right side as
f o fa 1a1a-u e-iAUf( u + s)* g(s) ds du + e-iAUf( u + s )*g(s) ds du
-a -1£ 0 0
When u is replaced by -v and then s is replaced by t + v in the first integral, this becomes
Therefore
1a 1a eiA<s-tlf(t)*g(s) ds dt = 1a 1a-
t eiAtf(s)*g(s + t)dsdt
+ 1a 1a-
t e-iAtf( s + t)* g( s) ds dt
Ellis et al.
Similarly
r r r r-t 1010 ei>.{s-t)g(t)*l(s)dsdt = 1010 ei>'tg(s)*l(s+t)dsdt
+ la la- t e-i>.tg(s + t)*l(s)dsdt
Therefore (1.8) can be rewritten as
q,:(A)~a(A) + ~:(A)q,a(A) - 2I
= la e-i>.t [g(t) + l(t) + la- t[l(s)*g(s+t) + 9(S)*l(S+t)]dS] dt
+ la e-i>.t [g(t)* + l(t)* + la- t[g*(s+t)l(s) + l(S+t)*9(S)]dS] dt
Therefore (1. 7) holds if and only if
g(t) + l(t) + l a- t[l(s)*g(s+t)+g(s)*l(s+t)]ds = 0 (O:=;t:=;a)
By (1.3) this becomes
or
21ak(t-s)g(s)ds _la- tg(s)*g(s+t)ds
+ 21a- t J.a g(u)*k*(s - u)g(s + t)du ds - l a-
t g(s)*g(s + t)ds
+ 21a- tl a g(s)*k(s+t-u)g(u)duds = 0 (O:=;t:=;a) o s+t
l a k(t-s)g(s)ds - l a- t g(s)*g(s+t)ds
+ la- t la g(u)*k(s - u)*g(s + t)duds
33
+ r-tla g(s)*k(s+t-u)g(u)duds = 0 (O:=;t:=;a) (1.9) 10 8+t
But a straightforward substitution along with (1.1) and the fact that k( -t) = k(t)* implies
34 Ellis et al.
that
/,4 k(t _ s)g(s) ds _14-t g(s)*g(s + t) ds
= /,4 k(t _ s)g(s)ds _/,4 g(s - t)*g(s)ds
= /,4 [k(t _ s) _ g*(s - t)Jg(s)ds
= -/,414 g*(u)k*(s - t - u)g(s) du ds (O:S t:S a)
Furthermore, substituting v for s + t, we have for 0 :s t ~ a,
14-
t 14 g( u)*k(s - u)*g(s + t) du ds
+ r-t 14 g( s)* k( s + t - u )g( u) du ds 10 8+t
= r 14 g(u)*k(v - t - u)*g(v) du dv 1t v-t
+ /,414 g(v-t)*k(v-u)g(u)dudv
After a reversal of integration in the second integral, the right side becomes
14 r g(u)"k(v - t - u)*g(v)dudv + 14 1u g(v - t)*k(v - u)g(u) dv du t 1v-t t t
By a substitution this becomes
14 r g(u)"k(v - t - u)*g(v)du dv + 1a r-t g(v)*k(v + t - u)g(u) dv du t 1v-t t 10
(1.10)
(1.11)
Interchanging u and v in the last integral, using the fact that k( u + t - v) = k*( v - t - u),
and then combining the two integrals, we convert this to
/,4 1a g( u)* k( v - t - u)* g( v) du dv
which therefore equals the left side of (1.11). Combining this fact with equality (1.10), we
conclude that (1.9) holds, which proves the theorem.
Ellis et al. 35
COROLLARY 1.2. Let k E Ll."xm(O,a) satisfy k(-t) = k(t)* for 0 ~ t ~ a,
and a88ume that g i8 a 80lution of the equation
get) -la k(t - s)g(s) ds = k(t) (0 ~ t ~ a).
Let
and
where
let) = -2k(t) + get) - 21t k(t - s)g(s)ds
Then detia(A) f. 0 and det q,a(A) f. 0 for A real.
PROOF. Take A real and suppose that ia(A) is not invertible. Then there is
a nonzero vector x such that ia(A)X = O. Since A is real, i:(A) = (ia(A)]*. Therefore
multiplying equality (1.7) from the right by x and from the left by X* yields
0= 2x*x f. O.
Therefore ia(A) must be invertible for all real A. A similar proof shows that q,a(A) is
invertible for A real.
1.2. Partitioned Integral Operators
Let a be a positive number, n a positive integer, and 7 = n~l' For any function
C(J with domain containing [0, a] and for 0 ~ p ~ n, we define a function C(Jp with domain
[0,7] by
(0~t~7)
36 Ellis et al.
Clearly, cp E Li"xm(o, a) if and only if CPP E Li"xm(o, r) for ° ~ p ~ n. Define S :
Li"xm(O,a) -+ [Li"xm(O,r)]n+1 by
(1.12)
Then S is an isometric isomorphism when [Li"xm(o, r )]n+l is normed by
n
IIcol(cpp);=oll = L IIcppll p=o
Similarly, for any function cp with domain containing [0, a] x [0, a] and for ° ~ p, q ~ n, we
define a function CPp,q with domain [0, r] x [0, r] by
CPp,q(t, u) = cp(pr + t, qr + u) (O~t,u~r)
In case cp(t, u) = k(t - u) for some function k with domain containing [-a, a],
cPp,q(t,u) = kp_q(t - u)
For any function k in Li"xm(-a, a) we define operators K on Li"xm(o, a) and Kp on
Li"xm(o, r), for -n ~ p ~ n, by
(Kcp )(t) = l a k(t - s )cp(s) ds (1.13)
and
PROPOSITION 1.3. Let k E Li"xm( -a, a), let n be a positive integer, and
let r = n~l' Then
(1.14)
where K is the operator defined in (1.19), Kn is the Toeplitz matrix (Kp_q);,q=o, and S
is defined by (1.1£). In particular, K and Kn have the same eigenvalues with the same
multiplicities as operators on Li"xm(o, a) and [Li"xm(o, r )]n+l, respectively.
Ellis et aJ. 37
PROOF. For any t.p in L;"xm(o, a) and for 0 :s; p:S; n and 0 :s; t :s; T, we have
(Kt.p)p(t) = (Kt.p)(pT + t) = 1a k(pr + t - s)t.p(s)ds
n r(q+l)r
= L 10 k(pT + t - s)t.p(s)ds q=O qr
n r = L 10 kp_q(t - s)t.pq(s)ds
q=O 0
n
= L (Kp_q t.pq)(t). q=O
Therefore S(Kt.p) = K:n(S(t.p)), which implies that K = S-lK:nS.
The matrix K:n in Proposition 1.3 determines an operator on
[L;"xm(o, T )]n+1, which we call a partitioned integral operator.
PROPOSITION 1.4. Let k E L;"xm(_a,a), let n be a positive integer, and
let T = n~l' Suppose that 'Y is a solution of the integral equation
'Y(t,u)-lak(t-sh(s,u)ds=k(t-u) (O:S;t,u:S;a), (1.15)
and for any integers p and q with 0 :s; p, q :s; n, let r p,q be the integral operator on
L;"xm(o, T) given by
Then
(1.16)
38 Ellis et al.
PROOF. For 8Jly e in Lrxm(o, r) 8Jld for 8Jly integers p 8Jld q with
o ::; p, q ::; n, it follows from (1.15) that
11' 11' n l<r+1)1' "}'p,g{t,u)e{u)du - L kp{t - S)-yo,g{s,u)e{u)dsdu
o 0 =0 rr
= 1r kp_g{t-u)e(u)du.
Therefore
Reversing the order of integration, we find that
n
r p,g - L Kp-r r r,g = Kp_g =0
(O ::;p,q::; n)
from which it follows that
{r p,q );,q=o - !C n {r p,g );,q=o = !Cn .
Taking the first column of both sides of this equation and rearranging, we obtain (1.16).
Equation (1.16) resembles the equation that defines the classical Szego poly
nomials. (See e.g. equation (1.4) in [EGL]). Accordingly, it is natural to consider the
polynomial n
P(z) = znJ + L Zirn_i,o, i=O
with operator coefficients arising from (1.16). (See also [A].) Let P be the "reversed"
polynomial n
P(z) = J + L ziri,o. i=O
Suppose J + ro,o is invertible. Since ro,o, ... , r n,O are compact, p(z) is invertible for all
but finitely m8JlY z in the interior of the unit disk. Therefore, since P(z) = zn p(z-l) for
z i= O,P(z) is invertible for all but finitely many z in the exterior of the unit disk. Each
Ellis et al. 39
z such that P(z) is not invertible is called a zero of P. The multiplicity of each zero is
defined as in [R]. (See also [GS] and Section X1.9 of [GGK].) We will denote by ve(P)
the total number of zeros of P, counting multiplicities, of P in the exterior of the unit
disk. The following theorem, whose proof will be postponed until later, relates ve(P) to
the number v_(I - K) of negative eigenvalues of I - K.
THEOREM 1.5. Let k be a continuous function from [-a, a] into the space
of m x m matrices, with k( -t) = k(t)*, let n be a positive integer, and let r = n~l. Let
ICn = (Kp_q);,q=o and IC~_l = (Kp_q);,~;o, where for -n ::; p::; n,Kp is the operator on
Lixm(o, r) defined by
(Kp~)(t) = lT k(pr + t - s)~(s)ds (O::;t::;r)
Assume that I - ICn and I - IC~_l are invertible and have the same number of negative
eigenvalues. Then (1.15) has a solution ,(t,s). For 0::; p::; n, define rp,o on Lixm(O,r)
by
and set
(r p,o~ )(t) = lT ,(pr + t, s )~(s) ds
n
P(z) = ZR 1+ L zir n-i,O j=O
Then P(z) is invertible for Izl = 1 and, counting multiplicities, the number of zeros of P
in the exterior of the unit disk equals the number of negative eigenvalues of 1- K.
Theorem 1.5 will be proved in Section 2.2 from more abstract theorems. Then,
in order to establish the main theorem, it will still remain to prove that, counting multi
plicities, the number of zeros of P in the exterior of the unit disk equals the number of
zeros in the upper half plane of det(I + g).
The next proposition will be needed in the proof of Theorem 3.3.
PROPOSITION 1.6. Let k be a continuous function from [-a, a] into the
space of m x m matrices, let n be a positive integer, and let r = n~l. Let IC n and JC~_l
40 Ellis et aI.
be the Toeplitz matrices described in Theorem 1.5. Then for n sufficiently large, I - ICn
and I - IC~_l have the same number of negative eigenvalues, counting multiplicities,
as operators on [Lixm(O,r)]nH and [Lixm(O,r)]n, respectively. Further, if 1- K is
invertible, then I -ICn is invertible for all n, and I -IC~_l is invertible for all n sufficiently
large.
PROOF. Throughout the proof, all eigenvalues will be counted according to
multiplicities. As in (1.12), we may define an isomorphism S2 = S2(n) from L~xm(O,a)
onto the Cartesian product [L~xm(o, r)]nH by
The norm
makes S2 into an isometry. H we consider K as an operator on L~xm(O,a) and ICn as an
operator on [L~xm(O,r)]nH, then just as in Proposition 1.3,
(1.17)
Consequently, 1- K and 1- ICn have the same eigenvalues as operators on their respective
L2 spaces. Given A#:O and a positive integer j, we may write (AI - K)i = Ai I - A where
A is an integral operator with a continous kernel (since K has a continuous kernel). It
follows that the eigenfunctions and generalized eigenfunctions of K for nonzero eigenvalues
are continuous and hence belong to L;xm(O,a) for 1 ~ p ~ 00. In particular, K has the
same nonzero eigenvalues with the same multiplicities whether it acts as an operator on
Lixm(o, a) or on L~xm(o, a). It follows from (1.14) and (1.17) that the nonzero eigenvalues
of ICn are the same whether the operator acts on [Lixm(O,r)]nH or on [L~xm(O,r)]nH.
There are equations similar to (1.14) and (1.17) relating /C~_l and K', where
(K'<p)(t) = l a-
r k(t - s)<p(s)ds.
Ellis et al. 41
Therefore, the nonzero eigenvalues and multiplicities of K:~_l are the same as an operator
on [Lr'xm(o,T)r or on [L~xm(o,T)r. Therefore, it suffices to prove the theorem when
K:~_l and K.n are viewed as operators on [L~xm(o, T )]n and [L~xm(o, T )]n+l, respectively.
Observe that
where
with
We claim that
~ -~n -
- K:~_l o
lim ~n = 0 n-+oo
(1.18)
(1.19)
in [L~xm(o, T )]nH. Since the mapping S2 = S2( n) is an isometry for each n, it will follow
from (1.19) that 1- K + S2(n)-1~nS2(n) may be made as close as desired (in norm) to
1- K by taking n sufficiently large. Since K is a compact operator, I - K has only a
finite number of negative eigenvalues, with finite multiplicities. Hence when n is large,
1- K + S2(n)-1~nS2(n) will have the same number of negative eigenvalues as 1- K,
with the same multiplicities. (Cf.[K], page 213.) It follows that for such n, the operator
1- K:n + ~n will have the same number of negative eigenvalues as I - K: n . In such a case,
(1.18) shows that I - K:~_l and 1- K:n will have the same number of negative eigenvalues.
Next we prove (1.19), or equivalently,
lim 3 n = 0 n-+oo
and lim [en Kol = 0, n-+oo
(1.20)
in the appropriate spaces. Let M = max {lIk(t)11 : -a ::; t ::; a} and let <Po, ... ,<Pn be any
elements of L~xm(o,T). Then
n-l n-l
L IIK_n+j<PnI12 < L IIK_n+jI1211<PnI12 j=o j=o
42 Ellis et al.
But
Therefore since a = (n + l)r,
and hence
This proves the first equality in (1.20). Similarly,
n
II[en Ko]col(<pj)j=oIl2 = ilL K n _j<pjI12 j=O
< (t, II K n _; II 111';11) , :0 (t, IIKn-;II' ) (t, 111';11')
< (t, M'r') llool(I';)l'~,II' so that
This implies the second equality in (1.20). Therefore I - Kn and I - K~_l have the same
number of negative eigenvalues for n sufficiently large.
Finally, suppose that I -K is invertible on L;nxm(o, a). Then I -Kn is invertible
on [L;nxm(o, r)]n+1 by Proposition 1.3. By the first part ofthe proof, I -Kn is invertible on
[L;nxm(o, r)]n+1. Since S2(n) is an isometry for each n, (1.19) shows that for n sufficiently
large, the operator I - K + S2(n)-16.nS2(n) is invertible and hence 1- Kn + 6.n is
invertible. By (1.18), I - K~_l is invertible on [Lrxm(o, T )]n. Therefore the first part of
the proof implies that I - K~_l is invertible on [L;nxm(o, r )]n for sufficiently large n.
Ellis et al.
CHAPTER 2. ORTHOGONAL OPERATOR-VALUED POLY
NOMIALS
43
In this chapter we prove Theorem 1.5. The proof will make use of a result
concerning Stein equations for operators on a Hilbert space.
2.1. Stein Equations for Operators
THEOREM 2.1. Let A be a bounded linear operator on a Hilbert space H
whose essential spectrum lies in the open unit disk, and let T = I - K, where K is a
compact selfadjoint operator on H. Suppose that T - A*T A is positive definite. Then
A has no eigenvalues on the unit circle, T is invertible and, counting multiplicities, the
number of negative eigenvalues of T equals the number of eigenvalues of A in the exterior
of the unit disk.
PROOF. Let
B=T-A*TA (2.1)
so that B is positive definite. We first prove the theorem under the assumption that the
spectral radius of A is less than 1. It follows from (2.1) by induction that for n ~ 1,
(2.2)
where n-l
Un = ~)A*)kBAk and Vn = (A*tTAn. k=O
Observe that for any x i= 0, (Unx, x) is positive and nondecreasing as a function of nand
that < Vnx, x > tends to zero because the spectral radius of A is less than 1, which implies
that !!An!! tends to 0 as n tends to 00. From this and (2.2) it follows that T is positive
definite. Thus the theorem is true under the assumption that A has spectral radius less
than 1.
Now consider the general case, in which the essential spectrum of A lies in the
open unit disk. Then there are only finitely many eigenvalues of A in the exterior of
44 ElJis et al.
the unit disk, each having finite multiplicity. Observe that there are no eigenvalues of A
on the unit circle. Indeed, if 1>'1 = 1 and Ax = >.x for some x i= 0, then (2.1) implies
that < Bx, x >= 0, which contradicts the fact that B is positive definite. Therefore, the
unit circle divides the spectrum of A into two spectral sets. Using the associated Riesz
projections, we decompose H into a direct sum:
where H+ and H_ are invariant under A, the spectrum of AIH+ lies in the open unit disk,
and the spectrum of AIH_ lies in the exterior of the unit disk. Furthermore, H_ is finite
dimensional, and dim H _ equals the number of eigenvalues of A, counting multiplicities,
in the exterior of the unit disk.
Let P+ be the orthogonal projection of H onto H+. Since AH+ ~ H+, we have
AP+ = P+AP+ and P+A* = P+A* P+
Therefore (2.1) implies that
P+TP+ - (P+A* P+)(P+TP+)(P+AP+) = P+BP+ (2.3)
It follows that the restrictions of P+TP+, P+AP+, and P+BP+ to H+ satisfy the hypotheses
of T, A, and B in the theorem, and the spectral radius of P + AP + is less than one. Therefore,
by the argument above, P+TP+ is positive definite on H+.
Similarly, let P_ be the orthogonal projection of H onto H_. Then AP_ =
P_AP_ and P_A* = P_A*P_, so
(2.4)
The restriction of P _AP _ to H _ is invertible, because its spectrum lies in the exterior of
the unit disk. Let R be its inverse. Multiplying (2.4) by R* from the left and R from the
right we have
Ellis et al. 45
Since R*{P_BP_)R is positive definite, it follows from our earlier work that the restriction
of -P_TP_ to H_ is positive definite and hence that P_TP_ is negative definite on H_.
Now let us decompose H into an orthogonal direct sum:
where L_{respectively, L+) is spanned by eigenvectors corresponding to negative (respec
tively, positive) eigenvalues ofT, and Lo = Ker{T). This is possible since T = I +K, where
K is compact. Observe that
because for x in H +
and we proved that P+TP+ is positive definite on H+. Therefore
(2.5)
Similarly,
because P_TP_ is negative definite on H_. Therefore
(2.6)
From (2.5) and (2.6) we conclude that Lo = {OJ and dimL_ = dimH_. Since Lo = Ker T
and T = I + K with K compact, it follows that T is invertible. Finally, the equation
dimL_ = dimH_ means that, counting multiplicities, the number of negative eigenvalues
of T equals the number of eigenvalues of A in the exterior of the unit disk.
Equation (2.1) is known as a Stein equation. The result in Theorem 2.1 is well
known for matrices. (See page 103 of [H], for example.) Theorem 2.1 can also be derived
from Theorem 7.1' in Chapter I of [D KJ.
46 Ellis et al.
2.2. Zeros of Orthogonal Operator-Valued Polynomials
Next we prove an analogue of Krein's first theorem for Toeplitz matrices whose
entries are operators on a Hilbert space. The proof is modeled after the proof for the
block-matrix case in Theorem 2.1 of [GL]. Given a Hilbert space H and a positive integer
n, we write Hn for the Hilbert space of n-tuples of vectors in H, with the standard inner
product derived from H. In the next theorem, we consider operators on Hn determined in
the obvious way by matrices whose entries are operators on H.
THEOREM 2.2. Let H be a Hilbert space and let Ak (-n :::; k :::; n) be
operators on H with Ao - I, AI, ... ,An compact and A_k = Ak for 0 :::; k :::; n. Let
Tn and Tn- 1 be the selfadjoint Toeplitz matrices (Ap_q );,q=o and (Ap_q );,~~o, respectively,
and assume that Tn and Tn- 1 determine invertible operators on the Hilbert spaces Hn and
Hn-l, respectively, such that Tn and Tn- 1 have the same number of negative eigenvalues.
Let X = col(Xj)'l=o be the {unique} solution of the equation
(2.7)
and define the operator polynomial P by n
P(z) = 2: zj Xn-j j=O
Then Xo is positive definite, P(z) is invertible for z on the unit circle and, counting
multiplicities, the number of zeros of P in the exterior of the unit disk equals the number
of negative eigenvalues of Tn.
PROOF. The operator X exists because Tn is invertible. Since Tn- 1 is also
invertible, we may write Hn = H x Hn-l and consider the associated factorization
( AO Tn = e (2.8)
where
Ellis et al. 47
From the hypothesis that Tn and Tn- 1 have the same number of negative eigenvalues, it
follows that Ao - CT;!1~ is positive definite. Moreover, it follows from (2.8) and (2.7)
that
( 1 -CT-1) = 0 1 n-1 col( oOjl)j=o
col( DOj 1)']=0
Therefore
x = (A _ I:*T-1 1:)-1 o 0" n-1"
and hence Xo is positive definite.
Equation (2.7) implies that
(2.9)
and
(2.10)
Taking adjoints in (2.10) and noting that Tn - 1 and XOI are selfadjoint, we obtain
(2.11)
Substituting (2.11) into (2.9) yields
(2.12)
Let Kx be the companion matrix
-XIXOI 1 0 -X2XOI 0 0
Kx=
1 -XnXol 0 0
and let 6. be the (n - 1) x n matrix defined by
~=C 0 :) I
48
Then
Ellis et al.
- row ( XOI Xi) Tn-l 6.* )
6.Tn _ l 6.*i (2.13)
Since 6.Tn- l 6.* = Tn - 2 , we find from (2.10) and (2.11) - (2.13) that
(.40 - Xi' \ A_,
KxTn-lKX = ~l An - 1
Therefore Tn - 1 satisfies the equati0n
Tn- 1 - KxTn-lKX = diag(Xo 1 , 0,··· ,0) (2.14)
Solving (2.14) for Tn - 1 and substituting repeatedly back into (2.14), it follows that
n-l
Tn- 1 - (KxtTn-lKx = ~)Kx)jRK~ (2.15) j=O
where
R = diag(Xol, 0,· .. , 0)
Let
(j=0,···,n-1)
Then
Let
(
0
ex= :
_X~lX:
I
(2.16)
Ellis et al. 49
and
(2.17)
Then by direct computation,
so that
(2.18)
From (2.16) - (2.18) and the easily verified facts that /{'X = DCxD and (CI)i1jJ(O) = 1jJ(j)
for 0 :::; j :::; n - 1, it follows that
Therefore
(K'X)i1jJ(O) = DCiD1jJ(O) = D(S'X tl (CI)iS'XD1jJ(O)
= D(S'X)-l(CDi1jJ(O) = D(S'X)-l1jJ(j)
(O:::;j:::;n-1)
Taking adjoints, we find that
(O::;j ::;n-1)
(2.19)
(2.20)
Using the fact that R = 1jJ(O)Xi)l¢>(O) and substituting (2.19) and (2.20) into (2.15), we
find that
Therefore Tn - 1 satisfies the Stein equation
where o
50 Ellis et aI.
From the hypotheses of the theorem it follows that Tn - [is compact. Therefore T;l - [
is also compact. Since X is the first column of T;l, we deduce that Xo - [,Xl,"', and
Xn are all compact. Therefore
Kx =K+S
where K is compact and o [
o [ o
[
o Since sn = 0, it follows that K~ is compact and hence that the essential spectrum of
K~ = {o}. We may therefore apply Theorem 2.1 to (2.21) with T = Tn - l and A = K~ . We conclude that K~ has no eigenvalues on the unit circle and the number of its eigenvalues
in the exterior of the unit disk equals the number of negative eigenvalues of Tn - l . Therefore
the same is true for Kx. Since the eigenvalues of the companion matrix Kx coincide with
the zeros of the polynomial P, with the same multiplicities (cf. Example 1.1.4 in [Rj), the
proof of the theorem is complete.
PROOF OF THEOREM 1.5. Equation (1.15) has the form of equation
(2.7), with Tn = 1 -lCn and X = col(rj,o + SOjl)']=o' Thus the polynomial P in Theorem
1.5 becomes the polynomial P in Theorem 2.2. However, in order to use Theorem 2.2, we
need to convert the hypotheses and conclusions in Theorem 1.5 from statements about Ll
spaces to statements about L2 (Hilbert) spaces.
In the proof of Proposition 1.6, we used the fact that the operator K has a
continuous kernel to show that the nonzero eigenvalues of K are the same (and have
the same multiplicities) whether K is considered as an operator on Lrxm(O,a) or on
L~xm(o, a). And we showed that similar statements hold for Kn and K~_l' Therefore the
hypotheses for Theorem 1.5 imply that the hypotheses of Theorem 2.2 are satisfied. The
conclusions of Theorem 2.2 in this case are about the zeros of the polynomial P( z) as an
Ellis et al. 51
operator on the space L~xm(O,r). It remains to show that these zeros are located in the
same places, with the same multiplicities, when P(z) is regarded as an operator on the
space L;nxm(o, r). It turns out. to be easier to work with the reversed polynomial
.P(z) = 1+ fo,o + zf1,0 + ... + znf n,O
By Theorem 2.2,1 + ro,o is positive definite on L~xm(O,r) and hence .P(O) is invertible
there. By Theorem 2.6 of [KMR] , the zeros of.P in the open unit disk are the reciprocals of
the zeros of P, with the same multiplicities, considered as operators on L~xm(o, r). The
same is true of .P and P on L;nxm(o, r )(by the sametheorem) because .P(O) is invertible on
L;nxm(o, r). The invertibility of p(O) on L;nxm(o, r) follows from the general fact, proved
below, that the zeros of.P are the same whether the operators .P(z) act on L~xm(o, r) or
on L;nxm(O,r).
To verify the preceeding assertion, we first observe that the integral operators
f p,O that appear in the coefficients of p( z) have continuous kernels, since the solution ,( t, s)
of (1.14) is obviously continuous (k being continuous) and the operators fp,o are formed
from f. Next, let zo be a zero of P for either L;nxm(o, r) or L~xm(o, r), and expand .P(z)
in powers of ( = z - zo:
where Co, ... ,Cn are all compact integral operators with continuous kernels. Let fo be in
the kernel of .P(zo), and let It, ... ,fr be an associated" Jordan chain". That is, suppose
that (I - Co)fo = 0 and
k
(I - CO)fk - 2.: Cjlk-j = 0 (1 ~ k ~ r) j=l
Then k
Ik = 2.: Cjlk-j (0 ~ k ~ r) j=O
52 Ellis et al.
Since the Cj have continuous kernels, it follows that the functions 10"" ,Ir are continuous
and hence are in both L~xm(o, r) and L?,xm(o, r). Thus p(zo) has the same Jordan chains
whether viewed as an operator on L~xm(o, r) or on L?,xm(o, r). Since such chains may be
used to determine the multiplicity of a zero of an operator polynomial, we conclude that P
has the same zeros, whether for each z,p(z) is considered as an operator on L~xm(O,r)
or on L?,xm(O,r). From the earlier discussion about the zeros of P and P, the same
statement holds for the zeros of P, except at the point z = O. Therefore the conclusions
of Theorem 2.2 apply when the operator polynomials act on L?,xm(o, r). Hence P(z) is
invertible for Izl = 1, and
But Proposition 1.3 implies that v-(I -Kn) = v-(I -K). We conclude that
2.3. On Toeplitz Matrices with Operator Entries
If E is a Banach space, then £1 (E) will denote the Banach space of all summable
sequences of vectors in E, with the norm 00
1I(~o,6'''')1I = L II~jll· j=O
The next theorem concerns an operator on £1(E) determined by a Toeplitz matrix whose
entries are bounded linear operators on E. The theorem is a generalization of a well-known
theorem involving a Toeplitz matrix with scalar entries.
THEOREM 2.3. Let E be a Banach space with the approximation property,
let Q be the polynomial n
Q(z) = L zjQj, i=O
where Qo - I, Q1,' .. ,Qn are compact linear operators on E, and let TQ be the Toeplitz
operator
Ellis et a1. 53
where Qj = 0 for j < 0 and j > n. If Q(z) is invertible for Izl = 1, then TQ is a left·
invertible Fredholm operator on tl (E) and the codimension of the range of TQ equals the
number of zeros of Q in the open unit disk, counting multiplicities.
PROOF. Clearly Q(z) is invertible for z in an annulus containing the unit
circle. It follows that the function z H Q(z)-l is analytic on an annulus containing the
unit circle. Hence the coefficients in the Laurent series for Q(z)-l form a convergent series
in the operator norm. That is, if
00
Q(zt1 == S(z) = L ziSj (Izl = 1), j=-oo
then 00
E IISjll < 00. (2.22) j=-oo
For Izl = 1, we have
The coefficient of zi must vanish for j f: 0, and hence
00
E Si-iQi = 6i,01 (-00 < j < (0). i=-oo
For any integer k,
00
E Sj-k-iQi = cj-k,oI = Cj,k I (-00 < j,k < 00). i=-oo
Letting m = k + i, we obtain
00
E Sj-mQm-k = Cj,k I (-00 < j,k < 00). m=-oo
Now restrict j and k to be nonnegative. Then, since Qm = 0 for m < 0,
00
E Sj-mQm-k = Cj,k I (0 5,j,k < 00) (2.23) m=O
54 Ellis et al.
The condition (2.22) shows that the Toeplitz matrix (Si-m)~=o determines a bounded
linear operator on fl(E). And (2.23) shows that this operator is a left inverse of TQ =
(Qm-k)~,k=O' Thus TQ is left-invertible in the algebra of operators on fI(E). In particular,
TQ is a semifredholm operator.
From the perturbation theory for semifredholm operators, we know that all
bounded linear operators sufficiently close to TQ are left-invertible semifredholm operators
with the same index as TQ.(See [TL], page 262.) Since E has the approximation property,
we may approximate the coefficients Qo, .. . ,Qn of Q as closely as desired by operators
Ro, ... ,Rn , respectively, such that Ro - I, R I , . .. ,Rn all have finite rank. Let
n
R(z) = L zi Rio j=O
and let TR be the Toeplitz matrix whose symbol is R, namely, TR = (Rm-k)~,k=O' where
Rj = 0 for j < 0 and j > n. Further, let A = (Aj_k)j,k~O, where Al = I and Aj = 0 for
j i= 1, and observe that
n n
TQ = L QjAj and TR = L RjAj j=O j=O
Clearly !lAII = 1 as an operator on £l(E), and
n n
j=O j=O
Hence we can choose the Rj to make TR a left-invertible semifredholm operator with the
same index as TQ. Moreover, since Q(z) is invertible for z in the compact set Izl = 1, we
may choose the coefficients of R to make R(z) invertible for Izl = 1.
Since the operators Ro - I, R1 , • •• ,Rn all have finite rank, there exists a topo-
logical direct sum
such that E(O) is a finite-dimensional subspace invariant under Ro, ... ,Rn , and E(l) is in
the intersection of the null spaces of the operators Ro - I, R1 , ... Rn. [Indeed, let X be
Ellis et al. 55
the union of the ranges of Ro - I, Rb . .. ,Rn and let Y be the intersection of their null
spaces. Since Y is closed and has finite codimension in E, X + Y is closed and there exists
a closed subspace W such that E = W ffi(X + Y). Also, since X n Y is a finite dimensional
subspace of Y, there exists a closed subspace Z in Y such that X + Y = X ffi Z. Then
E = W ffi X ffi Z, and we may take E(O) = W ffi X and E(l) = Z.]
Since E(O) and E(l) are invariant under the operator coefficients of R, they are
invariant under R(z) for each z. In fact, R(z) is the identity operator on E(l) for each z.
Likewise, the subspaces ll(E(O») and ll(E(I») oUI(E) consisting of all summable sequences
from E(O) and E(1), respectively, are invariant under TR, and TR is the identity operator
on ll(E(I»). So we may view TR as the direct sum of two operators, say,
TR = T(O) ffi TO),
where T(O) and T(l) are the restrictions of TR to II (E(O») and II (E(l)), respectively. The
operator T(O) is represented by a Toeplitz matrix whose entries are the restrictions to E(O)
of the entries in TR. The symbol of T(O) is the polynomial, call it R(O) , whose coefficients are
the restrictions to E(O) of the operators Ro, ... ,Rn. Since R(z) is invertible for Izl = 1,
and since E(O) and E(1) are invariant under R(z), it follows that for Izl = 1,R(O)(z) is
invertible as an operator on E(O).
Since E(O) is finite-dimensional and the symbol of T(O) is invertible on the unit
circle, Theorem 5.1 in Section VIII.5 of [GF] shows that T(O) is a Fredholm operator whose
index is the negative of the winding number of det R(O)(z) around the unit circle. That
is, ind T(O) = -m, where
m = arg [det R(O)(eiUt)ll::o
However, R(O) is analytic, so t L i~ winding number is just the number of zeros of det R(O)
in the open unit disk, counting multiplicitie:s. By Theorem 5.1 in [GS], the latter number
equals the number of zeros of the operator polynomial R(O) in the open unit disk, counting
multiplicities. Since R(z) is the identity operator on EO) for each z, it follows that the
operator polynomial R itself lms m zeros inside the unit disk, counting multiplicities.
56 Ellis et al.
Since the restriction T(l) is the identity operator on .e1(E(1»), the index of TR
must equal the index of T(O). This index is -m, as we saw in the preceding paragraph.
Recall, however, that TR is injective since it is left-invertible, so its index is the negative of
the codimension of its range. Thus TR is a Fredholm operator, and the codimension of its
range equals the number of zeros of the operator polynomial R in the open unit disk. Since
TQ has the same index as TR, we conclude that TQ is also a Fredholm operator. By making
the polynomial R sufficiently close to Q, we can insure that they have the same number of
zeros, counting multiplicities (by the operator version of the theorem of Rouche, proved in
[GS].) It follows that the codimension of the range of TQ, which equals the codimension of
the range of TR, also equals the number of zeros of Q inside Izl < 1. This completes the
proof.
CHAPTER 3. ZEROS OF MATRIX-VALUED KREIN FUNC
TIONS
In order to finish our preparation for the proof of our main theorem, we need
two results about Wiener-Hopf operators.
3.1. On Wiener-Hopf Operators
The first theorem is a continuous analogue of Theorem 2.3. The Toeplitz op
erator is replaced by the sum of the identity operator and a Wiener-Hopf operator, the
symbol of the operator is a Fourier transform rather than a polynomial, and the circle is
replaced by the real line.
THEOREM 3.1. Gi'ven g in L'{'xm( -00,00), let G be the Wiener-Hopf oper
ator defined on L'{'xm(o, 00) by
(Gcp)(t) = LX! get - s)cp(s)ds, (0 ~ t < 00) (3.1)
Ellis et al. 57
and let g be the Fourier transform of g. Suppose that (a) the support of 9 is contained in
[0,00), and (b) det(I +g(A)) =1= ° for -00 < A < 00. Then I +G is a left-invertible Fred
holm operator on Lr'xm(o, 00), and the codimension of the range of I + G on Lr'xm(o, 00)
equals the number of zeros of det(I + g) in the upper half plane, counting multiplicities.
PROOF. By Remark 2 in [GK, p. 229], hypothesis (b) implies that I + G is a
Fredholm operator, and in this case the negative of the index of I + G equals the winding
number of det(I + g( A)) as A varies from - 00 to 00 along the real axis. Moreover, condition
(a) insures that 9 is analytic in the upper half plane and I + g(A) ~ I as 1m A ~ 00.
Hence the winding number of det(I + yeA)) is just the number of zeros in the upper half
plane, counting multiplicities. Thus the theorem will be proved when we show that 1+ G
has a left inverse on Lr'xm(o, 00).
Since condition (b) implies that I + y( A) has an inverse for all real A, it follows
from Wiener's theorem for JR that I + Y is invertible in the Banach algebra of all functions
of the form
F(A) = cI + j(A),
where c E C and f E Lr'xm(-oo,oo), with
IIFII = Icl + 1: IIJ(t)1I dt
So we may write (I + g(A))-1 = 1- hU.) for some h in Lr'xm(-oo,oo). The identity
(I - h(A))(1 + yeA)) = I leads to
h(>.) + h(>,)y(>.) = y(>.),
and taking inverse Fourier transforms, we have
h(t) + 1: h(t-s)g(s)ds=g(t), ( -00 < t < 00).
For any v E JR, write
h(t-v)+ 1: h(t-s-v)g(s)ds = get-v)
58 Ellis et al.
and let '1£ = S + v, so that
h(t - v) + 1: h(t - u)g(u - v)du = g(t - v).
Since the support of 9 is in [0, 00),
h(t - v) + 100 h(t - u)g(u - v)du = g(t - v). (3.2)
Now let H be the operator on Lixm(o, 00) with h(t - s) as its kernel. We shall prove that
I + G has a left inverse, namely,
(I - H)(I + G) = I,
or equivalently,
Hr.p + H(Gr.p) = Gr.p, for all r.p in Lixm(o,oo). (3.3)
Now
(Hr.p)(t) + [H(Gr.p)](t) = 100 h(t - v)r.p(v)dv + 100 h(t - '1£) [1 00 g(u - v)r.p(V)dV]
With the order of integration changed in the second term, the expression above equals
100 [h(t - v) + 100 h(t - u)g(u - v) dU] r.p(v) dv = 100 g(t - v)r.p(v) dv = (Gr.p )(t),
by (3.2) above. This verifies (3.3) and completes the proof.
The next theorem establishes a useful intertwining between a Wiener-Hopf op
erator and a partitioned integral operator represented by an infinite Toeplitz matrix with
operator entries.
THEOREM 3.2. Let G be the Wiener-Hop! operator on Lixm(o, 00) with
kernel g, as in (9.1). Given any r > 0, define Gp on E = Lixm(O,r) by
(0 ~ t ~ r)
Ellis et al. 59
where gp(t) = g(pr + t) and e E E. Then the Toeplitz operator on i1(E) given by
is similar to the operator I + G. In fact,
(3.4)
where S is the isometric isomorphism from Lf'xm(o,oo) onto i 1(E) defined by Scp = (cpO,CPl,···) for cP in Lf'xm(o,oo) and cpp(t) = cp(pr + t) for ° ~ t ~ rand p ~ 0.
PROOF. The operator S defined in the theorem is obviously an isomorphism
onto il(E), and it is isometric because
The equality that defines the integral operator G in Theorem 3.1 may be decomposed into
a countable system of equalities, for p = 0,1,2, ... ,
00 l(qH)T (Gcp)(pr+t) = L g(pr+t-s)cp(s)ds
q=O qT (O~t<r)
00 r = L 10 g«p - q)r + t - u)cp(qr + u) du,
q=O 0
(3.5)
where s = qr + u. Using the gp and CPP defined above, we may write (3.5) in the form
p r (Gcp)(pr + t) = L 10 gp_q(t - u)cpq(u)du.
q=O 0
Then (3.5) may be written as a system of equations:
p
(Gcp)p(t) = L (Gp_qcpq)(t), p = 0,1, .... q=O
(3.6)
The relation (3.4) between 1+ G and TI+G follows easily from (3.6). We remark for later
use that if the support of 9 in Theorem 3.2 lies in [0, a] for some a > 0 and if r = n~l for
some n ~ 1, then the Gp in the theorem are 0 for p > n + 1, and (3.6) becomes
P iT (Gcp)p(t) = L gp_q(t - u)cpq(u)du, q=b(p,n) 0
p=O,l, ... (3.7)
60 Ellis et al.
where b(p, n) = max{O, p - n - I}.
3.2. Proof of the Main Theorem
It will be convenient for the proof to replace the function <II a in the main the
orem, stated in the Introduction, by eiaA<IIa( -A), which is the same as 1+ g(A). Thus we
count the zeros of det(I + g) in the upper half plane. Similar remarks apply to wa and
I +h.
OUTLINE OF THE PROOF. The prooffor the case when the kernel k is
continuous proceeds as follows:
11_(1 - K) = lIe(P)
= lIi(P)
= codimlmTI+f
= codimlmTI+G
= codimI m(1 + G)
= lIu [det(I + g)]
(Theorem 1.5)
(Theorem 2.6 in [KMR])
(Theorem 2.3, withQ = P)
(for n sufficiently large)
(Theorem 3.2)
(Theorem 3.1)
For the general case, we show that both 11_(1 - K) and lIu(1 + g) are stable under small
perturbations of k.
THEOREM 3.3. it Let k E Lr'xm( -a, a) with k( -t) = k(t)· for 0 ~ t ~ a.
Assume that there are solutions 9 and h of the integral equations
g(t) -loa k(t - s)g(s)ds = k(t) (O~t~a) (3.8)
and
h(t)-la h(s)k(t-s)ds=k(t) (3.9)
Ellis et al. 61
and let g(t) = h(t) = 0 for t outside [0, a]. Then, counting multiplicities, both the number
of zeros of det(I + g) in the upper half plane and the number of zeros of det(I + h) in the
upper half plane equal the number of negative eigenvalues of I - K.
PROOF. Throughout the proof, all eigenvalues and zeros will be counted ac-
cording to multiplicities. First assume that k is continuous. Then it is easy to show that
9 and h are continuous also. Since k(t - s)* = k(s - t), it foUows from (3.8) and (3.9) that
g(t)* -la g(s)*k(s-t)ds = k(-t) (O:::;t:::;a) (3.10)
and
h(t)* -la k(s-t)h(s)*ds = k(-t) (O:::;t:::;a) (3.11 )
Therefore Lemma 1.2 of [GH] implies that I -K is invertible as an operator on L;"xm(o, a).
Consequently there is a solution of the integral equation
-y(t,u) -la k(t - s)-y(s,u)ds = k(t - u) (0 :::; t, u :::; a) (3.12)
For any r.p E L;"xm(o, a), if we multiply (3.12) on the right by r.p(u) and integrate from 0
to a, we find that
la -y(t, u)r.p(u) du -la l a k(t - s)-y(s,u)cp(u)dsdu
= l a k(t - u)r.p(u)du (0 S; t,u S; a)
Reversing the order of integration in the double integral, we see that
fr.p-Kfr.p = Kcp
Therefore (I - K)(I + f) = I, so that
I + f = (I - K)-l
Thus Lemma 1.2 of [GH] implies that
-y(t,s) = g(t - s) + g(s - t)* + a(t,s) (OS;t,sS;r) (3.13)
62 Ellis et aI.
where
l min<t,.) a(t,s)= 0 [g(t-r)g(s-r)*-h(a+r-t)*h(a+r-s)]dr (3.14)
Let n be a positive integer to be chosen later and let r = n~l' Recall that for -n ::; p ::; n,
the operators Kp and r p,O are defined by
(Kp'1?)(t) = lor k(pr + t - s)cp(s)ds and (rp,ocp)(t) = lor 'Y(pr + t,s)'1?(s)ds
Let ICn = (Kp_q)o:Sp,qSn and IC~_l = (Kp-q)OSp,qSn-l. By Proposition 1.6,1 - ICn and
I - IC~_l are invertible and have the same number of negative eigenvalues for n sufficiently
large. Therefore the hypotheses of Theorem 1.5 are satisfied for n sufficiently large. Let
n
P(z) = zn 1+ L zjr n-j,O
j=O
Then Theorem 1.5 implies that for n sufficiently large, P(z) is invertible for Izl = 1 and
the number of zeros of P in the exterior of the unit disk equals the number of negative
eigenvalues of I - K. Let n
p(z) = 1+ E Zjrj,O
j=O
Then
(z f: 0)
Therefore p(z) is invertible for Izl = 1, and since p(O) = I, the number of zeros of P in
the open unit disk equals the number of zeros of P in the exterior of the unit disk (cf.
[KMR, Theorem 2.6]). It follows that for n sufficiently large, the number of zeros of P in
the open unit disk equals the number of negative eigenvalues of 1- K. Let TI+r be the
Toeplitz matrix defined by
with r p,O = 0 for p < 0 and for p > n. Then Theorem 2.3, combined with the preceding
results, implies that for n sufficiently large, TI+r is a left-invertible Fredholm operator and
Ellis et al. 63
the number of negative eigenvalues of I - K equals of codimension of the range of TIH.
Let THG be the Toeplitz matrix defined by
where for p 2:: O,Gp is the operator on L;nxm(O,r) defined by
Since 9 vanishes outside [0, aJ, Gp = ° for p < ° and for p > n + 1. We will next prove that
if n is sufficiently large, then THG is a left- invertible Fredholm operator and
codim Im THr = codim Im THG
For this, it suffices to prove that
lim IITIH - THGII = ° n--+oo
where the norm is the normofT1H-THG as an operator on C1(E) with E = L;nxm(O,r).
Then it is easy to verify, as in the proof of Theorem 2.3, that
n
IITI+r-TI+GII ~L Ilrp,o-Gpll + IIGn+1 11 p=o
So it suffices to prove
° Using (3.13), we find that for any t.p E L;nxm(o, r),
lI(ro,o - Go)t.pll = IT 1I1T[-Y(t,s) - g(t - s)Jt.p(s)dslidt
= IT "1T[g(s - t)*+a(t,s)Jt.p(s)dslldt
where a(t,s) was defined in (3.14). Let
M = max Ilg(t)11 and N = max Ilh(t)11 O~t~a O~t~a
(3.15)
64 Ellis et al.
Then since 9 and h vanish outside [0, a],
IIa(t,s)II ~ 1T [IIg(t - r)g(s - r)*11 + Ilh(a + r - t)*h(a + r - o9)II]dr
~ r(M2 + N 2 )
Therefore
II(ro,o - Go)cpli ~ 11'11' [IIg(09 - r)*11 + IIa(t,s)IIj IIcp(s)IIdsdt
~ r[M + r(M2 + N 2)] 11' IIcp(s) II ds
which implies that
(3.16)
Now consider IIr p,O - Gpll for 1 ~ P ~ n. Since g( t) = 0 for t outside [0, a], we have
g(s - pr - t)* = 0 for 0 ~ t, s ~ r and 1 ~ p ~ n. Therefore, for 1 ~ p ~ n and for any
cp E L~xm(O,a), we find using (3.13) that
II(rp,o-Gp)cpll = 11' "1Tb(pr + t - S)-g(pT + t - s)]cp(s)dslldt
= 11' lilT ap,o(t,s)cp(o9)dslldt
where for 1 ~ P ~ n and 0 ~ t,09:::; T,
rmin(PT+t,8) ap,o(t, 09) = 10 [g(pT+t-s)g(09*-r)-h(a+r-pT*-t)h(a+r-o9)]dr
Since
IIap,o(t,s)II:::; 11' IIg(pT + t - s)g(09 - r)*11 + IIh(a + r - pT - t)*h(a + r - o9)IIdr
~ r(M2 + N 2 )
we have
II(rp,o - Gp)cpli :::; 1T 1T IIap(t,09)llIlcp(o9)II do9dt
:::;T2(M2 + N2) lT IIcp(09)IIdo9
Ellis et al. 65
Therefore
Since r = n~I' this implies that
n
L IIrp,o - Gpll ~ nr2(M2 + N2) ~ r· a(M2 + N2) (3.17) p=1
Furthermore, for any cP E Lrxm(O,r),
so that
IIGn+1CPIi = l r "lr glen + l)r + t - s)Jcp(s)dsll dt
:51r l r Mllcp(s)1I dsdt = r· Mllcpll
(3.18)
Since r = n~I' the equality (3.15) follows from (3.16) - (3.18). Therefore for n sufficiently
large, TI+G is a left-invertible Fredholm operator and codim 1m TI+r = codim 1m TI+G.
Combining this with our earlier result, we have that the number of negative eigenvalues of
1- K equals the codimension of the range of TI+G. By Corollary 1.2, Theorem 3.1 and
Theorem 3.2, the latter number equals the number of zeros of det( I + g) in the upper half
plane. This completes the proof of the theorem in case k is continuous.
Now consider the general case in which k E L;nxm(-a,a). Then k can be
approximated arbitrarily closely in the norm of Lrxm( -a, a) by a continuous function kc
satisfying kc( -t) = kc(t)* for 0 :5 t :5 a. For such a function kc and for any cp E Lrxm(o, a)
we have
II(K-Kc)cpll :51a la IIk(t - s)-kc(t - s)1I1I cp(s)lIdsdt
~ l a l a IIk(t - s)-kc(t - s)1I dtllcp(s) II ds
~ la ias-s IIk(t) - kc(t)1I dtllcp(s)1I ds
~ i: IIk(t) - kc(t) II dt la IIcp(s) II ds
66 Ellis et al.
Therefore
Thus K can be approximated arbitrarily closely by an operator with a continuous kernel
ke satisfying ke( -t) = ke(t)* for ° ~ t ~ a. Since K is compact and 1 - K is invertible
with finitely many negative eigenvalues, if ke is chosen close enough to k, then 1 - Ke is
invertible with the same number of negative eigenvalues as 1 - K. For any such ke, let ge
be the solution of
(0 ~ t ~ a)
and let ge(t) = 0 for t outside [O,a]. Then ge = (1 - Ke)-l(ke) and 9 = (1 - K)-l(k).
From the equality
it follows that IIg - gell in Lixm(O,a) can be made arbitrarily small by choosing ke close
enough to k. Since 9 and gc vanish outside [0, al,
(9 - 9c)(.\) = l a [g(t) - gc(t)]eiAt dt
which implies that
119 - Ycll ~ a IIg - gcll
Therefore, since the determinant of 1 + Y does not vanish on the real line, it is possible
to choose ke close enough to k so that the determinant of J + Ye also does not vanish on
the real line. Then it follows from the analogue of Rouche's Theorem (see Theorem 2.2
and formula (1.3) in [GS]; see also [GGK]), that the number Vu of zeros of det(1 + y) in
the upper half plane equals the number of zeros of det(1 + Yc) in the upper half plane.
Therefore we have
v-(I - K) = v":(J - Kc) = vu[det(J + 9c)] = vu[det(J + y)]
Ellis et al. 67
This completes the proof of the part of the theorem involving det(I + g).
Finally, we will show that the part of the theorem involving det(I + it) follows
from the part about det(I + g). Let
f(t) = k( -t) = k(t)* (-a ~ t ~ a)
and let L be the integral operator on L;"xm(o, a) corresponding to f:
(Lcp )( t) = 1a l( t - s )cp( s ) ds (0::; t ::; a)
Taking the adjoints of equations (3.8) - (3.11), we have, respectively,
g(t)* -la g*(s)f(t - s)ds = f(t) (0::; t::; a) (3.19)
h(t)* -la l(t - s)h(s)* ds = let) (0 ~ t ~ a) (3.20)
get) -la £(s - t)g(s)ds = f(-t) (O::;t::;a) (3.21 )
h(t)-la
h(s)f(s - t)ds=f(-t) (0::; t ::; a) (3.22)
Equations (3.19) - (3.22) are precisely the equations that, according to Lemma 1.2 of
[GH], must have solutions h* and g* in order that 1- L be invertible as an operator on
L;"xm(o, a). Thus 1- L is invertible if and only if I - J{ is invertible. It follows easily that
L and J{ have the same nonzero eigenvalues with the same multiplicities. We now apply
the part of the theorem that we have proved, but we do so with k replaced by £. Then 9 is
replaced by h*. Thus we can conclude that the number of zeros of det(I + h*) in the upper
half plane equals the number of negative eigenvalues of 1- L, which equals the number of
negative eigenvalues of I-I<. Thus the proof will be complete if we show that det(I + h*)
and det(I + it) have the same number of zeros in the upper half plane. But
h*(>') = 1: h(t)*ei>.t dt
= (1: h(t)e-iXtdt) *
= h(-:\)*
68 Ellis et al.
Therefore
det(I + h*(A)) = det (I + h( -A))
Since A lies in the upper half plane if and only if - A lies in the upper half plane, the desired
result follows easily.
REFERENCES
[AG] Alpay, D. and I. Gohberg, On orthogonal matrix polynomials, in Orthogonal Matrix-valued Polynomials and Applications, I. Gohberg (ed.), Operator Theory: Advances and Applications, v. 34, Birkhauser Verlag, Basel, 1988.
[A] Atzmon, A., n-Orthogonal operator polynomials, in Orthogonal Matrixvalued Polynomials and Applications, I. Gohberg (ed.), Operator Theory: Advances and Applications, v. 34, Birkhiiuser Verlag, Basel, 1988, pp. 47-63.
[BG] Ben-Artzi, A. and I. Gohberg, Extensions of a theorem of M. G. Krein on orthogonal polynomials for the nonstationary case, in Orthogonal Matrix-valued Polynomials and Applications, I. Gohberg (ed.), Operator Theory: Advances and Applications, v. 34, Birkhauser Verlag, Basel, 1988.
[DK] Daleckii, Ju. 1. and M.G. Krein, Stability of Solutions of Differential Equations in Banach Space, Transl.Math.Monographs 43, American Mathematical Society, Providence, R.I., 1970.
[DJ Dym, H., Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontriagin spaces, interpolation and extension, in Orthogonal Matrix-valued Polynomials and Applications, I. Gohberg (ed.), Operator Theory: Advances and Applications, v. 34, Birkhauser Verlag, Basel, 1988.
[EGL] Ellis, R.L., I. Gohberg, and D.C.Lay, On two theorems of M.G. Krein concerning polynomials orthogonal on the unit circle, Int. Equations Op. Theory 11 (1988), 87-104.
[G] Gohberg, I., (ed.), Orthogonal Matrix-valued Polynomials and Applications, Operator Theory: Advances and Applications, v. 34, Birkhauser Verlag, Basel, 1988.
[GF] Gohberg, I. and LA. Feldman, Convolution Equations and Projection Methods for their Solution, A.M.S. Transl. Math. Monographs, v. 41, Amer. Math. Soc., Providence, 1974.
[GG] Gohberg, I. and S. Goldberg, Basic Operator Theory, Birkhauser Verlag, Basel, 1981.
[GGK] Gohberg, I., S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, I. Operator Theory: Advances and Applications, v. 49, Birkhauser Verlag, Basel, 1990.
[GH] Gohberg, I. and G. Heinig, On matrix-valued integral operators on a finite interval with a kernel depending on the difference of the arguments, Rev. Roumaine Math. Pures et Appl. 20 (1) (1975), 55-73 [Russian].
Ellis et aI. 69
[GK] Gohberg,1. and M.G. Krein, Systems of integral equations on a half line with kernels depending on the difference of arguments, Uspehi Mat. Nauk (N.S) 13 (1958), no. 2 (80), 3-72 [Russian]; A.M.S. Transl., Ser. 2, 14 (1960), 212-287 [English].
[GL] Gohberg, I. and L. Lerer, Matrix generalizations of M.G. Krein theorems on orthogonal polynomials, in Orthogonal Matrix-valued Polynomials and Applications,!. Gohberg (ed.), Operator Theory: Advances and Applications, v. 34, Birkhauser Verlag, Basel, 1988, pp. 137-202.
[GS] Gohberg, 1. and E. Sigal, An operator generalization of the logarithmic residue theorem and Roucht'l's theorem, Mat. Sb. (N.S.) 84 (126), 607-629 (1971) [Russian]; Math. USSR SB. 13 (1971), 603-625 [English].
[H] Householder, A.S., The Theory of Matrices in Numerical Analysis, Blaisdell Publishing Co., New York, 1964.
[K] Kato, T., Perturbation Theory for Linear Operators, Die Grundlehren der Mathematischen Wissenschaften, v. 132, Springer-Verlag, New York, 1966.
[K1] Krein, M.G., Continuous analogues of propositions about orthogonal polynomials on the unit circle, DoH. Akad. Nauk SSSR 105 (1955), no. 4, 637-640 [Russian].
[K2] Krein, M.G., Integral equations on a half-line with kernel depending upon the difference of the arguments, Uspehi Mat. Nauk (N.S.) 13 (1958), no. 5 (83), 3-120 [Russian]; A.M.S. Transl., Ser. 2, 22 (1962), 163-288 [English].
[K3] Krein, M.G., On the location of the roots of polynomials which are orthogonal on the unit circle with respect to an indefinite weight, Teor. Funkcii, Funkcional. Anal. i Prilozen 2 (1966), 131-137 [Russian].
[KL1] Krein, M.G. and H. Langer, Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight, and related extension problems, DoH. Akad. Nauk SSSR 258 (1981), no. 3, 537-541 [Russian]; Soviet Math. DoH. 23 (1983), no. 3, 553-557 [English].
[KL2) Krein, M.G. and H. Langer, On some continuation problems wich are closely related to the theory of operators in spaces lIz. IV: Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions, J. Operator Theory 13 (1985), 299-417.
[KMR] Kaashoek, M.A., C.V.M. van der Mee, and L. Rodman, Analytic operator functions with compact spectrum. I. Spectral nodes, linearization, and equivalence, Int. Eq. and Operator Theory 4 (1981), 504-547.
[L] Landau, H.J., Polynomials orthogonal in an indefinite metric, in Orthogonal Matrix-valued Polynomials and Applications, I. Gohberg (ed.), Operator Theory: Advances and Applications, v. 34, Birkhiiuser Verlag, Basel, 1988.
[M] Markus, A.S, Introduction to Spectral Theory of Polynomial Operator Pencils, A.M.S. Transl. of Math. Monographs, v. 71, American Mathematical Society, Providence, 1988. [Russian original, 1986.]
[R] Rodman, L., An Introduction to Operator Polynomials, Operator Theory: Advances and Applications, v. 38, Birkhiiuser, Basel, 1989.
70 Ellis et al.
[TLl Taylor, A.E. and D.C.Lay, Introduction to Functional Analysis, 2nd ed., Wiley & Sons, New York, 1980.
Robert L. Ellis Department of Mathematics University of Maryland College Park, Maryland 20742
Israel Gohberg School of Mathematical Sciences Raymond and Beverly Sackler
Faculty of Exact Sciences Tel Aviv University Tel Aviv 69978 Israel
David C. Lay Department of Mathematics University of Maryland College Park, Maryland 20742
Operator Theory: Advances and Applications, Vol. 58 © 1992 Birkhauser Verlag Basel
THE BAND EXTENSION ON THE REAL LINE AS A LIMIT OF DISCRETE BAND EXTENSIONS, II. THE ENTROPY PRINCIPLE
I. Gohberg and M.A. Kaashoek
71
In this paper it is shown that the maximum entropy principle, which identifies the band extension on the real line, may be derived from the corresponding result for operator functions on the unit circle.
O. INTRODUCTION
Let k be an m x m matrix function with entries in L2([-r, rJ). An m x Tn
matrix function f with entries in LI(lR) n L2(1R) is called a positive extension of kif
( a) f ( t) = k ( t) for - r ::; t ::; r,
(b) I - ip..,) is a positive definite matrix for each), E IR.
Here f denotes the Fourier transform of f. If (b) is fulfilled, then
(0,1) (I - f().)) -I = I - 9().),
where, is again an m X m matrix function with entries in Ll (JR) n L2(JR). A positive extension f of k is called a band extension if the function '"Y in (0.1) has the following additional property:
(c) ,(t) = 0 a,e, on JR\[-r,r].
It is known (see [DG]) that the band extension may also be characterized as the unique positive extension f of k that maximizes the entropy integral £(f), where
(0.2) £(f) = lim~ 100 10gdet(I - f().)) d)'. £10271" -00 [2).2+1
The main aim of the present paper is to establish the above mentioned maximum entropy characterization of the band extension by reducing it to the corresponding result for the discrete case, which concerns Fourier series on the unit circle with operator coefficients. We carry out this plan under the additional condition that there exists 6 > 0 such that the given matrix function k has continuous extensions defined on the closed intervals [-8,0] and [0,8]. Our reduction is based on partitioning of operators and does not use the usual discretization of the given k. Let us remark here that the maximum entropy principle for matrix and operator functions on the unit circle is well-understood
72 Gohberg and Kaashoek
and may be derived as a corollary of the abstract maximum entropy principle appearing in the general framework of the band method ([GKW3]). However, for the continuous case there are different entropy formulas ([AK], [Be], [Ch], [DG], see also [D], [MGD, and the maximum entropy principle does not seem to follow from the abstract analogue in the band method (see [GKW3] for an example).
In Part I of this paper it was proved that the band extension on the real line (viewed as a convolution operator) may be obtained as a limit in the operator norm of block Laurent operators of which the symbols are band extensions of appropriate discrete approximations of the given matrix function k. This result is the first main ingredient in our proof of the maximum entropy principle for the real line. The second main ingredient is Theorem I.5.1 [G K1], which expresses the entropy integral in terms of the multiplicative diagonal of the corresponding block partitioned operator. Also we employ in an essential way a notion of a generalized determinant for infinite dimensional operators that was introduced in [P] (see also Section 11.1 in [GKID.
The present Part II is divided in two chapters. The first two sections of Chapter 1 contain preliminary material; in the other two sections of this chapter some auxiliary results about the Perelson determinant are derived. The proof of the maximum entropy principle on the real line is given in the second chapter. Two types of approximations are considered, first from the outside of the band and next from the inside.
I. PRELIMINARIES
1.1 Some notation and terminology
In this section we introduce some notation and recall some of the terminology employed in the papers [GKl], [GK2].
By L2(R) and L2 ([a, b]) we denote the Hilbert spaces of square integrable em-valued functions on R and [a, b], respectively. Let a > ° be a positive number. By B(a) we denote the algebra of all bounded linear operators T on L2(R) such that the ablock partitioning of T is a L2([O, aD-block Laurent operator with symbol in the Wiener algebra on 1I' over the Hilbert-Schmidt operators on L2([O, aD. For the definition of a a-block partitioning we refer the reader to Section I.3 of [GK2] (see also Section I.5 of [GKl]). Block Laurent operators are considered in Section 1.1 of [GK2], and the Wiener algebra on 1I' over the Hilbert-Schmidt operators on a Hilbert space H is introduced in Section I.2 of [GK2]. The latter algebra is denoted by W(52, 1I'; H), where 52 stands for the operator ideal of Hilbert-Schmidt operators. See also Section 1.5 of [GK2] for further properties of the algebra B (a).
An identity operator is denoted by I; from the context it should be clear on which space its acts. Let H be a Hilbert space. We write C2(H) for the Hilbert space of all square summable sequences £ = (xd~o with elements in H. Inner product and norm on C2(H) are given by
00
(£,0 = l)Xj,Yj}, j=o
Gohberg and Kaashoek 73
11"11 ~ (t, II x; II' Y" The Hilbert space of all double infinite square summable sequences :r: = (Xi )~-oo with elements in H is denoted by ffl::ooH. Instead of £2(H) we also write fB'1 H.
1.2 The nonstationary operator Wiener algebra and multiplicative diagonals
Let H be a Hilbert space. By NSW(H) we denote the set of all double infinite operator matrices V = (Vij )0=-00 such that each entry Vij is a bounded linear operator on H and
00 (2.1) L .s~p IIVijl1 < 00.
/1=_OOJ-t=V
The set NSW(H) is an algebra under the usual operations of addition and multiplication. The element E = (8ijI)'ij=_00 is the unit in NSW(H). Here 8ij stands for the Kronecker delta. We shall refer to NSW(H) as the nonstationary operator Wiener algebra corresponding to H.
Each V E NSW(H) defines in a canonical way a bounded linear operator on Gl::ooH via the rule
00 V:r: = y -¢:::=> L Vijxj = Yi,
j=-oo i=O,±1,±2, ....
Note that NSW(H) is closed under taking adjoints, and thus NSW(H) is a *-subalgebra of the algebra of all bounded linear operators on G/::ooH.
Let V E NSW(H), and assume that V is positive definite as an operator on EB~ooH. The arguments used in Lemma 11.3.2 of [GKW2] show that V factors as
(2.2) V = (E + U)~(V)(E + U)*
where U = (Uij)'ij=_oo E NSW(H) and has the following properties
( Q ) Uij = 0 for i ~ j,
(f3) E + U is invertible in NSW(H) and the (i,j)-th entry of (E + U)-l - E is zero for i ~ j,
(r) ~(V) = (8ij~j(V)), where ~j(V) is a bounded linear operator on H for each j and 8ij is the Kronecker delta.
The factorization (2.2) with the properties (Q), (f3) and (r) is unique. The diagonal operator matrix ~(V) is called the right multiplicative diagonal of V.
PROPOSITION 2.1. Let V = (Vij )0=-00 be a positive definite operator in NSW(H). Then the k-th diagonal entry ~k(V) in the right multiplicative diagonal of V is given by
(2.3)
74 Gohberg and Kaashoek
where the operator:3 V[kl' Bk and Ck are defined by
(2.4)
(2.5)
(2.6)
(2.7)
00
V[krR2(H) -+ R2(H), (V[klx)j = LVk+i+1,k+i+1Xj, j=O
00
Bkx= LVk,k+i+1Xj; j=O
i = 0,1,2, ... ;
PROOF. Fix k E Z. From the factorization (2.2) it follows that
... O. ... )-1(1)
In other words, ~k(V)-l is the (1, I)-entry of the inverse of the operator V[k-1l' where V[k-ll is defined by (2.4) (with k -1 in place of k). Note that the operator V[k-ll may be partitioned as
( Vkk Bk) Ck V[kj ,
where V[kl' Bk and Ck are the operators defined by (2.4)-(2.6). But then one may use a Schur complement type of argument (cf., [BGK]' Remark 1.2; see also the proof of Lemma 11.2.1 in [GK1]) to obtain (2.3). 0
It will be convenient to consider two special cases. In what follows V = (l'ij)i,j=-oo E NSW(H), and we assume that V is positive definite. Recall (cf., [GK2], Section 1.1) that V is a H-block Laurent operator if Vij depends on the difference i - j only. In other words, l'ij = l'i-j for all i and j. Assume that V is such an operator. Then V has a well-defined symbol, namely
(2.8) Izl = 1. 11=-00
Since V is positive definite, the operator V(z) is a positive definite operator on H for each z E T, and hence (see [GKW2], Lemma 11.1.1) the operator function V(·) factors as
(2.9) V(z) = (I + u(z) )~o (I + U(z)) *, z E 11',
where ~O is a positive definite operator on H, for each z E T the operator 1+ U(z) is invertible, and the operator function U(·) and (I + U(-))-l - I are analytic on Izl > 1 (including (0), have the value zero at 00 and extend to continuous functions on Izl ~ 1.
Gohberg and Kaashoek 75
The factors in the factorization (2.9) are uniquely determined by the symbol VO, and one refers to ~o as the right multiplicative diagonal of V(·). By passing from (2.9) to the corresponding H-block Laurent operators one sees that the i-th entry in the right multiplicative diagonal of V does not depend on i and is precisely equal to the right multiplicative diagonal of the symbol of V.
Next, we assume that V is p-periodic. The latter means that Vi,j = Vi+p,j+p for all i and j. If p = 1, then we are back to the case considered in the preceding paragraph. Let H(p) denote the Hilbert space direct sum of p copies of H. The fact that V is p-periodic implies that we may view V as a H(pLblock Laurent operator. Indeed, put
( Vip,o
Vi=
V(i+l)p-l,O
Vip,p-l )
V(i+l)Ll,P-l '
i E Z,
and let V = (~-j)i,j=-oo' By identifying the spaces fIJ'::'ooH and fIJ'::'ooH(p) we see that
V may be identified with V. It follows that the right multiplicative diagonal of V may be obtained in the following way. First determine the right multiplicative diagonal ~ of the symbol of V. Since ~ is a positive definite operator on H(p) , the operator ~ factors as
(2.10) (~O
~ = (1 + A) ) (I +A)*,
~p-l
where A is a strictly upper triangular p x p operator matrix whose entries are bounded linear operators on H and the unspecified entries in the middle term of the right hand side of (2.10) are equal to the zero operator on H. By combining the factorization (2.10) with the result of the preceding paragraph we see that in the p-periodic case the right multiplicative diagonal of V is the double infinite p-periodic diagonal operator matrix whose diagonal entries ~i(V) are given by
(2.11) i=O, ... ,p-l.
1.3 A separation lemma for the Perelson determinant
Let A: Lr([a, b]) ---+ Lr([a, b]) be a Hilbert-Schmidt operator. For each h > ° let Mh be the averaging operator on Lr([a, b]) defined by
where
It I S; h,
It I > h.
a S; t S; b,
76 Gohberg and Kaashoek
The operator Mh is a Hilbert-Schmidt operator, and hence MhAMh is a trace class operator. We say that A belongs to the Perelson class (notation: A E P) if the limit
(3.1 )
exists, and in that case the quantity in (3.1) is called the Perelson trace of A (notation: TR(A)). If A belongs to the Perelson class, we put
(3.2) DET(I - A) := det(I - A) exp( - TR(A)).
Here det (I - A) is the second regularized determinant of I - A (see [G Krl], Section IV. 2). The quantity DET(I - A) is called the Perelson determinant of I-A. It can be shown (see the proof of Lemma 3.1 below) that for A E P we have
In fact, the latter identity is the definition of the Perelson determinant as it appears in [Pl. A review of the main properties of the Perelson trace and determinant is given in Section 11.1 of [GK1l. We shall need the following lemma.
LEMMA 3.1. Let Kl and K2 be Hilbert-Schmidt operators on L21([a, b]) from the Perelson class such that
(3.3)
Assume that Kl - K2 i<~ a trace class operator. Then
(3.4)
and equality holds in (3.4) if and only if Kl = K2.
PROOF. Condition (3.3) means that I - Kl and I - K2 are positive definite operators and the difference KI-K2 is positive semidefinite. Since the averaging operator Mh is selfadjoint, Kl ::::: K2 implies that M,J<lMh ::::: M,J<2Mh. Next, note that IIMhl1 ::; 1. To prove the latter inequality one can use the same arguments as are used in Section 10 of [LS] for scalar functions. There exists 0 < b < 1 such that
(KIf,!}::; (1- b)(f,!), f E L2([a, b]),
because I - Kl is positive definite. It follows that for each f E L2([a, b]) we have
(MhKIMhf,!) ::; (1 - b)(Mhf, M h!)
= (1 - b)IIMhf112 ::; (1- b)(f,f),
which implies that I - MhKl Mh is positive definite. So we have shown that
Gohberg and Kaashoek 77
Since MhKIMh and MhK2Mh are trace class operators, we may conclude that
(3.5)
Here det(I - A) is the usual trace class determinant.
It is known that Mh! --t ! if h 1 0 for each! E Lr([a, b]) (cf., [GGKJ, page 112). Thus (by [GKrl J, Theorem III.6.3) we have for v = 1,2 that M,J<vMh --t KII if h 1 0 with respect to the Hilbert-Schmidt norm. But then we can apply [GKrlJ, Theorem IV.2.1 to show that
and hence
v = 1,2,
DET(I - Kv) = det(I - Kv)exp(-TR(Kv))
= limdet(I - MhJ{vMh) exp( - TR(MhJ{"Mh)) hto
= limdet(I - M,J<"Mh), h10
v = 1,2.
By combining this limit formulas with (3.5) we obtain the inequality (3.4).
Now, assume that DET(I - K2) = DET(I - Kt). We have to show that KI = K2. Put e = KI - K2. By our hypotheses e is a trace class operator and e 2: o. From I - J{I + e = I - K2 we see that
where (I - KI)-Ie is a trace class operator. Hence, by [GK1J, Proposition 11.1.2,
DET(I - KI) = DET(I - K2)
= DET(I - Kr)DET(I + (I - Kr)-Ie).
It follows (see [GK1J, Proposition 11.1.1) that det(I + (I - KI)-Ie) = 1. Since I - KI is positive definite, we conclude that
(3.6)
But (I - Kr)-I/2e(I - Kr)-I/2 2: 0, and thus (3.6) implies that the eigenvalues of (I - J{r)-1/2e(I - Kr)-1/2 are zero, and hence this operator must be zero. The latter implies that e = 0, and thus KI = K2. 0
1.4 The Perelson determinant and factorization
For i, j = 1, ... ,N let Kij be a Hilbert-Schmidt operator on Lz'([a, b]). COll-
sider
( 4.1)
78 Gohberg and Kaashoek
Note that K is a Hilbert-Schmidt operator on L2:N ([a, b]). In what follows we aSSUlllP that 1 - K is positive definite. This implies that 1 - K factors as
(4.2) 1 - K = (I - F) (1 -DI
) (I - F*),
1-DN
where F is a strictly upper triangular N x N operator matrix whose entries are bounded linear operators on L2n([a, b]) and the off diagonal entries in the middle term of the right hand side of (4.2) are equal to the zero operator on L2:([a,b]).
PROPOSITION 4.1. Assume that the operators Kij, i, j = 1, ... , N, in (4.1) are in the Perelson class. Then K and the diagonal entries D I , ... , D N in (4.2) also belong to the Perelson clas<~ and
N
( 4.3) DET(I - K) = II DET(I - Dj). j=1
PROOF. Let Mh be the averaging operator introduced i~the previous section. Put (Kij)h = MhKijMh for each i and j. Furthermore, let Mh be the N x N diagonal operator matrix whose diagonal entries are equal to MIll i.e.,
Note that Mh is the averaging operator on L2:N ([a, b]). Now
N N tr(MhKMh) = 2: tr(I{ii)h --t 2: TR(Kii), h ! 0,
i=1 i=1
because of our hypotheses on K II , ... , K N N. SO we see that K belongs to the Perelson class and
( 4.4) TR(K) = TR(Kll) + ... + TR(I{NN)'
From [BarG] we know that the diagonal term in the right hand side of (4.2) is determined by the following identities:
( 4.5a)
(
1 - Ki+l,i+l -Ki+2,i+l
X .
-KN,i+l
Di = Kii + (Ki,i+1 Ki,i+2 ... Ki,N)X
-Ki+l,i+2 1 - Ki+2,i+2
} ?" )_1 - 1i+l,N -Ki+2,N
... 1 -~N,N ( ~i+I'i) Ki+2,i
. ,
KN,i
i = 1, ... ,N -1;
Gohberg and Kaashoek 79
( 4.5b)
From these identities we see that Dj - Kjj is a trace class operator for i = 1, ... , N. So we can apply [GK1], Propositions 11.1.1 and 11.1.2 to show that Dl,'" ,DN are in the Perelson class.
From (4.2) it follows that the entries of F are Hilbert-Schmidt operators. Indeed, write 1- D for the middle term in the right hand side of (4.2), and put H = I - (I - F*)-l. Then H is strictly lower triangular, and (4.2) implies that F - H = K -KH +FD-D. Thus F-H is a Hilbert-Schmidt operator. Since F is strictly upper triangular and H is strictly lower triangular, we may conclude that the entries of F are Hilbert-Schmidt.
Note that F is nilpoten~because F is strictly upper triangular. So F has no nonzero eigenvalue, and therefore det(I - F) = 1. Also FD is strictly upper triangular, and thus tr FD = O. By duality also det(I - F*) = 1. Now, use the multiplication rule for the second regularized determinant (see formula (2.5) in Chapter IV of [GKr1]). It follows that
( 4.6) det(I - K) = det((I - F)(I - D))e- trCF*
= det(I - D)e- trCF*,
where C = F + D - FD. From (4.2) we know that K = C + F* - CF*, and hence
CF* = F+D- FD+F* - K.
Recall that F and F D are strictly upper triangular and F* is strictly lower triangular. So the diagonal entries of CF* coincide with the diagonal entries of D - K. Therefore
(4.7)
N
trCF* = Ltr(CF*)ii i=l
N N = LTR(Dj) - LTR(Kid
i=l i=l
= TR(D) - TR(K).
For the third equality we apply (4.4) both to K and D.
By the result of the first paragraph of the proof (applied to both K and D) the operators K and D are in the Perelson class. But then we can use the second equality in (4.6) and the third in (4.7) to show that
(4.8) DET(I - K) = DET(I - D).
Since D is a diagonal operator matrix with diagonal entries in the Perelson class, it is clear from the definition of the Perelson determinant that the right hand side of (4.8) is equal to the right hand side of (4.3), and hence (4.3) is proved. 0
80 Gohberg and Kaashoek
II. MAIN RESULTS
11.1 Multiplicative diagonal and entropy integral
Let T be an element of the algebra 8(0') (see Section 1.1 for the definition),
and assume that I -T is positive definite. Let (Ti~})ij=-oo be the O'-block partitioning of T. We know that its symbol, ,
00
ST,u(Z)= L TSu}zv v=-oo
is in the Wiener algebra on 1I' over the Hilbert-Schmidt operators on L2'l([a, b]). Since 1- T is a positive definite operator, 1- ST,u(Z) has (see Section 1.2) a well-defined right multiplicative diagonal which we shall denote by ~o(I - T; 0'). From Proposition 1.2.1 we know that
(1.1)
where
are defined as follows:
n(u}: EBf Lr ([0, 0']) -+ EBf Lr ([0, 0']),
C(u):Lr([O,O']) -+ EBfLr([O,O']),
B(u): EBf Lr ([0, 0']) -+ Lr ([0, 0']),
00
(D(u)(CPj)~l)i = LTl~}cpj, i = 1,2, ... , j=l
00
B((T)(CPj)~l = LT~1cpj. j=l
In particular,
(1.2)
where Sl stands for the operator ideal of trace class operators.
Let J be an m x m matrix function with entries in L1 (JR) n L2(JR). The convolution operator on Lr(JR) associated with J will be denoted by Lf, i.e.,
(Lfcp)(t) = 1: J(t - s)cp(s)ds, t E JR.
From Proposition 1.4.1 in [GK2] we know that Lf E 8(0') for each 0' > 0, and thus ~o(I - Lf;O') is a well-defined operator for each 0' whenever 1- Lf is positive definite. The next theorem describe the entropy integral of J,
(1.3) £U) = lim (~ roo log det( I -1p..» d>.) , e!O 27r J -00 6"2 >.2 + 1
Gohberg and Kaashoek 81
in terms of D.o(I - Lf; a).
THEOREM 1.1. Let f be an m x m matrix function with entrie,~ in L1 (R) n L 2(R), and assume that there exists 6 > 0 such that f has continuous extension,~ defined on the closed intervals [-6,0] and [0,6]. Furthermore, as,mme that I - L f i,~
positive definite. Then D.oU - L f; a) - I is in the Perelson clas,~ and
(1.4)
In particular, the left hand side of (1.4) is independent of a.
Theorem 1.1 is just a reformulation of Theorem 1.5.2 in [GK1].
11.2 Approximation from the outside
Let k be an m x m matrix function with entries in L2([-T, T]), and assume that k satisfies the following additional condition:
(C) there exists 6 > 0 such that k has continuous extensions defined on the closed intervals [-6,0] and [0,6].
Furthermore, let f be an m x m matrix function with entries in L1 (lR.) n L2(1R.), and let f be a positive extension of k, i.e.,
(a) f(t) = k(t) for -T ::; t ::; T,
(b) I - L f is a positive definite operator.
In that case
(2.1) ((I - Lf)-1cp)(t) = cp(t) - i: ,(t - s)cp(s)ds, t E R,
for a unique m x m matrix function ,. We shall assume that f is not equal to the band extension b of k, that is, the support of, is not contained in [-T, T]. Our aim is to prove
(2.2) £(1) < £(b).
The proof of (2.2) will be based on two different approximations. The first is considered in the present section.
Put an = *T, and let (Ft})i,j=-oo be the an-block partitioning of Lf. The
fact that f is an extension of k implies that the operators FJ n), Ii I ::; n - 1, are uniquely determined by k. In fact,
(2.3) o ::; t ::; an, Ij I ::; n - 1.
In this section we consider the band
n
(2.4) 1- L zjFt)· j=-n
82 Gohberg and Kaashoek
By [ - c(n)o we denote the band extension of the discrete operator band (2.4), and we write c(n) for the operator on L2(JR) whose un-block partitioning is the L2([0, un])
block Laurent operator with symbol c(n)(.). According to Proposition 11.1.1 in [GK2] the function c(n)o is in the Wiener algebra on T over the Hilbert-Schmidt operators on L2([0, un]), and hence c(n) E 8( un).
Since [ - c( n) ( z) is a posi ti ve definite operator on L2 ([0, un]) for each z E T, the operator [ - c(n) is positive definite, and thus 110([ - c(n)j un) is a well-defined operator. By applying (1.1) and (1.2) with T replaced by c(n) we see that
(2.5)
Note that
(2.6)
and thus FJn) belongs to the Perelson class P because of condition (C) on k. In fact, (see [GK1J, Proposition 11.1.4),
(2.7) TR(FJn)) = ;n tr{ k(O+) + k(O-)}.
Since FJn) belongs to the Perelson class, formula (2.5) implies that the same holds true
for 110(I - c(n)j un) - [.
THEOREM 2.1. Let f be an m x m matrix function with entries in L} (JR) n L2(JR), and let f be a positive extension of k. Assume that f is not the band extension of k. Then there exist a positive integer nand ." > 0 such that
(2.9)
p= 1,2, ...
To prove the above theorem we need the following lemma.
LEMMA 2.2. For p = 1,2,3, ... we have
PROOF. Put e = pn, and consider the band
l
[- L ziF?). i=-l
We need the nonstationary operator Wiener algebra NSW(L2([0,utD). Note that the ul-block partitionin~ of [ - c(n) is an element of the latter algebra. In fact, the urblock partitioning of [ - c(n) is a positive extension in NSW(L2([0, Ul])) of the band (2.9). Let
(2.10)
Gohberg and Kaashoek 83
be the right multiplicative diagonal of the ITf-block partitioning of 1 - C(n). SiIH~f' 1 - CU) is the band extension corresponding to the band (2.9), the maximum f'ntropy principII' for the band (2.9) in NSW(L2([0,ITpJ)) implies (d., Section 2.4 in [GKW3]) that
(2.11)
Here we used that the diagonal entries of the right multiplicativf' diagonal of thf' ITt-block partitioning of 1 - C(I) do not depend on i and are all equal to 6.0(1 - CU); ITf). From the construction of the diagonal entries of the right multiplicative diagonal (d., formula (1.2.3)), it follows that
6.i(1 - c(n);ITf) - (1 - FJi») E SI, i E Z.
Also,
Thus for each i
6.i(1 - c(n); ITt) - 6.0(1 - C(t); ITi) E SI.
But then we can use the separation lemma for the Perelson determinant (Sf'f' Sf'ction 1.3) to conclude that
(2.12) i=0,±1,±2, ....
The next step is to compare 6.0(1 - c(n); ITn) with 6.i(1 - c(n); lTd for i = 0, ±1, ±2, .... Put
(2.13)
Both ~0(I - c(n);ITn) and R act on L2"([O,ITn]). We know (see (2.5») that R is a trace class operator. Let
(2.14)
be the ITi-block partitioning of 6.0(1 - c(n); ITn) and R, respectively. Denotf' the first matrix by 6.0. From (2.13) we see that
(2.15) R ·· - 6. (i,i) - (1 _ F,(I») n - 0 0 ' i = 1, ... ,po
Since FJI) belongs to the Perelson class, (2.15) implies that the same holds truf' for
1 - 6.~i,i), where i = 1, ... ,po From the result proved in the first paragraph of the proof of Proposition 1.4.1 it follows that 1 - 6.0 belongs to the Perelson class and
N
(2.16) TR(I - ~o) = LTR(I _ ~~i,i»). i=1
84 Gohberg and Kaashoek
Now use (2.15), (2.7), (2.11) and the fact that R is a trace class operator to show that
i=l i=l
i=l
= TR(FJn)) - tr(R)
= TR(FJn) - R)
= TR(I - ~o(I - c(n)j Un)).
So we have TR(I - ~o(I - c(n)j Un)) = TR(I - ~o). Since ~o(I - c(n)j Un) and ~o are unitarily equivalent, these operators have the same second regularized determinant. Hence we have proved that
(2.17)
Since ~o(I - c(n)j un) is positive definite, the same holds true for ~o, and so ~o factors as
(2.18) flo = (I + A) (Dl
) (1 + A*), Dp
where A is a strictly upper triangular p x p operator matrix and the off diagonal entries of the middle term in the right hand side of (2.18) are equal to zero. By Proposition 1.4.1 (applied to K = I - ~o) and (2.17) we have
p
(2.19) DET ~o(I - C(n)j un) = II DET Dj. i=l
Since the Un-block partitioning of I - c(n) is a L;n([O, unD-block Laurent operator, the ut-block partitioning of I - c(n) is p-periodic. From Section 1.2 we know that
So we have, by (2.12),
i = 1, ... ,po
By combining this with (2.19) we obtain
Gobberg and Kaasboek 85
Since f = pn, the lemma is proved. 0
PROOF OF THEOREM 2.1. First we show that there exists a positive integer n such that
(2.20)
The proof is by contradiction. Let, be as in (2.1). Assume I - c(n) = I - Lf. Recall
that I - eno, the symbol of the O'n-block partitioning of I - c(n) is the band extension of the band (2.4). It follows that the j-th Fourier coefficient of (1 - e(n)(.))-1 is zero for Ii I 2: n + 1. In other words,
n
(2.21) (I - e(n)(z))-1 = 1 - L ziGjn)
i=-n
for certain Gj n). Since I - c( n) = I - L f by assumption, (I - e( n) (.)) -1 is the symbol
of the O'n-block partitioning of I - Lf , and hence (2.21) implies that the set
must have measure zero. This holds for each n, and therefore
{(t,s)l!(t-s)#O,lt- sl2: r }
has measure zero. But then supp, C [-r, r], and thus f is the band extension of k, which contradicts our hypotheses on J. So for some n the inequality (2.20) holds true.
Since I - L f is positive definite, I - 2:~-oo zi F;n) is a positive extension
of the band (2.4) which, by (2.20), is different from the band extension I - e(n)(.). So we can apply the maximum entropy principle for discrete bands, to conclude that
Since
~o(I - C(n); O'n) - (I - FJn») E SI,
~o(I - Lf; O'n) - (I - FJn») E SI,
the difference ~o(I - c(n); O'n) - ~o(I - Lf; O'n) is a trace class operator, and we can use Lemma 1.3.1 to show that
From Theorem 1.1 we know that
86 Gohberg and Kaashoek
and so we have shown that
Put
(2.22)
Since (Tn = *7 and (Tpn = p~ 7, the proof is completed by taking 71 as in (2.22) and by applying Lemma 2.2. 0
11.3 Approximation from the inside
Let k and f be as in the first paragraph of Section 11.2. Again, put (Tn = *7,
and let (Ft})i,j=_oo be the (Tn-block partitioning of Lf . In this section we consider the band
n-l
(3.1) [- L zjFt)· j=-n+l
From (2.3) we know that the coefficients FJn) in (3.1) are uniquely determined by k and do not depend on the choice of f.
By [ - B( n) (.) we denote the band extension of the discrete operator band (3.1). We write jj(n) for the operator on L2'(lR.) whose an-block partitioning is the L~r([O, O'n])-block Laurent operator with symbol B(n) C·). According to Proposition 11.1.1 in [GK2) the function B(n)(-) is in the Wiener algebra on 'lI' over the Hilbert-Schmidt operators on L2'([O, O'n]), and hence jj(n) E B( (Tn).
Since [- B(n)cz) is a positive definite operator on L2C[O, O'n]) for each z E 'lI', the operator [ - jj(n) is positive definite, and thus l:t.o(I - jj(n); (Tn) is a well-defined operator. By applying (1.1) and (1.2) with jj(n) in place of T we see that
(3.2)
Since FJn) belongs to the Perelson class (see the previous section), formula (3.2) implies that the same holds true for l:t.o(I - jj(n); (Tn) - [.
THEOREM 3.1. We have
(3.3)
where b is the band extension of k.
Since (Tn = *7, we know from Proposition 1.5.2 in [GK2) that jj(n) belongs
to B( 7). Hence we may consider the right multiplicative diagonall:t.o(I - jj(n); 7) of the symbol of the 7-block partitioning of [ - jj(n). We shall need the following lemma.
Gohberg and Kaashoek
LEMMA 3.2. For n = 1,2, ... we have
(3.4) (DET ~o(I - iJ(n); (Tn)) l/(Tn = (DET ~o(I - iJ(n); T)) l/T.
PROOF. Let K be the operator on L2([0, T]) defined by
(3.5) o S; t S; T.
We know that for each n ;::: 1
(3.6)
Fix n ;::: 1. Let
(3.7)
be the (Tn-block partitioning of ~o(I - iJ(n); T), and consider the factorization:
o ) (I +A*),
Dn
87
where A is a strictly upper triangular n X n operator matrix. Since the (Tn-block partitioning of 1- jj(n) is a L2([0, (Tn])-block Laurent operator, we have
D1 = ... = Dn = ~o(I - jj(n); (Tn).
Let T be the trace class operator defined by the left hand side of (3.6), and let
G: ... :] be the (Tn-block partitioning of T. Then
(3.8) ( (n)) ~ii = Tii + I - Po . , i = 1, ... , n.
Hence I - ~11, ... ,I - ~nn are in the Perelson class, and we can apply Proposition 1.4.1 to show that
n
DET Ll = II DET D;, ;=1
where ~ is the n x n operator matrix in (3.7). Furthermore, by the result of the first paragraph of the proof of Proposition 1.4.1,
n
TR(I - ..6.) = LTR(I - ..6. jj ). ;=1
88 Gohberg and Kaashoek
On the other hand, by (3.8), (2.7) and (3.6),
n n
LTR(I - ~ij) = LTR(FJn) - Tji) i=l i=l
= ~ [;n tr{ k(O+) - k(O-)} - tr Tii]
n
= !.. tr{ k(O+) - k(O-)} - L trTii 2 i=l
= TR(K) - trT
= TR(I - ~o(I - ii(n)j r)).
Since ~ and ~o(I - ii(n); r) are unitarily equivalent, we conclude that
n
DET ~o(I - ii(n); r) = DET ~ = II DET Di i=l
Since Un = ~r, this yields (3.4). 0
PROOF OF THEOREM 3.1. From [GK2), Theorem 111.1.1 we know that
(3.9) (n-too).
Let (ii~~~)i,'j=_oo be the r-block partitioning of B(n), and let (Bi-j)i,'j=_oo be the r-block partitioning of Lb. Then formula (3.9) tells us that
00
(3.10a) L llii~n) - iivll-t 0 (n-too), v=-oo
(3.10b) (n -t 00). v=-oo
Note that B~n) is precisely the operator K defined by (3.5). It follows that
Gohberg and Kaashoek 89
where
F(n): Lr([O, r]) -+ EIl1Lr([O,r]),
Next we consider .6.0(1 - Lb; r), the right multiplicative diagonal of the symbol of the r-block partitioning of I - Lb. Note that Eo is precisely the operator K defined by (3.7). Thus
.6.0(1 - Lb; r) = 1- K + E(1 - T)F,
where E, F and T are the operators which one obtains if in the definitions of E(n), F(n) and T(n) the superscript (n) is omitted. From the limit in (3.l0a) we see that
IIT(n) - Til -+ 0 (n -+ 00).
The limit in (3.l0b) implies that E(n) -+ E and F(n) -+ F in the Hilbert-Schmidt norm if n -+ 00. So we see that
(3.11) lim E(n)(I - T(n»)F(n) = E(I - T)F n-+oo
in the trace class norm as well as in the Hilbert-Schmidt norm. In particular, we have (cf., [GKrlJ, Theorem IV.2.l) that
det(1 - K + E(n)(I - T(n»)F(n») -+ det(I - K + E(I - T)F),
tr(E(n)(1 - T(n»)F(n») -+ tr E(1 - T)F,
But then we may conclude that
n -+ 00.
lim DET .6.0(1 - E(n); r) = DET .6.0(/ - Lb; r). n-+oo
By Theorem 1.1,
and therefore
lim DET .6.0(/ - jj(n); r)l/T = expt:(b). n-+oo
n -+ 00,
90 Gohberg and Kaashoek
Finally, one uses Lemma 3.2 to get the desired limit in (3.21). 0
11.4 Proof of the maximum entropy principle
Let k and f be as in the first paragraph of Section 11.2, and let b be the band extension of k. We assume that f #- b, and we want to show that
(4.1) £(f) < £(b).
Put Un = ~T. Consider the operators c(n) and jj(n) introduced in Sections
II.2 and II.3, respectively. By definition, the Un-block partitioning of I - c(n) (resp., 1- jj(n)) is the Lr([O,un])-block Laurent operator whose symbol 1- c(n)(.) (resp., I _B(n)(.)) is the band extension of the discrete operator band (2.4) (resp., (3.1)). Since (3.1) is a part of (2.4), it follows that 1- c(n)o is also a positive extension of the band (3.1), and we can apply the maximum entropy principle for discrete bands (see [GKW3]) to show that
AO(I - c(n);Un):::; Ao(I - jj(n)jUn).
Now, by (2.5) and (3.2),
and thus (by Lemma 1.3.1) we have
So far n was arbitrary. Now take n and 7J > 0 as in Theorem 11.2.1. Then for p = 1,2, ...
(DET Ao(I - jj(pn); Upn)) l/CTpn ~ 7J + exp(£(f)).
On the other hand, by Theorem 11.3.1, the left hand side of the preceding inequality converges to exp(£(b)) if p -+ 00. Thus
exp(£(b)) ~ 7J + exp(£(f)),
and (4.1) is proved.
[AKj
[BarG)
REFERENCES
D.Z. Arov and M.G. Krein, Problems of search of the minimum of entropy in indeterminate extension problems, Funct. Anal. Appl. 15 (1981), 123-126.
M.A. Barkar and I.C. Gohberg, On factorizations of operators relative to a discrete chain of projectors in Banach spaces, Amer. Math. Soc. Tran<~l. (2) 90 (1970), 81-103. [Mat. Issled. 1 (1966), 32-54, Russian.]
Gohberg and Kaashoek 91
[BGK] H. Bart, I. Gohberg and M.A. Kaashoek, Minimal factorization of matrix and operator functions, OT 1, Birkhiiuser Verlag, Basel, 1979.
[Be] J.,J. Benedetto, A quantitative maximum entropy theorem for the real line, Integral Equations and Operator Theory 10 (1987), 761 ~ 779.
[Ch] ,J. Chover, On normalized entropy and the extensions of a positive definite function, J. Math. Mech. 10 (1961), 927~945.
[D] H. Dym, J Contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS 71, Amer. Math. Soc., Providence, RI, 1989.
[DG] H. Dym and I. Gohberg, On an extension problem, generalized Fourier analysis and an entropy formula, Integral Equations and Operator Theory 3 (1980), 143-215.
[GGK] I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Volume I, Birkhiiuser Verlag, Basel, 1990.
[GK1] I. Gohberg and M.A. Kaashoek, Asymptotic formulas of Szego-Kac-Achiezer type, Asymptotic Analysis, to appear.
[GK2] I. Gohberg and M.A. Kaashoek, The band extension of the real line as a limit of discrete band extensions, I. The main limit theorem, Proceedings Workshop on Operator Theory and Complex Analysis, Sapporo, ,Japan, 1991, to appear.
[GKW1] 1. Gohberg, M.A. Kaashoek and H.,J. Woerdeman, The band method for positive and contractive extension problems, J. Operator Theory 22 (1989), 109~ 155.
[GKW2] 1. Gohberg, M.A. Kaashoek and H .. J. Woerdeman, The band method for positive and contractive extension problems: An alternative version and new applications, Integral Equations and Operator Theory 12 (1989), 343~382.
[GKW3] I. Gohberg, M.A. Kaashoek and H.,J. Woerdeman, A maximum entropy principle in the general framework of the band method, J. Funci. Anal. 95 (1991), 231 ~254.
[GKrl] I.C. Gohberg and M.G. Krein, Introduction to the theory of nonselfadjoint operators, Transl. Math. Monographs 18, Amer. Math. Soc., Providence, RI, 1969.
[LS] L.A. Ljusternik and W.I. Sobolev, Elemente der Funktionalaysis, AkaclemieVerlag, Berlin, 1960.
[MG] D. Mustafa and K. Glover, Minimum entropy HeX) control, Lecture Notes in Control and Information Sciences 146, Springer Verlag, Berlin, 1990.
[P] A. Perelson, Generalized traces and determinants for compact operators, Ph.D. Thesis, Tel-Aviv University, 1987.
92
I. Gohberg School of Mathematical Sciences The Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University Tel-Aviv, Ramat-Aviv 69989, Israel
M.A. Kaashoek Faculteit Wiskunde en Informatica Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam, The Netherlands
Gohberg and Kaashoek
Operator Theory: Advances and Applications, Vol. 58 © 1992 Birkhiiuser Verlag Basel
VEAKLY POSITIVE MATRIX MEASURES, GENERALIZED TOEPLITZ FORMS, AND THEIR APPLICATIONS TO HANKEL AND HILBERT TRANSFORM OPERATORS
Mischa Cotlar and Cora Sadoskyl
93
The generalized Bochner theorem (GBT), that provides both integral representations for generalized Toeplitz forms in terms of positive matrix measures, and positive extensions of weakly positive ones, is the central result of a theory with many applications to Hankel and Hilbert transform operators.
The GBT unifies the proofs of the interpolation theorems of Nehari, Adamjan-Arov-Krein and Sarason, the lifting theorem of Sz. Nagy-Foias, and the prediction theorem of Helson-Szego, and, in its more general abstract form, leads to multidimensional versions of these results.
The theory is presented here as related to the matrix extension problems solved by Gohberg and collaborators, as well as with complete. positivity in C*-algebras.
INTRODUCTION
The study of weakly positive matrix measures and generalized Toeplitz kernels and forms started as an attempt to apply moment theory methods to the Hilbert transform [CS1,2]. In particular, it was shown in [CS2,3] that the Helson-Szego theorem on the boundedness of the Hilbert transform in L2(~) spaces, can be deduced from a generalized Bochner theorem (GBT) , which solves the following moment problem.
Given a scalar product < , > in the space P of trigonometric polynomials, when does there exist a 2x2 matrix measure [~ .. J, ~ .. > 0,
1J 1J-
i,j = 1,2, such that d,g> = E .. jf.g.- d~ .. , for f1,f2, the analytic and 1,] 1 J 1J
IAuthor partially supported by grants from the N.S.F. (USA).
94 Cotlar and Sadosky
antianalytic parts of f? The answer to this question lead to the notions in the title, and a theory of generalized Toeplitz forms has been developed in [CS1 - 19], [ArCS], [Ar1,2], [ArC], [A11,2], [B], [D01,2], [101,2], (d. also the bibliography of [CS11]).
This paper is an expanded version of two talks on generalized Toeplitz kernels and forms, delivered by the authors at the Vorkshop on Continuous and Discrete Fourier Transforms and Extension Problems, held at the University of Maryland in September, 1991. The basic facts and some of the recent results in the theory are surveyed, concentrating mostly on the authors' contributions, and on aspects related to the main subject of this volume.
The notions of generalized Toeplitz (GT) form and weakly positive matrix measure are introduced in Section 1, where their lifting properties are obtained through the generalized Toeplitz theorem (GBT, Theorem 1), and some of their applications summarized. In Section 2, the notion of positivity in the set 12(P) of 2x2 matrices with polynomial entries is
systematically explored, to obtain, from the GBT, the Helson-Szego theorem for the Hilbert transform, as well as the Nehari theorem for Hankel operators. In Section 3, the abstract lifting theorem for abstract Hankel and essentially Hankel forms is derived from the Gelfand-Naimark-Segal (GNS) construction and the Vold-Kolmogorov decomposition. The lifting and n-conditional lifting theorems for forms acting in multiparametric scattering structures are stated in Section 4, where some examples on big and middle Hankel operators in L2(T2) are given.
The relations between the work presented here and that of some other authors remain to be clarified. In particular, the relation between the order properties of 12(P) in Section 2 with the notion of complete
positivity in C*-algebras (cf. [E] and [Arv]), as well as the relation between the theory of Hankel forms in Fock spaces provided by the GBT, and that developed by Janson, Peetre and Rochberg [cf. [JPR]). Although the relation of some of the questions discussed here with the band extension and four blocks problems treated by Gohberg and his collaborators is becoming more clear (cf. Section 1 and other papers in this volume), the connection should be further explored in order to improve results through "cross- fertilization."
Cotlar and Sadosky
1. LIFTING PROPERTIES OF GENERALIZED TOEPLITZ FORIS AND VEAKLY POSITIVE lATRIX IEASURES.
Here P denotes the vector space of all trigonometric polynomials spanned by the functions en(t) = exp(int), n f H, and P1,P2 are the
subspaces of the analytic and antianalytic polynomials, spanned by
95
{en' n ~ o} and {en' n < O}, respectively. Also, p2 = P + P stands for the
space of pairs F = (F1,f2), F1,F2 f P, and u for the shift operator defined
in P by
uf(t) = exp(it)f(t), (1.1) and in p2 by
( lola)
I(T) is the set of the finite complex measures p defined in T, I+(T) = {p f I(T): p ~ O}, and 12(T) denotes the set of 2x2 matrix measures,
~ = [~ .. ], where, for i,j = 1,2, ~ .. f I(T). Correspondingly, writing IJ IJ
~ ~ 0 when [~ij(A)] is a non-negative definite matrix for each Borel set A
in T,
(1. 2)
The Generalized Bochner Theorem (GBT) (see Theorem 1 below) establishes a relation between sesquilinear forms (i.e., not necessarily positive scalar products) < , > in P, and sesquilinear forms « , » in p2, which leads to integral representations of the forms in P.
A form < , > in P is said positive if <f,f> ~ 0 Vf f P, and Toeplitz if
<f,g> = <uf,ug>, Vf,g f P. (1. 3)
Similarly, a forms « , » is positive in p2 if ( F,F » ~ 0, VF f p2, and Toeplitz if
2 « F,G » = « uF,uG », VF,G f P . (1. 3a)
96 eotIar and Sadosky
Every p f I(T) defines a Toeplitz form < , >p' by
<f,g>p := p(f,g) = ~fgdp (1. 4)
and < , >p is positive iff p ~ o. The Herglotz-Boch theorem asserts that,
conversely, a form < , > in P is Toeplitz and positive iff < , > = < , >p
for some p f I+(T), and this p is unique. Similarly, every p = [p .. J f 12(T) defines a Toeplitz form
1J H ~ in p2, by ~ , Hp
: = L p .. (f . g .) = L r f . g . dp .. . . -1 2 1J 1 J .. -1 2 J 1 J 1J 1,]- , 1,]- ,
(1. 5)
and « , »p is positive iff p ~ 0 (see Section 2). In turn, the vector
version of the Herglotz-Bochner theorem asserts that a form « , » in p2 is Toeplitz and positive iff « , » = « , »p for some (unique) p f I;(T).
REMARK 1. If P f M;(T) and p = P11 + P22' then p ~ 0 and Pij = wij(t)dp
(see Section 2), so that
«F ,G» p = ~ G * ( t ) [ W ij ( t ) J F ( t ) dp ( t ) (1. 6)
for all F,G f p2.
Now, every f f P has a unique representation f = P1f + P2f,
Plf f Pi' P2f f P2' so that f ~ ~f = (P1f, P2f) is a linear injective map of
Pinto p2, and P can be identified with the subspace ~P of p2. Through this map, every form « , » in p2 induces an associated form < , > in P, defined by
<f,g> := «f,~g», Vf,g E P. (1. 7)
Cotlar and Sadosky 97
Observe that if « , » is Toeplitz in p2, this < , > need not be Toeplitz in P, and only satisfies the weaker condition
<f,g> = <uf,ug> provided ~(uf) u(~f), r(ug) u(rg). (1.8)
This leads to the following concept. A form < , > in P is called generalized Toeplitz (GT) if it
satisfies the weaker condition (1.8), i.e, if for each i,j = 1,2, the restriction of the form to p, x p, is Toeplitz in the sense that, if both
1 J pairs (f,g) and (uf,ug) belong to Pi x Pj then <f,g> = <uf,ug>, i,j = 1,2.
On the other hand, the associated form < , > in P can be positive without « , » being positive in p2, and, in such case, « , » is called weakly positive (VP).
In particular, every ~ = [~ijJ f 12(T) defines a Toeplitz form
« , »~ in p2, and a corresponding associated form < , >~ in P,
L ~ .. ((P ,f)(rg)) , '-1 2 1J 1 J 1,J- ,
f G* (t ) [!JI ij (t) ] F (t ) dp (t) (1. 9)
(1. 9a)
The form < , > defined by (1.9) is a GT form and, if it is also ~
positive, the measure ~ is called weakly positive (VP) and denoted by ~ ~ O. Thus, if ~ f 12(T), then
while
~ ~ 0 (::::}« F,F »/1 = L ~,,(f.r.) > 0, 'IF (f1,f2) f p2, (1.10) ,. "-1 2 1J 1 J 1,J - ,
~~O(::::}«F,F» = L ~ .. (f.r.)~O ~ "-1 2 1J 1 J 1 ,J - ,
(1.10a)
98 Cotlar and Sadosky
Let us remark that scalar products « , )~, for dp(t) = dt in
(1.6), appear also in the work of Dym, Gohberg and their collaborators (cf. [ElG] and other papers in this volume).
Our subject is based on the extension of scalar products in P to positive Toeplitz scalar products in p2.
The basic result is the Generalized Bochner theorem (GBT):
THEOREll. [CS3]. For a sesquilinear form < , > in P the following conditions are equivalent: (1) <, > is a positive Toeplitz form. (2) There exists a positive matrix measure ~ = [~ij] ( .;(T) such that
< , > = < , >~ , i.e.: <f,g> = ~ij(fg) whenever f ( Pi' g ( Pj , for
i,j = 1,2. (3) There exists a positive Toeplitz form « , ) in p2 such that
<f,g> = « ~f,~g ), Vf,g ( P and ~ as in (1.7). (4) <, > is positive and < , > = < , >~ for some ~ ( 12(T).
For Toeplitz forms in P, the GBT reduces to the Herglotz-Bochner theorem.
COROLLARY 1. [CS3] (a) (Lifting property of GT forms.) If < , > is a positive CT form in P, then, by identifying P with ~P, < , > extends to a positive Toeplitz form« , ) in p2, i.e.: <f,g> = « ~f,~g).
(b) (Lifting property of weakly positive matrix measures.) If v = [vij ] ( 12(T) is weakly positive, v ~ 0, then there exists a positive
matrix measure
~ = [~ .. ] (12+(T) such that < ,> = < , >Il and, consequently, IJ v ~
~11 = v11 ~ 0, ~22 = v22 ~ 0, ~12 = v12 + hdt, ~21 v21 + lidt (1.11)
Cotlar and Sadosky 99
REMARK 2. Since there is a one-to-one correspondence between sesquilinear
forms in P and kernels K: II x II ~ (, given by K(m,n) = <em,en>, the notions
above can be given in terms of kernels, instead of forms. For instance,
< , > is positive iff K is positive definite, < , > is Toeplitz iff K(m + 1,
n + 1) = K(m,n), Vm,n t ll, and < , > is a GT form iff K is a GTK, i.e.:
K(m + 1, n + 1) = K(m,n) only where sign(m + 1) = sign m and sign
(n + 1) = sign n.
Observe that, while, for a given form, the measure ~ ~ 0 in the
Herglotz-Bochner theorem is unique, the matrix measure ~ = [~ijJ in the GBT
is not unique. In fact, ~11 and ~22 are uniquely determined but not so ~12'
and there are parametrization formulas, as well as Schur type constructions,
describing all such [~ .. J (cf. [Arl, 2J, [Afl, 2J ), that generalize the IJ
classical parametrization of the Nehari theorem given by Adamjan, Arov and
Krein [AAK1J.
In the special case when the weakly positive matrix measure
v = [vij ] in Corollary l(b) satisfies Vii = v22 = (I + l)p,
v12 = v21 = (M - l)p, for a measure p ~ 0 in T and a fixed construct M ~ 1,
the result leads to precised versions and new variants of the Helson-Szeg6 theorem for one or two weights, a related theorem of Koosis, and the
prediction results of Helson-Sarason and Ibragimov-Rozanov (cf. [ArCSJ). The GBT provides also refinements to the Fefferman-Stein duality
theory and to the study of Carles on measures (cf. [ArCSJ), as well as to
harmonizable stochastic processes (cf. [CS6J). The results mentioned here are also proved valid for GT forms in
~, either by reduction to the case in T (as done in [ArC1J and [CS 9J), or through the method of local semigroups of isometries [BrJ, a notion
appearing naturally in the consideration of GT forms, providing a unified treatment of the GT forms and the M.G. Krein extension method (cf. also
[L] ) . The lifting property of Corollary l(b) has its own development In
two directions. The first is suggested by the Adamjan-Arov-Krein (A-A-K)
theorem [AAK2], and concerns the so-called conditionally positive scalar
100 Cotlar and Sadosky
products, which are positive in certain subspaces of finite codimension. This generalization of the lifting theorem through conditionally positive extensions leads to weighted versions of the A-A-K theorem, as well as of the Helson-Szego theorem [CS 17].
On the other hand, the restriction of a positive GT form to P1 x P2 gives a Hankel form BO: P1 x P2 ~ (, while its restrictions to
P1 x P1 and P2 x P2 give two Toeplitz quadratic seminorms, 11.11 1 and 11-11 2,
such that the positivity of the form is equivalent to I BO (f ,g) I ~ Ilf 111 IIg211
for f f P1, g f P2• Corollary l(a) expresses essentially that BO can be
extended to a Toeplitz form B: P x P ~ (, such that IB(f,g) I ~ Ilflll IIgl1 2 for
all f,g f P, and B = BO in P1 x P2. In this form, Corollary l(a) is valid
in the more general situation when P is replaced by an arbitrary pre-Hilbert space H, P1 and P2 by two subspaces V1 and V2, and the shift by a linear
isomorphism u of H onto H, such that uV1 C V1 and u- 1V2 C V2, and is a
result on extension of u-Hankel and essentially u-Hankel forms (see Section 3).
Such abstract lifting theorems provide as immediate corollaries the intertwining lifting theorem of Sz.Nagy and Foias [Sz.N-F] and the interpolation theorem of Sarason [Sa], and new n-dimensional versions of such theorems (cf. [CS 16]).
Moreover, the GBT and Corollary 1 lead to new n-dimensional and noncommutative (quantized) versions of the theorems of Nehari, A-A-K and Helson-Szego [CS 17,18], as well as to non-unitary versions of the Sz.Nagy and Foias theorem, closely related to the Grothendieck Inequality [CS 15J.
There are also nonlinear lifting results [CS 13] connected with the local nonlinear commutant lifting theorem of Ball, Foias, Helson and Tannenbaum [BFHT] and to the multilinear analysis of Coifman and Meyer, as well as with ideas of Gelfand and Kostuchenko.
Finally, the weakly positive matrix measures have been applied to multivariate prediction problems [Dol,2J, and to linear systems and colligations in Krein spaces [MJ.
Cotlar and Sadosky 101
2. THE GBT AND THE THEOREMS OF HELSON-SZEGO AND NEHARI.
In this section we consider in more detail the notion of positivity in p2, and its application to the theorems of Helson-Szeg6 and
Nehari. In particular, we show that the GBT (Theorem 1) can be expressed In terms of order properties that underlie the notion of complete positivity in
C*-algebras (cfr. [Arv] , [Da] , [EJ). Given a *-algebra A, for each n E ~, let In(A) be the set of the
nxn matrices f = [fijJ, with entries f ij E A, and let Mn(A') be the set of
the nxn matrices e = [e .. J, with entries e .. : A ~ C, linear functionals In IJ IJ A. Then, M (A) is also a *-algebra under the algebraic matrix operations, n
and each e E Mn(A') defines a linear functional in Mn(A) by
e(f) e([f .. J) IJ ~ e .. (f .. ), Vi i..J IJ IJ i,j
(2.1)
An element f E A is said positive in A if f = g*g, for some g E A,
and similarly for the *-algebra Mn(A). An f E Mn(A') is said positive if
e(f) ~ 0 whenever f is positive in Mn(A).
Vhen n = 2, every e E M2(A') defines a sesquilinear form « , » in
A2 A x A, given by
f([g.*f.J) J 1
~ e .. (g. *f.) i..J IJ J 1
i,j (2.2)
and the order properties of M2(A') can be expressed in terms of scalar
products.
Since P is an algebra of continuous functions with *-operation
f*(t) = f(t), taking A = P, the GBT relates with the order in M2(P') as
follows. The Fejer-Riesz lemma says that f E P satisfies f(t) ~ 0 for all t iff f = gg, for some g E P1' and iff f is a positive element of P. In 12(P)
we have the following properties frequently used in the application of the GBT.
102 Cotlar and Sadosky
(A) For a given f = [f ij ] £ 12(P) the following conditions are equivalent:
(A1) f is a finite sum of positive elements of 12(P).
(A2) f is a finite sum of elements of the form [filjJ, fl,f2 £ P.
(A3) [f .. (t)J is a positive matrix for all t £ T. 1J E .. f .. ~.~. is a positive element of P for all pairs ~1~2 £ P. 1J 1J 1 J
As a consequence follows (B) i = [i .. ] £ 12(PI) is positive iff 1J
(2.3)
for every pair fl,f2 £ P, and iff there exists ~ = [~ij] £ I;(T)
such that i(f) = ~(f) for all f £ 12(P). (Here the measures ~ij are
considered as linear functionals in P, element in 12(PI).)
(C) i = [i .. ] £ 12(P') is weakly positive, 1J
i.e., ~ is considered as an
i.e. : E .. i .. (fI) > 0 1J 1J 1 J -whenever fl £ P1' f2 £ P2' iff (2.3) holds for fl £ P1 and f2 £ P2' and
iff there exists a weakly positive v = [vij ] £ 12(T) such that
i(f) = v(f) whenever f = [filj] for fl £ P1' f2 = P2·
(D) ~ = [~ij] £ 12(T), considered as an element of 12(P') is positive in
12(P') iff ~ £ I;(T), and iff
for all Borel sets A in T. In the special case when ~11 = ~22 = the
Lebesgue measure, ~ is positive in 12(P') iff d~12 = ~(t)dt,
Cotlar and Sadosky 103
The last assertion is related to the following property in general
C*-algebras: a t A satisfies II all ~ 1 iff [aij ] is positive in 12(A) for
a11 = a22 = 1, a12 = a21 * = a (cf. [E]).
Property (A) is a consequence of a corresponding property of general *-algebras (see [T], IV, 3), with the exception of the implication (A3) ~ (A2) which follows from the more refined property:
PROPOSITION. ([CS 4J, [CS 8J). Let I;(P) = {f = [fijJ t 12(P): f12 f PI'
f21 = I12 t P2}· If f = [f ij ] t I;(P) is positive in 12(P) , then f is the
limit (in the norm of the supremum) of finite sums of elements of the form [fiIj] t 12(P), where fl t PI' f2 t P2. Consequently, every weakly positive
matrix measure ~ t 12(T), considered as an element of 12(P'), satisfies
~(f) ~ 0, whenever f t I;(P) and f is positive in 12(P).
2 SKETCH OF PROOF. By (AI) ~ (A3), fl1 ~ 0, If121 ~ f11f22' and, by
Fejer-Riesz, fll = glgl' f22 = g2g2' for gl,g2 t Pl· Since f12' gl and g2
are analytic, setting g. = p.h., where p. is the Blashke product of g., this 11111
continuous analytic functions and 11(t) 1 $ 1. Thus f = [¥I.~.J + [¢.¢.], 1 J 1 J
- 2 . where ¥II = 1¥11 = ¥l2 = h2' ¢1 = (1 - 111 )hl' ¢2 = 0. Slnce ¥l1'¥l2'¢1'¢2 are
limits of analytic polynomials, the assertion follows. 0
The Proposition and property (A3) are analogues of the Fejer-Riesz
property in 12(P).
On the other hand, from the definition of Toeplitz and GT forms it is easy to deduce the two following properties, that can be adopted as definitions.
(E) Let < , > be a Toeplitz form in P, and set f(en) = <fn,fO>' fn(t)
= exp(int). Then f extends to a linear functional £ in P, such that <f,g> = £(fg) , for f,g f P.
104 Cotlar and Sadosky
(F) Let < , > be a GT form in P, and set 111(en-m) = <In,lm> if n ~ 0,
m ~ 0, 122(en-m) = <en,em> if n < 0, m < 0, 112(en-m) = <en,em> if
n ~ 0, m < 0, 121 (en-m) = <en,em> if n < 0, m ~ 0. Then each lij
extends to a linear functional i .. in P, such that <f,g> = i . . (fg) if 1J 1J
f £ Pi' g £ Pj , for i,j = 1,2. The extensions ill and 122 are unique,
but not so 112 and 121'
(G) From (F) and (C) follows that the GBT can be deduced from either one of the lifting properties (a) and (b) of Corollary 1. In fact, by (F), every positive GT form is given by a weakly positive matrix l = [lijJ £ 12(P'), which, by (C) can be replaced by a weakly positive
matrix measure v £ 12(T), and, by the lifting property, v can be
replaced by a positive matrix measure ~ £ M;(T).
The Herglotz-Bochner theorem is known to be a consequence of the Fejer-Riesz lemma, property (E), and the Krein version of the Hahn-Banach theorem for positive linear functionals. Indeed, if < , > is a positive Toeplitz form in P, then, by (E), <f,g> = l(fg), with l(fI) ~ 0, so that, by Fejer-Riesz, l is positive in P and extends to a positive linear functional in C(T), furnishing the desired positive measure ~.
Similarly, the GBT follows from properties (F), (C), (D) and the Krein-Hahn-Banach extension theorem. Indeed, as observed above, it is enough to prove the lifting property (b) in Corollary 1. But, if ~ is a weakly positive matrix measure, then, by (C) and (F), its restriction to the subspace M~(P) of 12(P) is positive in the subspace, extends to a positive
functional ~ in 12(P) by K-H-B, and, by (D), ~ £ M;(T).
REMARK 3. Here we consider scalar products in P and p2 because of the simplicity of the functions f £ P. But since P is not a Banach algebra, to have a closer relation with Banach algebras, one should work, as Gohberg and his collaborators do, in the Viener spaces V, V1 and V2, instead than in
P,P1 andP2 (d. [DG12]' [EGJ).
Cotlar and Sadosky 105
An important special case of Corollary 1 is when the weakly positive matrix measure v = [vijJ is of the form v11 = v22 = (I - l)p,
v12 = v21 = (I + l)p, for a positive measure p and a constant I ~ 1. In
this case, condition (1.11) becomes:
P is absolutely continuous, dp = w(t)dt, and, for h £ H1,
2w(t) ~ Reh(t) ~ Ih(t) I ~ 212w(t), larg h(t) I ~ ; - £1 a.e. (2.5)
so that w N Ihl [CS 3J. If r is the Hilbert transform, i.e. the bounded operator in L2(T)
satisfying rf = -i(P1f - P2f) for all f £ P, then two bounded functions can
be defined, u = log(w/lhl) and v = arg h, with rv = loglhl, since log h = loglhl + i arg h for h t H1. Then (2.5) is easily seen to be equivalent to each of the following:
3w real valued, such that WNW and Irwl ~ 12w a.e. (2.6)
3u,v ( Lm, IIulim ~ CI , IIvilm ~ ; -£1' such that w = exp(u + rv). (2.7)
A classical problem in analysis was to characterize the measures p ~ 0 in T such that r be bounded in L2(p), with norm ~ M,
(2.8)
Condition (2.8) can be rewritten as
(2. 8a)
for
P11 = P22 = (I - l)p, P12 = P21 = (I + l)p. (2.8b)
106 Cotlar and Sadosky
Thus the ~ = [~ij]' for these ~ij' is weakly positive and of the special
type described above, so that the measures p are absolutely continuous, dp = wdt, and characterized by each of the conditions (2.5), (2.6) or (2.7). Condition (2.7) is the necessary and sufficient condition in the Helson-Szego theorem, with further control of the constants, and (2.5) and (2.6) give other equivalent characterization of these measures ([CS 3,5]).
An important advantage of the argument above is that it solved the previously open problem of characterizing the pairs of positives measures Pl,P2 such that r: L2(Pl) ~ L2(P2)' continuously, with control of the norm
([CS 4,5]).
Moreover, the characterizations (2.5), (2.6) extend to LP, 1 < P < m, as follows. (For more details on this subject, see [CS 14]).
COROLLARY 2. [CS 5]. r acts continuously in LP(p), 1 < p < m, with
* norm ~ I, iff dp = w(t)dt and for all 0 ~ rp f LP , IIrpll ~ 1, there are
a f LP*, lIall ~ CII' and h f H1, such that
(2.5a)
where l/p* = 11 - 2/pl.
For dp = dt, w = 1, Corollary 2 reduces to the classical M. Riesz Inequality for the Hilbert transform.
Let us see now that the GBT provides a refinement of the Nehari theorem on Hankel forms.
A Hankel form is a sesquilinear form B: P1 ~ P2 satisfying any of
the following equivalent conditions:
(i) B(uf ,g) = B(f, u- 1g), Vi f Pi' g f P2;
(ii) 3 a linear functional l: P2 ~ (, such that B(f,g) = l(fg);
Cotlar and Sadosky 107
(iii) 3 a sequence p: II ~ (, such that B(e ,e ) = P for all + m n m-n
m ~ 0, n < 0, where em(t) = exp(int).
Observe that (i) makes sense because
(2.9)
Every positive Toeplitz form B: P x P ~ ( defines a Toeplitz quadratic seminorm by IIfilB = B(f,f)1/2, which converts P into a pre-Hilbert
seminormed space. If BO: Pl x P2 ~ ( is a Hankel form and Bl ,B2 are two
positive Toeplitz forms, we write
Vf E Pl , g E P2. (2.10)
If < , > is a positive GT form in P then, by (G), its restrictions to Pi x Pi and P2 x P2 coincide with positive Toeplitz forms B1,B2, while
its restriction to Pi x P2 is a Hankel form BO' which, by Schwarz
inequality, satisfies Bo ~ (B 1,B2). Thus, it is the same to give a positive
GT form in P or to give a Hankel form BO and two positive Toeplitz forms Bl
and B2, satisfying BO $ (B i ,B2) in Pi x P2. From the GBT follows
COROLLARY 3. (1) A Hankel form BO: Pi x P2 ~ ( satisfies BO ~ (B i ,B2) in Pi x P2, for
Bl ~ 0, B2 ~ 0, Toeplitz forms in P, iff there exists a Toeplitz form
B: P x P ~ ( such that B = BO in Pi x P2 and B ~ (B 1,B2) in P x P.
loreover, setting B(f,g) = ~12(fg), ~21 = ~12' B1(f,g) = ~ll(fg),
B2(f,g) = ~22(fg), for all f,g E P, it is [~ijJ E ';(T).
108 Cotlar and Sadosky
(2) In the special case when B1(f,g) = B2(f,g) for all f,g l P,
B(f ,g) = f fg~dt, for ~ l LOO , II~IIID ~ 1.
Part (2) of Corollary 3 is the classical Nehari theorem, and ~ is called a symbol of BO'
The notion of positive GT forms can be formulated in an abstract setting as follows. A pair of systems [H1;u1;Yl], [H2;u2;Y2] is an
algebraic scattering structure if Hl and H2 are vector spaces, u1: H1 ~ H1,
u2: H2 ~ H2 are linear isomorphisms, and Y1 ( H1, Y2 ( H2 are subspaces
satisfying
(2.11)
For Hl = H2, a Hilbert space, and u1 = u2, a unitary operator in that space,
the structure is a classical Lax-Phillips scattering system. Let Bl and B2 be two fixed Toeplitz positive sesquilinear forms in H1,H2,
respectively, i.e., forms satisfying
An abstract Hankel form in an algebraic scattering structure is a
sesquilinear form B: Y1 x Y2 ~ ( satisfying B(u1f,g) = B(f,u;lg), Vf l Y1,
g l Y2, and an abstract GT form in such a structure is a Hankel form B
satisfying B ~ (B1,B2) in Y1 x Y2, for B1,B2 as in (2.12).
In what follows, for simplicity it will be assumed that Hl and H2
are Hilbert spaces, Bl and B2, their scalar products, u1 and u2 , unitary
operators in 11,H2, respectively, and V1 and V2, closed subspaces satisfying
Cotlar and Sadosky
(2.11). In the next section it is shown that Corollary 3 extends to this abstract setting, but since the notions of this section relate to order properties of *-algebras that do not translate to the abstract situation, another type of ideas will be required.
3. GNS CONSTRUCTION, VOLD DECOIPOSITION AND ABSTRACT LIFTING THEOREIS.
109
The Herglotz-Bochner theorem is known to be closely related to the Gelfand-Naimark-Segal (GNS) principle. If < , > is a positive Toeplitz form in P, then it converts P into a Hilbert space, i.e., there exist a Hilbert space H and a linear map J: P ~ H such that <f,g> = <Jf,Jg>H' and JP is
dense in H. Setting U(Jf) = J(uf), U extends to a unitary operator in H, so
that {n H Un} is a unitary representation of 0, and e = JeO is a cyclic
element, i.e., the elements Une span H, and <en,em> = <Une,Ume>. This is
called the GNS construction, and setting p(A) := <E(A)e,e>, E(A) the
spectral measure of U, the Bochner theorem follows: <f,g> = p(fg).
If < , > is a positive GT form, instead of a positive Toeplitz one, then it still defines a Hilbert space H' and a map J: P ~ H', but now U', given again by U'(J£) = J(u£), need not be unitary. U, is an isometry with domain JP1 + J(u- 1P2), and range J(uP1) + J(P2). This isometry extends
to a unitary operator U in a larger space H.
THEOREM 2. (GNS construction for GT forms [ArCS]). If < , > is a positive Cl form in P, then there exist a Hilbert space H, a unitary operator U in H, a linear map J: P ~ H and a cyclic pair e1 = JeO' e2 = Je_ 1,
en(t) = exp(int), such that, for i,j = 1,2,
Setting Pij(A) = <E(A)ei,ej >, for E the spectral measure of U, the
matrix measure P = [PijJ E I;(T) satisfies the CD1.
110 Cotlar and Sadosky
REtARI4. This theorem extends for operator-valued forms <f,g> f L(N), N a Hilbert space, by replacing P by P(N), the set of trigonometric polynomials f(t) = Enen(t), with coefficients en f N, and defining in P(N) the scalar
product
<f ,g> := L «en,em>en,em>N· n,m
The pair JeO,Je_ 1 in Theorem 2 is cyclic because
(3.2)
Pi e uP1 = {ceO}' and '2 e u- 1P2 = {ce_ 1} are one-dimensional subspaces.
Replacing Pi and P2 by the abstract Vi and V2 of an algebraic scattering -1 structure (see the end of Section 2), then Rl = Vi - uV1, R2 = V2 - u V2
need not be one-dimensional, but it is still a cyclic set. In fact, the Vold-Iolmogorov decomposition gives, for i = 1,2,
n 0 1 u· R. CD H. CD H. III 1
where
H~ = n u~V., 1 n~O 1 1
1 1 0 0 so that u.H. = H. and u.H. = H., and every fl· f HI· can be written as 11111 1
f. = 1
(cf. [es ]) .
~ n=-m
n 0 1 001 1 u.f . +f. +f., f . fR., f. fH., f. fH., 1 n,l 1 1 n,l 1 1 1 1 1
THEOREt 3. (Analogue of the GNS construction for the Hankel forms)
(3.3)
(3.3a)
(3.3b)
If B: Vi K V2 ~ ( is a Hankel form in an algebraic scattering
structure [H1jU1jV1] , [H2jU2jV2] with fixed positive Toeplitz forms Bl and
B2, and B ~ (B1,B2) in Vi K V2, then there exist a unitary operator U zn a
Hilbert space N, and two linear maps, J1: Hi ~ Nand J2: H2 ~ N, such that,
Cotlar and Sadosky 111
REMARK 5. As observed in Remark 2, it is sometimes more convenient to state these results in terms of kernels K: II x II ~ ( instead of forms in P. In particular, for positive definite kernels K there is a GNS theorem as Theorem 2, and, as observed in Remark 4, this holds also for operator-valued kernels K(m,n) E L(N). If A is a C*-subalgebra of L(H), H a Hilbert space, and K is a kernel assigning to each pair m,n Ella bounded linear operator K(m,n): A ~ L(H), then this kernel is called completely positive definite if for every pair of finite sets {mj } C ll, {(j} C H, and linear operators
i?j: A ~ L(H),
The GNS extends also to completely positive definite Toeplitz kernels, leading to an important theory. The corresponding development for GT kernels should be of independent interest.
REMARK 6. If K is a positive definite kernel in II x ll, so is exp K, and the second quantization in Fock spaces exp H is known to allow to obtain the GNS construction of exp K from that of K, and similarly for the generalized GNS construction of Theorem 2. Of special interest are the characterizations of the so-called S-positive definite Toeplitz kernels, corresponding to representations in exp H of this type, and having as cyclic element the vacuum exp O. The analogue problem for GT kernels seems to be difficult and has still to be solved.
REMARK 7. For H1 = H2 and u1 = u2 = the identity, Theorem 3 reduces
essentially to the Beatrous-Burbea [BeBuJ generalization of a theorem of Bergman-Schiffer. The 2-parameter version of Theorem 3 given in the next section provides also a 2-parameter analogue of the Beatrous-Burbea result.
4. MULTIPARAMETER AND n-CONDITIONAL LIFTING THEOREMS, THE A-A-K THEOREM AND APPLICATIONS IN SEVERAL VARIABLES.
The lifting theorems of the previous two sections extend to multiparameter scattering structures. In order to avoid notational complications we develop here only the two parameter case, that reveals the
112 Cotlar and Sadosky
if f1 and f2 are given by (3.3b), then J1f1 = ~n Unfn,1 + f~,
J2f2 = ~nUnfn,2 + f~, and such that, if B'(f1,f2) := <J1f 1, J2f2>N' for
all f1 f H1, f2 f H2, then B': H1 x H2 ~ ( is a sesquilinear form satisfying
B' = B in V1 x V2, and B' ~ (B1,B2) in H1 x H2 .
• oreover, if E is the spectral measure of U, and, for each pair f1 f V1, f2 f V2, ~f f (A) := <E(A)J1f1, J2f2>N' then B' admits the
l' 2
integral representation
(3.4)
+ L J en(t)d~ 1 + J eO(t)d~ 1 1· n f2,fn,1 f1,f2
This integral representation can be precised further, but we shall not go into the matter here.
Vhen H1 and H2 are Hilbert spaces under the scalar products B1 and
B2, then Theorem 3 can be extended further, replacing the condition (2.11)
by the weaker condition
+ - + + -1 - - () H. = V. $ V. $ V., with u.V. C VI" u· V. C V. 3.5 1 III 1 1 III
i = 1,2, and the Hankel forms by the so called essentially Hankel forms B: V1 x V2 ~ (, that satisfy
where Pi is the orthoprojector of Hi onto Vi' i = 1,2.
The general abstract lifting theorem thus obtained for essentially Hankel forms is logically equivalent to the Sz.Nagy-Foias lifting theorem for intertwining contractions, and contains the Sarason theorem as a special case [CS 16]. Moreover, as stated in Section 4, this lifting theorem for Hankel forms extends to 2-parameter scattering and other abstract settings.
Cotlar and Sadosky
essential features of the extension, so that the multiparameter statements will be apparent.
For simplicity, we consider in this section the scattering structures [Hi; ui ; Vi]' i = 1,2, where H1,H2 are Hilbert spaces under
scalar products Bl'B2, respectively, IIflli = Bi (f,f)1/2, i = 1,2, and
u1 f L(H1), u2 f L(H2) are unitary operators satisfying (2.11).
Ve consider here the extension of the Lifting Theorem 3 to the case when there is another pair Tl f L(H1), T2 f L(H2) acting in the
scattering structure, satisfying
(4.1)
and the commutation condition
ia T·U· = e U·T· for some a f ~, i = 1,2.
1 1 1 1 (4.2)
The subspaces
V~ = {f f V.: k f V., Vk f 7l} u.f
1 1 1 1
and V: = {f f V.: k
f Vi' Vk f 7l} T.f 1 1 1
113
for i = 1,2, play an important role and, in what follows, it is assumed that
(4.4)
A sesquilinear form B: Hl x H2 ~ ( is called Toeplitz in the
scattering structure [Hi; ui,Ti'Vi], i = 1,2, if, for all fl f H1, f2 f H2,
114 Cotlar and Sadosky
Similarly, a form B: V1 x V2 ~ ( is called Hankel in the scattering
structure if, for all f1 f V1, f2 f V2,
(4.5a)
In this context, B ~ (B1,B2) means that IB(f1,f2)1 ~ IIf1111 IIf2112'
Le., that B is bounded with norm IIBII ~ 1.
Given a bounded form B: V1 x V2 ~ ( we call a pair of forms
B': H1 x H2 ~ (, B": H1 x H2 ~ ( a lifting pair for B if IIB'II ~ IIBII,
IIB"II ~ IIBII and
B' = B in V1 x V~ and B" = B in V x V;. (4.6)
THEOREI 4. (Two-parameter lifting theorem in scattering structure [CS 10,12], cf. also [CS 13]). Civen a scattering structure[H1; U1,T1; V1],
[H2; U2,T2; V2], as described above, every bounded Hankel form B has a
lifting pair of Toeplitz forms B' ,B".
This theorem applies, in particular, to the structure given in L2(Td), d ~ 2, by the shift operators, in the three following special cases, providing versions of the Nehari and Helson-Szeg5 theorems in several dimensions. Special case I. H1 = H2 = L2(T2), u1 = u2 = u, defined by
uf(x,y) = eixf(x,y), T1 = T2 = T, defined by Tf(x,y) = eiYf(x,y),
V1 = H2(T2) = {f f L2: f(m,n) = 0 if m < 0 or n < O}, V2 = vt. Here
V; = {f f L2: f(m,n) = 0 if n ~ O}, V; = {f fL2: f(m,n) = 0 if m ~ o}. 2 2 Special case II. H1 = H2 = L (T ), u1 = u2 = u, T1 = T2 = T, as above,
2 A. 2 A •
V1 = {f f L : f(m,n) = 0 If n < O}, V2 = {f f L : f(m,n) = 0 If n ~ O}.
Here V; = V2 and V; = {O}.
Special case III. H1 = H2 = L2(T2; w(x,y)dxdy), for some positive
integrable weight w. V1 and V2, subspaces of L2(T2;w), are defined as in I
Cotlar and Sadosky 115
or II. The Hankel forms in special case I are called big Hankel forms in
L2(T2), and those in special case II, middle Hankel forms. The Nehari theorem for big Hankel forms can be stated in terms of BI0(T2) functions, in the sense of A. Chang and R. Fefferman (cf. [CS 19]).
Every bounded Hankel form B: V1 x V2 ~ ( is associated with a
bounded Hankel operator T: Vi ~ V2, through
For every n f ~, the singular numbers of B are defined by
= inf{IITIEII: E subspace of V1, codim E ~ n} (4.8)
In particular, sO(B) = sO(T) = IITII = IIBII·
For a bounded form B: Vi x V2 ~ (, acting in a scattering
structure [Hi; ui ; Vi]' i = 1,2, a form B(n) is called an n-conditional
lifting of B, if there is a subspace I C V1 of codimension ~ n, such that
(4.9)
THEOREt 5. (n-conditional lifting theorem in one-parameter scattering structures [CS 17]).
Civen a scattering structure [Hi; u1; V1], [H2; u2; V2], every
bounded Hankel form B has, for each n f ~, an n-conditional Toeplitz lifting B(n). Hence, for each n,
sn(B) = inf{IIT - Tn ll : Tn Hankel operator of rank ~ n} (4.10)
where T is the Hankel operator given by (4.7).
116 Cotlar and Sadosky
REMARK 7. In the case Hi = H2 = L2(T), u1 = u2 = u, the shift, Vi = H2(T),
V2 = H~(T), Theorem 5 reduces to the classical Adamjan-Arov-Krein (A-A-K)
theorem [AAK2J. Theorem 5 which is based on a result in [TrJ, also applies . 222 2 In the case Hi = L (Tj~l)' H2 = L (Tj~2)' Vi = H (Tj~l)' V2 = H_ (Tj~2)'
providing two-weighted versions of the A-A-K theorem [CS 17J.
Combining Theorems 4 and 5 follows THEOREM 6. (Estimates for the singular numbers of Hankel operators in the bidimensional torus [CS 17]). Let T: Vi ~ V2 be a bounded Hankel operator,
Vi and V2 be as in special cases I or II. Then for every n f ~,
and (4.11)
liP Til = So (P T) :: s (P T) < s (T), T TnT - n
where Pu and PT are the orthogonal projections of V2 onto V; and V;,
respectively. In particular, every compact Hankel operator in L2(T2) 28
zero.
The abstract multiparameter lifting theorem also provides the extension of the Helson-Szeg6 theorem (see Section 2) to the multidimensional torus. In fact, in the circle, the theorem asserts that the Hilbert transform r is bounded in L2(Tjw) iff log W f BMO(T), with norm comparable to the operator norm of r. In [CS 12,14 ] it is proved that the double Hilbert transform r = r r is bounded in L2(T2jw) iff log f BMO(T2), x y
with norm comparable to that of the operator norm of r. This result also extends, through the concept of simultaneously u-boundedness of operators, to the characterization of the weights W for which r acts continuously in LP(T2jw) for 1 < p < w [CS 14J.
Finally, the preceding results extend to m and md, d ~ 1, as well as to the symplectic space ((d, [, ]), where the shifts are replaced by the so-called twisted shifts in m2d . This provides noncommutative versions of Theorems 2, 5 and 6, and corresponding theorems in Fock spaces [CS 12,18J.
Cotlar and Sadosky
[HK1J
[HKJ
[AllJ
[Al2J [Ar 1 J
[Ar 2J
REFERENCES
V.I. Adamjan, D.Z. Arov and I.G. Krein, Infinite Hankel matrices and generalized Caratheodory-Fejer and I. Schur problems, Func. Anal. Appl .. , £ (1968), 1-117 (in Russian). _______________ , Analytic properties of Schmidt pairs of a Hankel operator and generalized Schur-Takagi Problem, lat. Sbornik 86 (1971), 33-73. P. Alegria, On the Adamjan-Arov-Krein and Arocena parametrizations: a constructive and generalized version, Acta Cientif. Venez. 39 (1988), 107-116. ____ , Doctoral Dissertation, Univ. del Pais Vasco, 1992. R. Arocena, On generalized Toeplitz kernels and their relation with a paper of Adamjan, Arov and Krein, North Holland lath. Studies, 86 (1984), 1-22. ____ , On a geometric interpretation of Schur parameters, Rev. U.I.A. 34 (1988), 150-165.
117
[Ar CJ R. Arocena and I. Cotlar, Continuous Generalized Toeplitz kernels in m, Portugaliae lath. 39 (1980) 419-434.
[Ar CSJ
[Arv J
[BFHTJ
[BeBu J
R. Arocena, I. Cotlar and C. Sadosky, "Veighted inequalities in L2 and lifting properties, Adv. lath. Suppl. Studies, 7A (1981), 95-128. V.B. Arveson, Subalgebras of C*-algebras, Acta lath. 123 (1969), 141-22. J.A. Ball, C. Foias, J.V. Helton and A. Tannenbaum, On a local nonlinear commutant lifting theorem, Indiana Univ. lath. J. 36 (1987), 693-707. F. Beatrous and J. Burbea, Positive-definiteness and its applications to interpolation problems of holomorphic functions, Trans. Amer. lath. Soc. 284 (1984), 247-270.
[BrJ R. Bruzual, Local semigroups of contractions and some applications to Fourier representation theorems, Int. Eqs. lOp. Theory 10 (1987), 780-801.
[CS lJ I. Cotlar and C. Sadosky, A moment theory approach to the Riesz theorem on the conjugate function for general measures, Studia, lath., 53 (1975), 75-101.
118
[CS 2J
[CS 3J
Cotlar and Sadosky
__________ , Transformee de Hilbert theoreme de Bochner et Ie probleme des moments, I, C.R. Acad. Sci., Paris, A 285 (1977), 433-435, 661-665. __________ , On the Helson- Szeg6 theorem and a
related class of modified Toeplitz kernels, Proc. Symp. Pure lath. AIS, 34 (1979), 383-407.
[CS 4J , Characterization of two measures satisfying the Riesz inequality for the Hilbert transforms in L2, Acta Cient. Venez., 30 (1979), 346-348.
[CS 5J , On some LP versions of the Helson-Szeg6 theorem, in Conf. Harmonic Analysis in honor of A. Zygmund (Eds.: W. Beckner, A.P. Calderon, R. Fefferman, P.W. Jones), Wadsworth Int. lath. Series (1982), 306-317.
[CS 6J , Generalized Toeplitz kernels, stationarity and harmonizability, J. Anal. Math., 44 (1985), 117-133.
[CS 7J , A lifting theorem for subordinated invariant kernels, J. Func. Anal., 67 (1986), 345-359.
[CS 8J , Lifting properties, Nehari theorem and Paley lacunary inequality, Rev. lat. lberoamericana, ~ (1986), 55- 71.
[CS 9J , Prolongements des formes de Hankel generalisees en formes de Toeplitz, C.R. Acad. Sci., Paris, 1 305 (1987), 167-170.
[CS 10J , Nonlinear liftings theorems, integral
[CS 11 J
[CS 12J
[CS 13J
representations and stationary processes in algebraic scattering systems, Operator Th.: Adv. k Appl. 41 (1989), 97-123. __________ , The Generalized Bochner theorem in
algebraic scattering structures, in Analysis at Urbana, II, Cambridge University Press, London, 1989. __________ , Two- parameter lifting theorems and double Hilbert transforms in commutative and non-commutative settings, J. lath. Anal. k Appl. 150 (1990), 439-480.
, Nonlinear lifting theorems, Integral representations and stationary processes in algebraic scattering systems, Op. Th.: Adv. k Appl. 41 (1989), 97-123.
Cotlar and Sadosky 119
[CS 14]
[CS 15]
[CS 16]
[CS 17]
[CS 18]
[CS 19] [Da]
[Do 1]
[Do 2]
[DG 1]
[DG 2]
[E]
rUG]
[HS]
[JPR]
[I]
[10 1]
__________ , The Helson- Szego Theorem in LP of the
bidimensional torus, Contemporary lath. 107 (1990), 19-37. __________ , Lifting of Hankel forms bounded by non
Toeplitz norms, Int. Eqs. lOp. Theory, 14 (1991), 501-532. __________ , Transference of Metrics induced by
unitary couplings, J. Funct. Anal. to appear. _________ , Weighted and two-dimensional Adamjan-Arov-Krein theorems and Sarason commutators, Institut littag-Leffler Report No. 24, 1990/91. _________ , The Adamj an- Arov- Krein theorems in abstract and regular representations of ~2 and the symplectic plane, Institut littag-Leffler Report No. 34, 1990/91. _________ , preprint.
E.B. Davies, Quantum Theory of Open Systems, Academic Press, London, New York, 1976. I. Domiguez, A matricial extension of the Helson-Sarason Theorem and a characterization of some multivariate linearly complete regular processes, J. lultivariate Anal. 31 (1989) 289-310. _____ , lultivariate prediction theory and weighted inequalities, Op. Th. Adv. k Appl. (1990). H. Dym and I Gohberg, Extensions of matrix valued functions and block matrices, Indiana Univ. lath. J. 31 (1982), 733-765. ________ , Unitary interpolants, factorization indices and infinite Hankel block matrices, J. Funct. Anal. 54 (1983), 229-289. E.G. Effros, Aspects of non-commutative order, Lecture Notes in lath. 650, Springer-Verlag, Berlin, New York, 1970. R. Ellis and I. Gohberg, Orthogonal systems related to infinite Hankel matrices, J. Funct. Analysis, to appear. H. Helson and G. Szego, A problem in prediction theory, Ann. lath. Pure Appl., 51 (1960), 107-138. S. Janson, J. Peetre and R. Rochberg, Hankel forms and Fock spaces, Rev. lat. Ibero Amer. 8 (1987), 61-138. S. larcantognini, Unitary colligations in Krein spaces, J. Operator Th. (1990). I.D. loran, On Intertwining Dilations, J. lath. Anal. Appl. 141 (1989), 219-234.
120 Cotlar and Sadosky
[10 2] ____ , On Commuting Isometries, J. Operator Th. 24 (1990), 75-83.
[Ne] Z. Nehari, On bilinear forms, Ann. of lath. 68 (1957), 153-162. [Ni] N.K. Nikolskii, Treatise on the Shift Operator, Springer-Verlag,
Berlin-Heidelberg-New York, 1986. [Sa] D. Sarason, Generalized interpolation in Hm, Trans. Amer. lath.
Soc. 127 (1967), 179-203. [Sz. N-F] B. Sz. Nagy and C. Foias, Analyse harmonigue des operateurs de
l'espace de Hilbert, lasson-Akad. Kiado, Paris and Budapest, 1970.
[T] N. Takesaki, Operator Algebras. Springer-Verlag, Berlin, New York, 1979.
[Tr] S. Treil, The theorem of Adamjan-Arov-Krein: Vector variant, Publ. Seminar LOll Leningrad 141 (1985), 56-72. (In Russian).
I. Cotlar Facultad de Ciencias Universidad Central de Venezuela Caracas 1040, Venezuela
C. Sadosky Department of lathematics Howard University Washington, D.C. 20059, USA
Operator Theory: Advances and Applications, Vol. 58 © 1992 Birkhliuser Verlag Basel
REDUCTION OF THE ABSTRACT FOUR BLOCK PROBLEM
TO A NEHARI PROBLEM
J.A. Ball, 1. Gohberg and M.A. Kaashoek
121
The abstract version of the four block extension problem is reduced to an abstract Nehari problem of which the solution may be obtained via the band method. It is also shown that the maximum entropy solution of the four block problem may be obtained in this way from the maximum entropy solution of the associated Nehari problem.
O. INTRODUCTION
This paper concerns the abstract version of extf'nsion problems of the fol
lowing type. Consider the 2 x 2 matrix function
(0.1 ) ""(t) = (f(t) g(t) + :p(t)) t R ~ h(t) k(t) , E,
where f, g, hand k are given functions in L1 (R). The problem is to find :p E L1 (R) such
that :p(t) = 0 a.e. on -00 < l < 0 and
(0.2) sup II~(A)II < l. AER
Here Cil denotes thf' Fourier transform of <fI, and the norm in (0.2) is the usual operator
norm for matrices. The above problem, which is usually referred to as a four block problem,
the analogous problem with the real line replaced by the circle, and other matrix-valued
versions of the problem appeared in the middle of the eighties in mathematical system
theory in connection with problems of Hoo-control (see [1], [2J and [5]). The existence of
a solution (also for the optimal case) was proved in [4J by using the commutant lifting
theorem. In a more general setting (involving many blocks) a description of all solutions
has been given in [3J (see also [6]). In the paper [3J also the maximum entropy solution was
singled out, and the special role played by the maximum entropy solution was clarified.
122 Ball et al.
The four block problem has been treated in [10] by means of the band method, and the
same approach was used in [13] to solve other versions of the four block problem. For
rational matrix-valued functions an alternative solution based on an all pass embedding
method may be found in [9]. Another direction in the theory of the four block problem
concerns thp computation of the minimal norm, i.e., the smallest possible value of the left
hand side of (0.2) for all possible choices of <p in (0.1). The papers [7) and [8] develop a
skew Toeplitz theory to treat such issues.
In this paper we solve an abstract version of the four block problem by
reducing it to an abstract Nehari problem of which the solution may be obtained via the
band method (see [10) - [13]). The method of solution described in the present paper
may be applied to many concrete cases. For the problem (0.1), (0.2) discussed above it
yields the following result. First, note that for the problem (0.1), (0.2) to be solvable it is
necessary that
(i) ~(.\) := 1 - 11(.\)12 - Ih(.\)12 > 0,
(ii) 6(.\) := 1 -lh(.\)12 - Ik(.\)12 > o.
Assuming that (i) and (ii) hold, introduce the functions
where
Yl(t) = {{log(1 + ~(.\)-ll1(.\W)}V(t), 0,
let
and consider
t > 0, t < 0,
t > 0, t < 0,
Here IV denotes the inverse Fourier transform of the function I. Then there exists ~ as in
(0.1) such that (0.2) holds if and only if (i) and (ii) are satisfied and the Hankel operator
ron L2([0, ,)())) defined by
(ru)(t) = 100 /,( -t - S)lL(S) ds, 0 ~ t < 00,
has norm strictly less than one. In the latter case all solutions of the problem (0.1), (0.2)
are obtained by taking
(0.3)
where z is any element of Ll(R)such that
(0.4) z(t) = /,(t) (t < 0), sup Iz('>') I < l. >'ER
The proof of our main theorem uses elements of the band method (see [10]
[13]). The band method may also be applied directly to the four block problem considered
here. However, for examples of interest, the present adapt ion of the band method leads
to simpler formulas at the end. We also show that the triangular extension (see the next
section for the definition) of the four block problem corresponds to the triangular extension
of the associated Nehari problem. For the concrete example mentioned above the latter
means that the maximum entropy solution for the problem (0.1), (0.2) is obtained by
taking z in (0.3) to be the maximum entropy solution of the Nehari problem (0.4).
The present paper consists of two sections (not counting this introduction) ..
The first section contains the main theorems; the proofs are given in the second section.
1. MAIN THEOREMS
To state the main theorems we need some notation and terminology.
Throughout this paper N is an algebra with the following triangular structure:
(1.1)
The latter means that N is an algebra with an involution *, a unit e, and the spaces
appearing in the decomposition (1.1) are subalgebras and have the following properties:
124 Ball et aI.
(ii) (.tV2)* = N2, (iii) Nt#2 = .N2,Nd =.N2, (0 = i, u).
We put
(1.2)
We write PRrl (resp., PNl ) for the projection of N onto N2 (resp., Nt) along the space
Nu (resp., N;!). In a similar way by interchanging the roles of land u, one defines the
projections pRru and PNu' By PNd we denote the projection of N onto Nd along the space
N2+N2. We shall also assume that N satisfies the following additional requirements:
(EO) The principal submatrices of a positive definite element in N4x4 are positive
definite in the corresponding algebras.
(E1) The positive definite elements in N admit a left and a right spectral factor
ization relative to the decomposition (1.1).
Recall that an element b in a·-algebra A is called positive definite (notation: b > 0) if
b = c·c with c invertible in A. Furthermore, bEN is said to admit a left (resp., rigbt)
spectral factorization relative to (1.1) if b factors as b = c*c where c is an invertible element
of.N such that c and its inverse c-1 are both in Nt (resp., Nu ). Elements in N that admit
a spectral factorization are automatically positive definite in.N. Axiom (E1) requires that
the converse implication is also true.
We say that an element "pl E Nt has a strictly contractive extension <p if
(a) <p-"ptEN;!,
(b) e-<p*<p>O.
The problem to find all (strictly) contractive extensions of <Pi is called a Nebari problem.
Put B = ,N2 x 2, and set E = (~ ~ ). We call an element G E B strictly
contractive if E - G*G is positive definite in B.
(1.3)
THEOREM 1.1. Given
cp = ('I'll '1'21
Ball et al.
there exists x E N2 such that
(1.4) G = ('Pll ip2l
is strictly contractive if and only if
(1.5)
(1.6)
'Pi + X) ip22
125
and the element,pe := PNl {Xl (ipe + w )X2} has a strictly contractive extension in N. Here
(1.7a)
(1. 7b)
(1.7c) ( + A-I * )-1 A-I * W = e ipH L..l. ipH ipH L..l. ip2l ip22·
Furthermore, in that case all X E N2 such that G in (1.4) is strictly contractive are given
by
(1.8)
where 7/Je + y is a strictly contractive extension of 7/Je.
If Gin (1.4) is strictly contractive, then G is said to be a strictly contractive
extension of cP (in (1.3)). Note that x in (1.8) may be rewritten as
(1.9) -1(.1 )-1 X = Xl 'f'e + Y X 2 - ipe - w.
Thus Theorem 1.1 tells us that the general form of the (1,2 )-entry in a strictly contractive
extension of cP is equal to X 1l ipX21 _W, where ip is an arbitrary strictly contractive extension
of 7/Je. In this way one may derive a linear fractional description of all strictly contractive
extensions of cP by substituting in (1.9) the linear fractional representation of all solutions
ip = 7/Je + Y of the corresponding Nehari problem.
We call G E B a triangular extension of cP in (1.3) if G is a strictly contractive
extension of 7/J and the (1,2) entry of G(E - G*G)-l is in Ne.
126
(1.10)
Ball et al.
THEOREM 1.2. Let II> be as in (1.3). Then II> has a triangular extension
C = (<Pll <pe + x) E B = N'lX2 <P21 <P22
if and only if (1.5), (1.6) hold and the equation
(1.11)
has a solution z ENe such that z-l ENe and PNdZ is positive definite in Nd . Here
(1.12)
with Xl, x2 and w as in (1. 7a,b,c). Furthermore, in that case II> has a unique triangular
solution C which one obtains by inserting
(1.13) -1( -1 )*[P (.1.* )]* -1 X = -<Pf - w + Xl Z Nu 'l-'e Z X 2
into (1.10).
Theorem 1.2 tells us that C in (1.10) is the triangular extension of II> if
and only if 9 := X1(X + <pe + W)X2 is the triangular extension of 'l/Je, i.e., 9 is a strictly
contractive extension of 4'£ and g(e - g*gt 1 E N£ (cf., [11], Theorem 1.2.1). From the
proof of Theorem 1.2 we shall derive a left spectral factorization for E - CCO, when C is
t he triangular extension of II>.
2. PROOFS OF THE MAIN THEOREMS
Throughout this section N is the algebra with triangular structure (1.1)
introduced in the previous section, and N is assumed to satisfy axioms (EO) and (El). As
before B = N2x2. We shall need the direct sum decomposition
(2.1 )
where
B~={ (: ~) EB 1 a,dEN2},
Bd = { (~ ~) E B I a, d E Nd },
B~={ (~ !) EBla,dE~}.
Ball et al. 127
Note that (2.1) is a triangular structure on B. It will be convenient to consider also the
decomposition:
(2.2)
where
Be = { (; !) E Bib ENe},
Ru = { (~ ~) E Bib E N2 }.
For later purposes we mention the following facts:
(2.3) BuB~ c Bu, B~Bu c Bu,
(2.4) BeB~ c Be, B~Be c Be,
(2.5) BuBd C Bu, BdBu C Bu,
(2.6) B£Bd C Be, BeBd c Be,
Now, let M = B2 x 2 , and consider
(
0 0
M = (0 Bu) = { 0 0 1 0 0 0 0
o 0
:t)={(T * * ] M~ = (~~ a22 * I bENe, 0 a33
0 0
;J={(T 0 0 n Md = (~d a22 0 I aii ENd, 0 a33
0 0
~)={[ 0 0 0
M O = (B~ a22 0 ~ ) I,EN., 3 B; * a33
* * a44
(! 0 0
n M4 = (;: ~) = { 0 0
ICEJVfl}, 0 0 0 0
ai; E N2, i=I, ... ,4},
i=I, ... ,4},
aii E JVfl, i=I, ... ,4},
128 Ball et al.
where the *'s denote arbitrary elements of N. We have
(2.7)
and by using (2.3) - (2.6) it is straightforward to check that M with the decomposition
(2.7) is an algebra with band structure, that is (see [10] -[13]),
(2.8)
and the following multiplication table holds
Ml Mg Md Mg M4 Ml Ml Ml Ml MO + M Mg Ml Mt Mg Me M~
Md Ml Mg Md Mg M4 Mg MO + Me Mg M~ M4 M4 M M~ M4 M4 M4 .
where
The involution * in (2.8) is the natural involution on M induced by the involution on N.
In the sequel we use freely the notation introduced in the first three para
graphs of Section 1.1 of [12] for the present M. Note that
where B+ = B~ + Bd and B_ = B~ + Bd. Let «I» be as in (1.3), and put
(eO 'I'll rpi)
K = ~ : '1'21 '1'22 = (~ «I»). 'I'll '1'21 e 0 «I» E rpi '1'22 0 e
We have K E Me. Now let G E B = N2X2 be an extension of 4>, i.e., G - 4> E Bu. Put
Ball et al. 129
Then KG is an extension of K, that is, KG - K belongs to M 1+M4.
LEMMA 2.1. Let C E B. Then C is a strictly contractive (resp., triangular)
extension of~ if and only if KG is a positive (resp., band) extension of K.
PROOF. Use axiom (EO) and the following identities:
(2.9)
(2.10)
A straightforward reasoning gives the desired results (see [11], Section 1.2 for further de
tails) .•
Note that any extension of K is of the form KG for some C E B.
LEMMA 2.2. Let G E B. Then E - GG" admits a left spectral factorization
relative to (2.1) if and only if KG admits a left spectral factorization relative to (2.7).
PROOF. Assume 1- GG· = H" H with H±l E B_ = B!.. +Bd. Since
(2.11 ) ( E C) (E - CC" 0) (E 0 ) KG = 0 E 0 E G" E '
we see that KG factors as KG = L"L with L = (:J. ~). Now
Thus KG = L"L is a left spectral factorization of KG relative to (2.7).
Next, let KG = L" L be an arbitrary left spectral factorization of K relative
to (2.7). Thus
L=(~ ~), with A,lP,D and DX in B_. Note that DD x = DXD = E. Thus D is invertible (and
D±l E B_). From
(2.12) ( E G) ( A" C") (A 0) C" E - 0 D" CD'
130 Ball et aI.
we see that G* = D*C and D* D = E. Because of the latter identity, we have D* = D-1 ,
and thus C - DG* = O. Now use (2.12), and rewrite (2.11) as
O)=(E -G)(A* C*)(A 0)( E 0) E 0 E 0 D* C D -G* E =(~ ~r(~ ~).
So E - GG* = A * A, where A E B_. Finally, note that
( AX 0) = (A 0 )-1 (A-l 0) C X D X C D * * .
Hence A-I = A X E B_. So E - GG* has the desired spectral factorization .•
Consider the matrices
(2.13) ( eO <pu) Al = ~ ~ <P21 ,
<Pll <P21 e
<P22 ) o , e
and let ~ and 6 be defined by (1.5) and (1.6), respectively.
LEMMA 2.3. We have Al > 0 (resp., A2 > 0) if and only if ~ > 0 (resp.,
6 > 0). Furthermore,
(2.14)
(2.15)
whenever ~ or 6 is invertible.
<Pll~ -1<P21
e + <P21 ~ -1 <P21 _~-1<P21
-6-1<p21
e + <Pi1 6-1 <P21
<P22 6- 1<p21
-<Pll~-1 ) -<P21~ -1 , ~-1
PROOF. To prove the statements concerning AI, use axiom (EO) and a
factorization of the type (2.9) with G = (<Pll). For A2, replace (2.9) by (2.11), and <P21
apply with G = ( <P21 <P22 ) .•
PROPOSITION 2.4. There exists
(2.16)
<Pll <P21
e o
Ball et al. 131
such that B > 0 and the (1,4 )-entry of ii-I is zero if and only if ~ and 8 are positive
definite. In that case there exists only one such B which one obtains by taking
(2.17) b ( A -1 * )-1 A-I * = - e + 'PllLl 'Pll 'r~ll Ll 'P21 'P22·
PROOF. Assume B in (2.16) is positive definite. Then, by axiom (EO), the
matrices Al and A2 are positive definite. So, by Lemma 2.3, we have ~ > 0 and 8 > o. Conversely, assume that ~ and 8 are positive definite. We have to solve a
band extension problem in the algebra Ai of 4 x 4 matrices with entries in N for the band
given by o 'Pll
e 'P2l
'P;1 e lf~2 0
To do this we apply the band method. Consider the following subspaces
Then
(2.18)
.c1 = {(aij);,j=1 E N 4 x4 I aij = 0 for j - i -:; 2},
.cg = {(aij)L=1 E N 4X4 I Gij = 0 for j - i > 2 or j -i -:; O},
.cd = {(aij)L=1 E N 4 x4 I aij = 0 for j i i},
.c~ = {(a;j);,j=1 E N 4X41 aij = 0 for j - i 2: 0 or j - i < -2},
.c~ = {(aij)L=1 E N 4 x4 I aij = 0 for j - i 2: -2}.
and (2.18) defines a band structure on M (which is different from the one associated with
(2.7)). Note that
o
o
We shall apply Theorems 1.1.3 and 1.1.4 in [10] for the band structure defined by (2.18).
Since 6 > 0, we know from Lemma 2.3 that Al > 0, and hence (use Axiom (EO)) we may
132 Ball et aI.
conclude that e + 'Pll~-l'Pil is positive definite in N. Now, apply axiom (E1) to show
that we may choose Xl as in (1.7a). We may also factor 6-1 as
(2.20)
where u is invertible in N. Put
o 6-1
-'Ph6- l
-'P226- l
o 0) o 0 eO' o e
(
Xl 0 0 0) n-= 0 u 0 0
o 0 eO' o 0 0 e
Then Y E C3 := Cg+Cd and y-l '- C_ := C4+Cg+Cd. Furthermore, the diagonal ofY is
equal to f)* 15, and hence positive definite in Cd. So we can apply Theorem 1.1.3 in [10J to
show that
(2.21 ) B := (y-l)* 15* 15y-l
is a band extension of Ko (in (2.19)) relative to the band structure (2.18). Since Y is lower
triangular, it follows that the (1,4)-entry of B is equal to the adjoint of (y-l)41 , which
one computes to be the element given by the right hand side of (2.17).
To prove the uniqueness of B we first note that axioms (EO) and (E1) imply
that the positive definite elements in N4X4 admit left and right spectral factorizations
relative to (2.18) (cf., the proof of Lemma 2.5 at the end of this section). But then we
can apply Theorem 1.1.4 in [10] to show that Ko has a unique band extension relative to
(2.18) .•
The element bin (2.17) is also given by the following formula
(2.22) b * c-l (+ * c-l )-1 = -'Pll 'P2l v 'P22 e 'P22 v 'P22 ,
which one may prove by direct checking.
PROOF OF THEOREM 1.1. We divide the proof into five parts.
Part (a). Let G be a strictly contractive extension of CPo Then KG > 0 by
Lemma 2.1. Thus, by axiom (EO) the matrices Al and A2 are positive definite. Therefore,
by Lemma 2.3, we have ~ > 0 and 6 > O.
Ball et al. 133
Part (b). Assume ~ > 0 and {) > O. By Lemma 2.3, the block matrices Al
and A2 are positive definite in N3x3. It follows that
( All 0) o e > 0,
Thus, by axiom (EO), the (l,l)-entry of All and the (3,3)-entry of A21 are positive definite.
But then we can apply axiom (E1) to show the existence of the factorizations (1.7a) and
(1. 7b).
Part (c). Assume ~ > 0 and {) > 0, and let G be as in (1.4) with x E N2. We may apply Proposition 2.4. Let jj be as in (2.16) with b as in (2.17). Recall that jj is
also given by (2.21), and note that b = -w, where w is given by (Lit). It follows that
KG = MI + B + M{,
where
lvh = (H o 0
~ e+~+w) 00'
o 0
By (2.21) we have jj = (H-I)* H- I with
where u is as in (2.20). We see that
H*KGH = H*MIH + E + HMtH*,
where E is the unit of 8 2 x2 and
(0 -XI'l/"P22U* 0 Xl1/;)
H*M H = 0 0 0 0 I 0 0 0 0
o 0 0 0
with 1/; = 'Pi + X + w. In this way we derive that G is a strictly contractive extension of <fl
if and only if
( , -XI'l/''P22u* 0 T) (2.23) -u'P2~1/;*xi e 0 0 e
1fi*xi 0 0
134 Ball et al.
is positive definite. Note that
( -U<P221P*xi ) * -1 • • e - (-Xl1PY'22 u* Xl1P) 1P*xi = e - Xl1P( e + Y'22 0 Y'22)1P Xl
= e - (Xl1PX2)(Xl1PX2)*'
where we use (1.7b). It follows that G is a strictly contractive extension of ~ if and only if
(2.24)
Part (d). Let G be a strictly contractive extension of~. Put
1Pt. = PNJXl(Y't. + W)X2}. We have XlXX2 E Nu , because X E N2 and Xl,X2 in Nu . Thus
(2.25)
From (2.25) we see that 1Pt. + y, where
(2.26)
is a strictly contractive extension of 1Pt.. Thus 1Pl has a strictly contractive extension in N
and x has the desired representation.
Part (e). Let .,pi + Y be a strictly contractive extension of .,pt. Define x by
(1.8). Since xII and xa1 are in Nu , we have X E Nu . Let G be as in (1.4) with this choice
of x, and put '1/.' = Y'l + X + w. Then (2.25) and (2.26) hold, and hence we have also (2.24).
Thus, by the result of part (c), G is a strictly contractive extension of~ .•
PROOF OF THEOREM 1.2. If cP has a triangular extension, then, by
Theorem 1.1, we have ~ > 0 and 0> O. So in what follows we may assume that both ~
and 0 are positive definite. Furthermore, we assume that Xl, X2 and ware as in (1.7a,b,c).
We split the proof into three parts. Sometimes we write a-* for the adjoint of a-I.
Part (a). Bv Lemma 2.1 it suffices to look for a band extension of
(2.27)
Axioms (EO) and (E1) imply (see Lemma 2.5 below) that the positive definite elements in
M have a left and right spectral factorization relative to the band structure (2.7). Hence
(see Theorem 1.1.1 in [11]) the band extension of K (assuming it exists) is unique.
Ball et al. 135
Part (b). In this part we apply Theorem I.1.1 in [11] to construct a band
extension for K assuming (1.11) has the desired solution. We seek
(2.28)
such that
(2.29)
and
(a) aii E Ni , dii E Nt (i = 1,2), Cu E Nu (b) aii and d ii are positive definite in Nd (i = 1,2);
(c) aii 1 E Ni , dii 1 E Nt (i = 1,2).
One computes that (2.29) is equivalent to
and the following seven equations:
These seven equations may be rewritten in the following way:
(2.30a) 4'21
e o
4'22) (a22) ( e ) ( re ) o C12 0 + 0 , e C22 0 0
136 Ball et al.
o CPll CPt) (all) ( e) (n~ ) CP21 CP22 a21 0 + 0
e 0 Cll 0 0' o e Cu 0 n~
(2.30b)
where r~, n~ E N2 and n~ E N2 are free to choose. From l2.l5) we see that
(2.31 )
Now factor 6-1 as 6-1 = u*u with u±1 E Nu , which is possible because of axiom (El).
Then
(2.32) -* + 0 u a22 = u ur u.
Note that ur~ E N2. We seek a22 E Nt. So u-*a22 E Nt. It follows that we must take
r~ E N2 so that PN~U = -ur~. In other words,
(2.33)
Write U = Ud + uO with Ud E Nd and Uo E N2. Then
(2.34)
Since u±l E N u , we conclude that at21 E Nt and a22 > 0 in N. SO we found a22, and from
(2.31) we see that
(2.35)
Next, we solve (2.30b). By using (2.14) we see that (2.30b) may be rewritten
as:
(2.36a)
(2.36b)
Ball et al. 137
The first row in (2 .36b) gives:
where w is defined by (1.7c). Now use (1.7a) to get
(2.37)
The second row in (2.36b) and (2.37) yield:
A-I * ( A-I *) + A-I * ( 0) a21==-'P21 U 'Pll'Pe Cu- e +'P21 U 'P21'P22 Cu 'P21 U 'Pll e + n u
A-I * ( A -1 * ) + A -1 * (+ A -1 * )-1 == 'P21 U 'PllWCu - e + 'P21U 'P21 'P22 Cu 'P21 U 'Pll e 'Pll U 'Pn all·
Thus
It follows that
where
Now use that
It follows that
A -1 * ( A -1 * )-1 (A * )-1 A U 'Pll e + 'Pll U 'Pn 'Pn == e - U + 'Pn 'Pll u.
+ 'P;2'P21~ -1 'Pi1 (e + 'Pll~ -1 'P~1 )-1 'Pll ~ -1 'P21 'P22Cu
* • ( ~ • )-1. == -'P22'P22 Cu - 'P22'P21 U + 'Pll'Pll 'P21'P22 Cu
* ( * )-1 == - 'P22 e - 'P21 'P21 'P22 Cu'
138
Thus
= ('Pi + w)*all + [e - 'P;2(e - 'P21'P;I)-I'P22]cu
= ('Pi + w)*all + [e + 'P;25-1'P22r1cu'
N ow use (1. 7b) to show that
(2.38)
Now let us assume that (1.11) has the desired solution z. Put
(2.39)
Then atll E Nf , because Ti and z have this property. Furthermore,
(2.40)
and thus Pd( all) is positive definite in Nd. Next define
(2.41)
(2.42)
Cu := -X2[PNj7jJi z )]d1 E Nu ,
a21 := 'P21~-I'P~lWCu - (e + 'P21~ -1'P;I)'P22Cu
,,-1 • (+ ,,-1. )-1 + 'P22 U 'P11 e 'P11 U 'P11 all·
Ball et al.
Then we know from (2.38) that (2.36a) holds for some n~ E NJ. Furthermore, from (1.11)
we see that
(2.43)
and thus
So (2.37) holds true with n~ = xII s~ E N~. Finally, define
Then Y in (2.28) has the desired properties, and thus we can apply Theorem 1.l.1 in [11]
to show that
Ball et al. 139
is the desired band extension.
It remains to compute the (1,4)-entry of B. Note that
(2.44) ~ ~) eO·
o e
It follows that
So let us solve
Then
Thus Bl4 = 'Pe + x, where
and hence x is of the form (1.13).
Part (c). In this part we assume that K has a band extension B = KG.
By Theorem I.1.1 in [11] this implies that the equation (2.29) has a solution Y with the
propertif's (a), (b) and (c) mentioned directly after (2.29). Put z = xl*andll as in (2.39).
Then z±l ENe, and Pd(Z) = dl*(Pdall)dl, which is positive definite in Nd. From (2.38)
we bee that
and (2.37) yields
140 Ball et al.
By combining these two identities we see that (1.11) has a solution with the desired prop
erties .•
LEMMA 2.5. A positive definite element in B2x2 admits a left and a right
spectral factorization relative to the band structure (2.7).
PROOF. Let A E B2x2 be positive definite. Then, by axiom (EO), the
principal submatrices of A are positive definite. Hence we can factor A as A = LDL*,
where Land D are of the form
L = (: ~ ~ ~) * * eO'
* * * e
(d1
D= 0 o o
with db d2 , d3 and d4 positive definite in N. By axiom (El) we have
d --" i = xixi' -±l .N. Xi E l·
Now,
C-O 0 j) EM-, M=L :
X2 0 M-1 E M_, 0 X3 0 0 X4
and A = MM" is a right spectral facturization relative to (2.7) .•
REFERENCES
[1] J.C. Doyle, Synthesis of robust controllers, Proc. IEEE Conf. Dec. Control, 1983. [2] J.C. Doyle, Lecture notes in advances in multivariable control, ONR/Honeywell Work
shop, Minneapolis, 1984. [3] H. Dym and I. Gohberg, A new class of contractive interpolants and maximum entropy
principles, in: Topics in operator theory and interpolation (Ed. I. Gohberg), OT 29, Birkhauser Verlag, Basel, 1988, pp. 117-150.
[4] A. Feintuch and B.A. Francis, Distance formulas for operator algebras arising in optimal control problems, in: Topics in operator theory and interpolation (Ed. I. Gohberg), OT 29, Birkhauser Verlag, Basel, 1988, pp. 151-170.
[5] B.A. Francis and J.C. Doyle, Linear control theory with an Hoc optimality criterion, SlAM J Control and Optimization 25 (1987), 815-844.
[6] C. Foias, On an interpolation problem of Dym and Gohberg, Integral Equations and Operator Theory 11 (1988),769-775.
[7] C. Foias and A. Tannenbaum, On the four block problem I, in: Topics in Operator Theory, Constant Apostel Memorial Issue (Ed. I. Gohberg), OT 32, Birkhauser Verlag, Basel, 1988, pp. 93-112.
Ball et al. 141
[8] C. Foias and A. Tannenbaum, On the four block problem II, the singular system, Integral Equation Operator Theory, 11 (1988), 726-767.
[9] K. Glover, D.J.N. Limebeer, J.C. Doyle, E.M. Kasenally and M.G. Safonov, A characterization of all solutions to the four block general distance problem, SIAM J. Control and Optimization 29 (1991), 283-324.
[10] I. Gohberg, M.A. Kaashoek en H.J. Woerdeman, The band method for positive and contractive extension problems, J. Operator Theory 22 (1989), 109-155.
[11] 1. Gohbprg, M.A. Kaashoek en ILJ. Woerdeman, The band method for positive and contractive extension problems: an alternative version and new applications, Integral Equations Operator Theory 12 (1989), 343-382.
[12] I. Gohberg, M.A. Kaashoek en H.J. Woerdeman, A maximum entropy principle in the general framework of the band method, J. Funct. Anal. 95 (1991), 231-254.
[13] 1. Gohberg, M.A. Kaashoek en H.J. Woerdeman, The time variant versions of the Nehari and four block problems, Springer Lecture Notes, 1496, pp. 309-323.
J.A. Ball Department of Mathematics Virginia Tech Blacksburg, VA 24061, U.S.A.
1. Gohbprg School of Mathematical Sciences The Raymond and Heverly Sackler Faculty of Exact Sciences Tel-Aviv University Tel-Aviv, Ramat-Aviv 69989, Israel.
M.A. Kaashoek Faculteit Wiskunde en Informatica VrijP U niversiteit De Boelelaan l081a 1081 HV Amsterdam, the Netherlands.
142 Operator Theory: Advances and Applications, Vol. 58 © 1992 Birkhiiuser Verlag Basel
THE STATE SPACE METHOD FOR INTEGRO-DIFFERENTIAL EQUATIONS OF
WIENER-HOPF TYPE WITH RATIONAL MATRIX SYMBOLS
A. B. Kuijper 1
This paper concerns systems of integro-differential equations of convolution type on the half-line for which the symbol is a rational matrix function. The equations are studied via certain singular input/output systems. For maximal operators associated with the equations, which act between certain Sobolev spaces on the half-line, explicit conditions for the operators to be invertible and to be Fredholm are derived, as well as explicit formulas for inverses, generalized inverses and the Fredholm characteristics. All conditions and formulas are expressed in terms of the matrices appearing in the singular system corresponding to the equations, and in matrices that are related to the system.
1. INTRODUCTION AND MAIN THEOREMS
During the past ten to fifteen years a new method has been developed to deal with prob
lems in mathematical analysis involving rational matrix-valued functions. This method, which is
referred to as the state space method, originated in [BGK I] and [BGK2] and is based on the
idea of realization, which comes from mathematical systems theory. For a rational m x m
matrix function W which is analytic at infinity, a realization ([Ka]) is a representation in the
form
WO .. )=D +CO._A)-l B, (1.1)
where A is a square matrix of size n xn, say, and B, C and D are matrices of sizes n xm, m Xn
and m x m, respectively. The representation (1.1) allows one to reduce analytic problems for W
to linear algebra problems involving only the matrices A, B, C and D (see [BGK4] and the refer
ences given there).
In this paper the state space method is used to study systems of integro-differential equa
tions of the following type:
q(idd )CP(t)+jJ.(t-s)CP(s)ds=f(t), O<t<oo. t 0
(H)
1 Research supported by the Netherlands organization for scientific research (NWO).
Kuijper 143
N . Here q(A.) = LX) AJ is a given polynomial with m x m matrix coefficients and k (t) is an m x m
)=0
matrix function with entries in L I (IR) whose Fourier transform is rational. Thus, the symbol of
equation (H),
W(A)=q(A)+k(A),
is an arbitrary rational matrix function with no restriction on the behaviour at infinity. Such a
function cannot be represented in the form (1.1), but it admits (see [GK I]) a representation
W(A)=D +C(AG -A)-I B, (l.2)
where A, B, C and D are matrices as before, G is a square matrix of the same size as A and the
pencil AG -A is IR-regular. The latter means that the pencil AG -A is invertible for all AE IR.
In general, the matrix G will not be invertible. On the other hand, because of the freedom one
has in the choice of the matrices in (1.2), one may always take D =1 m (see [GKI]).
We use the representation (1.2) with D =1 m to analyse equation (H) in terms of the
singular input/output system
1 iGp'(t)
f{t)
=Ap(t)+Bq,(t),
= Cp(t) + <I>(t). (1.3)
Since the feedthrough coefficient is equal to the m x m identity matrix, we may interchange in
(1.3) the role of input and output, and derive the inverse system:
1 iGp'(t)
q,(t)
=A x p(t) +Bf(t),
=Cp(t)+!(t), (1.4)
whert' A x =A -Be. This approach allows us to obtain explicit descriptions for the Fredholm
and invertibility properties of equation (H), and to derive explicit formulas for the solutions in
terms of the matrices in (1.2) and the spectral properties of the pencils AG - A and AG - A x
associated with the systems (1.3) and (1.4).
In the following subsections, we shall explain in more detail which equations are con
sidered, and how the state space method is used to obtain solutions.
1.1. The equations. Before turning to the half-line equation (H), we first need some
definitions. In the sequel, (:1 m )' will denote the space of ([: m -valued tempered distributions
([S]), i.e., in the dual of the space :/ m of ([: m -valued rapidly decreasing functions on IR. In the
setting of (:/ m )' we define an integra-differential operator i w : (:/ m )' ~ (:/ m)' associated
with the functions q and k in (H), as follows:
0.5)
144 Kuijper
Here ! and * denote the operator of differentiation and convolution on (:/ m) , • respectively.
and equality has to be interpreted in distributional sense. More concretely, we have L w (I/» = f for a tempered distribution f if and only if
(I/>, ~(-i)jXJ aU> + j e (s-t)a(s)ds) = (f,a) (1.6) j=O -00
for all test functions a E :/ m. Here a (j) is the j'h pointwise derivative of a. For more details
on the distributional differentiation ! and the convolution operator * on (:/ m)' we refer to [S].
The solutions cp of the half-line equation (H) are required to be cr m -valued distributions
with support in 1R + :::: [0,00), i.e.,
(1.7)
Then, replacing the integral by a convolution, the left hand side of (H) is a well-defmed element
of (:/ m )'. In the sequel we use the notation (:/ m) , (IR +) for the subspace of (:/ m)' of those
distributions which satisfy (1.7). Similarly, one defines (:/ m)' (IR - ) as the subspace of elements
in (:/ m)' which have support in (-00,0]. The equality sign in (H) means that the tempered dis
tribution in the left hand side of (H) coincides on (0,00) with the tempered distribution f in the
right hand side of (H), in other words, (1.6) holds for all a E :/ m with supp a c 1R + •
We shall analyse equation (H) in terms of (possibly unbounded) operators Tw acting
from a domain in H';' (IR +) into H';'+r [IR + I. Here sand r are integers, H';' (IR +) is the space of
distributions in the Sobolev space H';'(IR) (cf. [H]) with support in 1R +, and H~r[IR+I is the
image of H';' (IR) under the quotient mapping
Now, for prespecified rand s the operator Tw (H';' (IR+) ~H';'+r [IR+]) is defined by
-Tw(cp)=q+ Lw(I/», CPE D(Tw)·
A _
Here the subindex W refers to the symbol W 0 .. ) = q (A.) + k(A.) of (H), and L w is the full line
operator defined in (1.5). Observe that T w is the restriction of the operator
q + i w : (:/ m)' ~ (:/ m)' [IR + 1 to the largest possible domain in H';' (IR +) such that
1m Tw cH~r[IR+J. As a consequence. using the continuity of the operator Lw and the con
tinuity of the embeddings from H ';' (IR + ) and H ~r [IR + 1 into (:/ m )' and (:/ m) , [IR + I. respec
tively. the operator Tw is closed. The operator Tw will be referred to as the maximal half-line
integro-differential operator with symbol W (acting between H ';' (IR + ) and H ';'+r (IR + I ).
Kuijper 145
Throughout this paper we shall assume the integers -s and r + s to be nonnegative. In
this case the right hand side in T w (q,) = [f] is a (C m -valued function for which the ftrst r +s-1
pointwise derivatives exist as absolutely continuous functions on compact subintervals of (0,00),
and for which the ftrst r +s pointwise derivatives exist as L 2 -functions. On the other hand, the
solution q, we look for is al10wed to be a distribution and need not be a function. -
1.2. Singular linear systems. It turns out that the operators L w and T w also appear as
input/output maps of certain singular linear input/output systems. For the full line operator L w this connection can be explained by using the Fourier transformation :J in (:1 m) '. Applying:J
to both sides of Lw(q,)=f yields W(A):Jq,=:Jf. Hence, defining pE(:1m)' via
:Jp = (A.G - A) -\ B:J q" the well-known properties of the Fourier transformation in (:1 m)' (see,
e.g., [S]) imply that
i~GP=AP+Bq" f =Cp+q,. dt
The corresponding result for the half-line equation (H) reads as follows.
(l.8)
THEOREM 1.1. Let sand s + r be nonnegatil'e integers. Let the maximal half-line
integro-differential operator T w (H ~ (IR + ) ~ H ~r [IR + I ) have a rational symbol given by
(1.9)
Assume thut "A.G -A is IR-regular, and let [f]E H~r[IR+I. Then the equation Tw(q,)=[f] and
the singular linear system
1 i.!!...Gp =Ap+Bu, dt
[y] =C[p]+[u], (I]
are equivalent in the following sense. If4>e D(Tw ) is a solution o/the equation T w (4))=[f],
then the system (I] with input u=4> has output [Y]=[f]. Conversely, let q,EH~(R+) and
assume that the system (I] with input u = 4> has output [y] = [f ]. Then 4> E D (T w) and
Tw(q,)=[f]·
The statements in the above theorem can be made' more precise as follows. Assume that
T w (q,) = [f] for an element q, E D (T w). Then, the ftrst equation in (I] with u = q, has a unique
solution p in the space (:1 n )' , and this solution has the property that Cq + (p) + q + (ell) = [f] in
the sense of (:1 m)' llR + I. Conversely, if q, E H~ (IR +) is a distribution for which there exists a
p E (:1 n ) , satisfying the equations in (I] with u = q, and [y] = [f], then 4> e D (T w) and
Tw(cp)=[f].
The proof of the above theorem is in Section 6.
Now, let us see how Theorem 1.1 can be used to solve equation (H) or, equivalently, to
146 Kuijper
find the solutions of T w (<I» = [fl. The first step is to apply the quotient mapping q + to the first
equation in (~]. This yields
i.!£.G[P]=A[P]+B[<I>]' [f]=C[p]+[<I>]. dt
By interchanging the roles of the input [<I>] and the output [f ] in ( 1.1 0), one obtains
i.!£.G[P]=AX[P]+B[f]' [<I>]=-C[p]+[f]. dt
(1.10)
(1.11)
Here A x =A - Be. Next, one solves the first equation in 0.11) for [p]. At this stage the spec
tral properties of 'AG -A x enter into the analysis. Inserting [p] into the second equation of
(l.ll), one finds [<1>]. The final step is to determine representatives pE(ym)' of [p] and
<I> E H ~ (IR + ) of [<I>] satisfying the equation i!!G P - A P = B <1>. This final step turns out to be pos-dt
sible only under additional conditions involving the spectral properties of both the pencil 'AG - A
and the pencil 'AG - A x. These conditions correspond to conditions for the solvability of equa
tion (H).
1.3. Description of results. To state some of our main theorems we need certain spec
tral projections related to the matrix pencils 'AG - A and 'AG - A x appearing in (1.10) and (1.11).
Assume both pencils to be regular. Let V be any subset of cr, and let reV) be a Cauchy contour
around all zeros of d (A) := det (AG - A) in V, excluding all other zeros of d (.). Put
Pv = _1_. S G('AG -A)-I dA. 2m nV)
The matrices P Ii are projections (see [St], [GK I D, and can be viewed as a generalization of the
usual Riesz projections. We shall need the projections P _, P + and Po which appear by taking
V equal to the open upper half plane, the open lower half plane and the real axis, respectively.
Also, we need P 00 =1 - P, with P =P Q: =P _ +P + +P o. Finally, let E and Q be defined by
E = _1-. S(1- A -I )('AG -A)-I d'A, 2m r
Q= _1-. S(A-A -I )G(AG -A )-1 d'A. 2m r
Here r is a Cauchy contour around all zeros of d (-) in the complex plane. The septet
will be referred to as the data for the decomposition of the pencil 'AG - A. We also need
( 1.12)
Kuijper 147
(1.13)
the data for the decomposition of the pencil 'AG - A x , which one obtains from the above fonnu
las by replacing A by A x , where A x = A - Be. The matrices in (1.12) and (1.13) will be used
without further reference.
We now can state the main theorems for the maximal operator related to the half-line
equation (H). Recall that the m-dimensional Dirac-distribution vO E (:1 m)', where v E cr m , is
defined by (v. o,a) = v T a(O). Furthermore, if g is a cr m -valued function, then g + is the function
g + (t) = g (t) for t ~ 0 and g + (t) = 0 for t < 0; often g + is considered as a distribution.
THEOREM 1.2. Let s S;O and s +1' ~O be integers. Let the maximal half-line integro
differential operator T If (H ~ (IR + ) ~ H ~r llR + J ) have a rational symbol W given in the real
i:ed form (1.9), with associated decomposition data (1.12)-(1.13). Assume that the pencil
'AG - A is IR -regular. Then the operator T w is invertible if and Oll/Y if
(i) 'AG -A x is lR-regular;
(ii) Ce (Qx)J P':;,B =Ofor j > 1';
(iii) rank~=-ms;
(iv) (Cn =Im P~ 83Ker P _ 83lm~.
Here ~ is the mutrix (B QB ... Q-s-I B). Assume that the above conditions are fulfilled.
Choose a left inver.lc' ~(-I) of the matrix~. Then the inverse of Tw is the operator from
H ~+r fIR + J into H ~l (IR + ) defined by
r+s (Tw )-1 (q+(f» =f+ + r,CEx(iQx)} P~B(f(j»+ +
}=O
+ ± Ce(iQX)}p':;,B(~)j-r-s«(f(r+s»+)+ }=r+HI dt
+<fe<t-s)f(s)ds-CE x e-it0. X x)+ + -£IVj(i~)jO+ o }=O ~
+ -£1 r+si!,-Ice (iQx) v+1 p~Bf(V-j) (0)( ~)j 0, j=o v=j dt
where f E H ~r (IR), the rectors x and I'} , j = 1,2, ... , -s-1 , are given by
x=-n 1 Rr.s(Bj), (vI v; ... V~s-I )T =i~(-l)n2Rr,s(Bj),
and e is the function
t < 0,
t > O.
(1.14)
(1.15)
148 Kuijper
In the above formulas n 1 is the projection of t£ n onto 1m P ~ along its natural complement in (iv). nz is the projection oft£n onto 1m ;/3 along its natural complement in (iv). and the map
ping Rr.s :H;+r(lR)-7t£n is given by
R r.s(g) = i n-s (n x )-/+:flp~ (inx )j+l g (j) (0) +i j P~eiSnx g (s)ds. j=O 0
THEOREM 1.3. Let s ~ 0 and r + s ~ 0 be integers. Let the maximal half-line integrodifferential operator T w ( H ~ (IR + ) -711 ~~r [IR +]) have a rational symbol W given in the real
i::ed form (/.9). with associated decomposition data (/,/2)-(/./3). Assume that the pencil
'A.G - A is IR -regular. Then the operator T w is Fredholm if and only !f
(i) A.G -A x is lR-regular;
(ii) ce (nx)j P~B =0 for.i > r;
Assume that the above conditions are fulfilled. Then the kernel and image ofTw are given by
Ker Tw = {(Cee- itnX x)+ - -~IVj(i~)jo I j=O dt
-s-l X E 1m P ~, x + i ~ nl Bv 1 E Ker P _ },
j=O
dim Ker T w = dim (1m P ~ (') (Ker P _ + 1m ;/3» + dim (Ker P _ (') 1m ;/3) + dim Ker :JJ.
1m T w = {q + (f) I .f E H ~+r (IR), R r.S (Bf) E 1m P ~ + Ker P _ + 1m ;/3 },
codim 1m T w = codim (1m P ~ + Ker P _ + 1m ;/3),
ind Tw =dim 1m p~ -dim 1m P _ -ms.
(1.16)
(1.17)
(1.18)
(1.19)
(1.20)
Here the mapping R r,s : H~+r(lR) -7 <r n and the matrix ;/3 are as in Theorem 1.2. Finally, choose a generali::ed inverse (in the weak sense);/3+ of;/3. and let At and A! be generalized inverses (ill the weak sense) of the operators
A 1 := (I - P ~ ) OCer P _ +Im ;/3: Ker P - + 1m ;/3 -7 Ker P ~ ,
A 2 := P - ~m ;/3 : 1m ;/3 -7 1m P - ,
respectively. Then the operator Ttv : H ~r [IR +]-7 H ~ (IR + ) given by
Kuijper
r+s , , Tii,,'(q+(j» =f+ + lo:ce(iQ X )l P:"B(jU»+ +
j=fJ
+ i Ce(iQX)jp:"B(!!.--)/-r-s«(j(r+s»+)+ j=r+s+\ dt
00 -l'-l d + (Je (t-s)f(s)ds _CEXe-itO: x)+ + I. v j(i-)j ~+
o j=fJ ~
+ -~\ r+si:-1CEX(iQX)V+l P:"Bf(V-j)(O)(!!.--)j~. j=O V=j dt
149
(1.21)
where f E H~r(IR). the function e is as in Theorem 1.2 and the vectors x and Vj.
j = 1.2 •...• -s-1. are given by
x=P~ A r (l-P~ )Rr.s(Bf)-P~Rr,s(Bf). (vr vi ... V~s-\ )T =;jJ+ A'1P -Ar(l-p~ )R,.,s(Bf).
is a generalized inverse of Twin the following sense:
(1.22)
For s =0 the conditions (iii) and (iv) in Theorem 1.2 reduce to the matching condition
([ /I = 1m P ~ $ Ker P _ .
Also. the formulas (1.14)-(1.21) simplify considerably in this case. because all summations with
upper bound -s -1 vanish and the matrix jJ has size 0 x O. When s < O. the right hand side of for
mula (1.14) does not depend on the specific choice of the left inverse .%' (-I) of jJ. Indeed. for an
injective operator X all left inverses coincide on 1m X. Thus. formula (1.14) is fully determined
by the realization (1.9) of Wand the right hand side [f 1 of equation (H). Furthermore. the con
ditions (i) and (ii) may be replaced by two equivalent conditions (i') and (ii'):
(i') det W(A) *0 for A E IR;
(ii') W(A)-IA-r is bounded.
Note that condition (ii') is automatically fulfilled for r sufficiently large. It seems to be an open
question whether conditions (i') and (ii') are also necessary and sufficient for the (possibly
unbounded) integro-differential operator T w to be Fredholm if the kernel function k is a general
m Xm matrix-valued function with components in L 1 (IR). Finally, we point out that the
Fredholm indices Ker T w, codim 1m T w and ind T w depend only on the parameter s.
The results obtained in this paper have various predecessors. By taking q (A.) == 1m. the
equations studied here reduce to Wiener-Hopf integral equations of the second kind with a
rational symbol. Equations of the latter type have been explicitly solved in [BGK1] by using the
150 Kuijper
realization (1.1) (see also [GGK], Chapter XIII and Section XVIII.S). In fact, the results in this
paper may be viewed as generalizations of the earlier results in [BGK I]. The class of equations
(H) also covers systems of ordinary differential equations with constant coefficients,
corresponding to both monic and nonmonic matrix polynomials, for which explicit formulas for
solutions in terms of standard triples have been obtained in [GKLR] and [GLR]. In [GKLR] and
[GLR] the right hand side is taken sufficiently smooth and the solutions are functions not neces
sarily restricted to a prescribed function space. First kind Wiener-Hopf equations with rational
matrix symbols, i.e., the case when q(A)=O, were explicitly solved earlier in [Ro] by using the
realization (1.1) with D =0. A standard assumption in [Ro] is that W (A)A r is proper, which
allows one to remain within the framework of bounded operators. In the present paper, for the
first time all the equations mentioned above are treated in terms of the state space method from
one point of view.
Realizations of type (1.9) have been used earlier in [GK I] and [GK2] to solve Toeplitz
equations and singular integral equations with rational matrix symbols. In both cases the sym
bol is regular on the full Cauchy contour associated with the equations. In our case an essential
difficulty is that on the extended real line the symbol has a singularity at infinity.
An inversion theory of scalar integro-differential equations of convolution type, not
necessarily with a rational symbol, is developed in [Ta], [VGl] and [VG2] (see also [VG3]
[VG5]). In these papers the main emphasis is on generalizations of the factorization method
developed in the earlier work of [Kr] for second kind Wiener-Hopf integral equations (see also
[GKr] for systems of second kind equations). More precisely, under the assumption that the
symbol of the equation admits a certain factorization, Talenti [Ta] shows that the natural opera
tor corresponding to the equation acting between well-chosen Sobolev spaces is Fredholm.
Furthermore, Talenti describes the Fredholm characteristics of the operator in terms of the fac
torization. Under the assumption of a slightly more general factorization of the symbol, it is
shown in rVG I ]-[VG2] (see also [VG3]-[VG5]) that one can associate to the equation an in a
certain sense invertible operator acting between well-chosen scales of Sobolev spaces. The fac
torizations in [Ta] and [VG 1]-[VG5] always exist for rational functions without poles and zeros
on the real line. Another approach for first kind equations with positive definite symbols, which
also works for the matrix case, appears in [Rl] (see also [R2]). All the above mentioned refer
ences present a method of obtaining solutions, e.g. via factorization, but, in general, they do not
give explicit formulas.
1.4. Descriptions of contents. This paper contains a large part of the author's thesis
[Ku]. Not counting the present introduction, the paper consists of seven sections. Section 2 con
tains preliminaries on the spectral properties of matrix pencils. Also, the decomposition data for
a pencil are introduced in Section 2. Sections 3 and 4 deal with singular differential equations,
i.e., differential equations of the types appearing in the first parts of (1.8) and 0.10). The
Kuijper 151
general solutions are described in tenns of the data for the decomposition of the pencil associ
ated with the equation. Section 5 concerns preliminaries on realizations. A number of auxiliary
results that will play an important role later are presented here. Section 6 contains the proof of
Theorem 1.1. The proofs of Theorems 1.2 and 1.3 is in Section 7. Finally, Section 8 contains an
example illustrating the earlier results.
2. PRELIMINARIES ON MATRIX PENCILS
Given two square matrices G and A of size n x n say, the expression AG -A, where A is
a complex parameter, is called a (linear matrix) pencil. This section concerns the spectral pro
perties of AD -A. We begin with some definitions. A pencil is said to be regular if its deter
minant is not identically equal to zero. For a nonempty subset ~ of the Riemann sphere
([ 00 = ([ u {co}, a pencil AG - A is ~-r('gular jf AG - A (or just G if A = co) is invertible for each A
in ~. Clearly, ~- regularity of a pencil implies regularity. Whenever necessary an n x n matrix
is identified with the operator on ([ n of which the matrix relative to the standard basis in ([ /I is
the given one.
Three matrices that are closely related to linear pencils will occur frequently in this
paper. Let AG -A be a regular matrix pencil. We define the left separating projection P, the
right equivalence matrix E and the associate matrix Q of the pencil AG - A (corresponding to 00)
by
E = ~ J (I_~-l )(~G _A)-l d~, ~1t1 r-
Here r is a Cauchy contour such that all the zeros of det (AG - A) are in the interior domain ~+
of r. Also, we assume that 0 is in the interior domain of r. Apart from this, the particular
choice of the contour r is irrelevant.
The above terminology has been taken from [GKI]. In [GKI], Theorem 2.1, it is shown
that the matrix E is invertible and that the matrix P is idempotent. Furthennore, in [GKI] the
following identities are proved (cf. [GK I], Lemma 2.2 and Theorem 2.1):
AE = QP + (I-P), GE = P + Q(I-P). (2.1)
Note that these identities imply that (AG - A)E leaves the subspaces Ker P and
1m P invariant. In fact, the matrices P, E and Q induce a decomposition of the pencil AG - A of
152 Kuijper
the following form:
(2.2)
[0 1 0 1 0= 0 O 2 : Ker P EiHm P ~Ker P EBIm P, (2.3)
where 1..0 I -II is a (L1, u f)-regular pencil and 1..- 0 2 is a (<r ~ \ L1+ )-regular pencil. More
over, P and E are the unique operators such that the partitions in (2.2) and (2.3) hold true. These
properties justify the terminology for the matrices P, E and O. Note that, in contrast to [GKl],
we define the matrices P, E and 0 only for a Cauchy contour r around all the zeros of det
(AG -A).
The next two lemmas provide additional information about the matrices P, E and O.
Recall that the index of nilpotency of a nilpotent matrix N is the smallest integer 0 such that
N° =0. For a rational m x m matrix function W with determinant not identically equal to zero,
the zeros of W are the points in <r ~ where the rational matrix function W- I (A) := W (A) -I has
its poles. For a regular pencil AG - A this implies that the zeros inside the complex plane are
precisely the points where det (AG - A) = O. The order of the pole at infinity of a rational matrix
function W is the smallest number v ~ 0 such that A -v W (A) is analytic at infinity. Finally, the
spectrum of an operator (a matrix) X will be denoted by cr(X).
LEMMA 2.1. Let AG - A be a regular n x n matrix pencil. Let P and 0 be the left
separating projection and the associate matrix of the pencil AG - A. respectively. Then the
matrix 0(1 - P) is nilpotent and the index of nilpotency of 0(1 - P) is equal to the order of the
pole at infinity of the function (AG _A)-I plus one.
PROOF. The proof is based on the decomposition of the pencil AG -A in (2.2) and (2.3).
Recall that the pencil An I -II is invertible for all AE ruL1+. Since all the zeros of A.G -A in
<r are in L1+, formula (2.2) implies that the pencil An I -I) is also regular on <r \ L1+. Thus,
1..0 I -I) is invertible for all A in <r, and hence 0 I is a nilpotent matrix. Moreover, taking
inverses on both sides of the equation (2.2), one immediately sees that the index of nilpotency of
the matrix 0) equals the order of the pole at infinity of the function (AG - A) -1 plus one.
Using the decomposition in (2.3), the same remarks hold for the matrix O(l-P), and the lemma
is proved. [J
LEMMA 2.2. Let AG -A be a regular n x n matrix pencil. For an arbitrary subset Vof
<r. define the matrix P v by
P v = _1_. f G(~G _A)-J d~, 2m reV)
Kuijper 153
where reV) is a Cauchy contour around all zeros of 'AG -A in V such that all other zeros of
'AG - A are outside reV). Let P and Q be the leji separating projection and the associate matrix
of the pencil 'AG - A, respectively. Then, P v is an idempotent matrix satisfying
PP V =P V P =P v · Furthermore, QP I' =P v Q , and the spectrum of Q 11m P v coincides with
the set of ::.eros of the pencil 'AG - A inside V.
PROOF. The well known contour integration arguments together with the generalized
resolvent identity for pencils,
show that P ~7 = P v and PP 1'= P \ P = PI" Next, use equation (2.2) to compute P v with
respect to the decomposition ([ n = Ker P EEl 1m P. This yields
2][i
where P (Q 2; V) is the spectral projection of the matrix Q 2 corresponding to the part of a(Q 2)
inside V. To justify the last step, note that the (1,1) block element in the integrand is analytic,
which is a consequence of the nilpotency of the matrix Q I. Now, the above calculations,
together with (2.3) imply that QP I, =PI,Q. Moreover, it follows that a(Qbm PI') consists of
the eigenvalues of the matrix Q 2 inside V. Formula (2.2) now proves the lemma. 0
We shall refer to the matrix P If defined above as the leji spectral projection correspond
ing to V of the pencil 'AG - A. Note that it is not a priori assumed that 0 E V. In case V = ([, then
Pv=P, the left separating projection of 'AG-A. If V=([+ «[_), i.e. V is the open upper
(lower) half plane, we use the notation P + (P _) for the corresponding left spectral projection.
If V = IR we write PI' = P (). Finally, we introduce the leji spectral projection corresponding to
00, P = =1 - P. We shall henceforth rcfcr to the ordered sct of seven operators
(2.4)
a~ the data for the decomposition of the pencil 'AG -A.
When the matrix G is the n x 11 identity matrix, then the zeros of the pencil 'AG - A are
precisely the eigenvalues of thc matrix A, and we have P =1 n' E = I nand Q =A. In this case
Lemma 2.1 is a lemma about empty matrices and the second lemma is nothing else but a corol
lary of the classical Riesz theory. The extension of this theory to regular (matrix and operator)
pencils is due to in [St]. See also [GGK], Section IV.I.
154 Kuijper
3. SINGULAR DIFFERENTIAL EQUATIONS ON THE FULL·LINE
Let W - A be a given n x n regular matrix pencil. In this section we consider the equa-
tion
i!{.Gp-Ap=«I> dt
(3.1)
in the context of the space of tempered distributions (.:/ n) , , the dual of the space .:/ n. In partic
ular, given an element «I> in (.:/ n)' , we look for all solutions of equation (3.1) in the same space.
An element p E (.:/ n)' is caBed a solution of equation (3.1) if
-i(p,G To.') = (p,A To.) + (<<1>,0.),
where a.' indicates the classical derivative of the function a..
THEOREM 3.1. Let W -A be an n Xn regular matrix pencil, and let
(E, 0, P; p _, p +, Po, P 00) be the data for the decomposition of W - A. Let «I> be an arbitrary element of (.:/ n )'. Then the general solution of equation (3.1) in the space (.:/ n )' is given by
where x is an arbitrary vector in 1m Po, the functions I and p are given by
{ iEe -itO. P + , t < 0,
I(t) = -'E -itnp 0 I e _, t>,
N . . pCA) = - ~EOJ P oo/..) ,
j=O
(3.2)
L (<<I» = Ee -int P 0 F E (.:/ n )' with F a primitive in (.:/ n )' of -ie iD.t P 0 «1>, and N is the order of the pole at infinity ofCAG _A)-I.
PROOF. The proof is in five steps.
Step (;). In the first step we show that the right hand side of (3.2) is a well-defined tem
pered distribution and a solution of (3.1). From Lemma 2.2 we know that I (I) is an exponen
tially decreasing function for t ~±oo. Hence by standard results from the theory of distribu
tions, the convolution in (3.2) is well defined in (.:/ n )'. Furthermore, Lemma 2.2 shows that
e in, p 0 is a '(; 00 matrix function bounded in norm by a polynomial. Thus, f = ie iD.t P 0 «I> is a
tempered distribution, and we may choose a primitive F of f in (:I n )' (cf. [S], Theorem n,I and
Theorem VII,VI). Next, choose a vector x from the subspace 1m Po, and define the distribution
p by the right hand side of (3.2). Then p is weIl-defined. To prove that p satisfies (3.1), fllst
recall the foIlowing identities from Section 2:
Kuijper 155
P 00 =I-P, P y =PP y =PyP, VE {+,O,-j.
Put Pp =£-1 p(i~)'I>, and PI =£-1 (l * '1». Then, as ON+I (/-P)=O by Lemma 2.1, we have dt
d N. d' N+I . d' i-Op p = - ~ (iO)J+I P 00 (-)J+I 'I> = - ~ (iO)J P 00 (_)J 'I> =
dt j:(J dt j=1 dt
Furthermore, standard properties of the convolution product and the differentiation in the space
(:In)' show that
Using formula (2.1), a combination of the above two formulas yields
The next step is to consider the third term in (3.2). Compute that
i~GL(CP)-AL(CP)=i~GEe-iOt PoF -AEe-iOt PoF = dt dt
=iGE(-iQ)e-iOt PoF -iGEe-iOt ie iOt Po'l>-AEe-iOt PoF =P 0'1>.
Finally, a straightfoward calculation shows that the fourth term in (3.2) does not contribute to
the left hand side in (3.1) at all, as
Here we used the results of Section 2, again. A combination of the above arguments now prove
that p is a solution of (3.1), indeed.
Step (ii). Fi \ a primitive F of -ie itO P 0 cP and a vector Xo in 1m Po, and put Po the distri
bution as in (3.2) with X=Xo. Suppose that Po E (:In)' is another solution to equation (3.1).
Then Po - Po is a solution of the homogeneous equation
156
j!!.-CP =Ap. dt
Kuijper
(3.3)
Thus, to finish the proof of Theorem 3.1, it suffices to show that any solution of (3.3) in the
space (:1' n )' is of the form
p(t) = Ee -int Px, -00 < t < 00, (3.4)
where x is a vector in 1m Po. As a preparation to solving equation (3.:n, we first study two spe
cial cases of equation (3.3) in the space (JJ")' of n-dimensional distributions (see [S], [H]), a
space which contains the space (:1' n ) , •
Step (iii). In this step we consider equations of type
. d X 1- \jI=\jI, dt
(3.5)
where X is a nilpotent matrix of size k x k, say, in the space (JJ k )'. We call \jI E (JJ k)' a solu
tion of (3.5) if
where JJ k is the space of ([ k -valued infinitely differentiable functions with compact support
(see [S]), and a' denotes the pointwise classical derivative of the function a. Clearly, \jI=O is a
solution to equation (3.5). We claim that it is, in fact, the only solution of (3.5). Indeed, let N
be the order of nilpotency of the matrix X. A premultiplication of the equality in (3.5) with - - -
X N- 1 yields X N- 1 \jI=O. Thus X N- 1 (~\jI)=O. But then, a premultiplication of identity (3.5) dl - -
with X N- 2 shows that XN - 2 \j1=0. Continuing like this, we eventually have that \jI=Xo\jl=O,
which proves the claim.
Step (iv). Now, consider an equation of the type
. d y 1-\jI = \jI, dt
(3.6)
where Y is a k x k matrix, in the setting of (JJ k)'. We call \jI E (JJ k)' a solution of (3.6) if
The general solution to the above equation is well known (see, e.g., [S], Theorem V,IX) and
consists of the solutions of equation (3.6) interpreted in the usual sense,
-00 < t < 00.
Step (v). Let p be a solution in the space (:1' n)' of equation (3.3). Put PI =PE-1 P and
Kuijper 157
P2 =(/-P)E-1 p. Then p=Ep I +EP2, and a premultiplication of the identity in (3.3) with P,
respectively 1 - P shows that
(3.7)
Here we used the formulas in (3.1), the fact that ill' =po. (cf. Lemma 2.2), and the trivial
observation that Pp I = PI, PP2 =0. Now, recall that (:I n)' C (.2' n)' . So, P I and P 2 are distri
butions in (.2) n)' which satisfy the equations in (3.7) in (.2) n)' -sense. But then, as 0.(/- P) is a
nilpotent matrix by Lemma 2.1, Step (iii) of the proof shows that P2 =0. Furthermore, it fol
lows from Step (iv) that the general solution to the first equation in (3.7) is given by
PI (t) = Pp I (t) = e-iQPt Px = e-iO.t Px, -00 < t < 00,
with x an arbitrary vector in cr n. Thus, P = E P I + E P 2 = E P I is of the form
p(t) = Ee-itO Px, (3.8)
for some vector x in cr n. Put P e = P + + P _. Then P = Po + P e' and we may write P = Po + P e •
Here P v is the distribution in (.2) n )' given by the right hand side of (3.8) with P v instead of P,
v E {O,e}. Now, recall from Lemma 2.2 that
(J(o.~mP )rdR=0. e
As a consequence, Po is a continuous function bounded in norm by a polynomial. In other
words, using a well known characterization of tempered distributions (see [S], Theorem
VII, VI), Po is an element of (Y n )'. But then, PeE (Y n )' as well, and by the same characteri
zation of (Y n )' we must have that P e X = O. We conclude that P = Po is of the form indicated in
the theorem. This finishes step (v). [J
In case the distribution cp is a tempered function, i.e., a function bounded in norm by a
polynomial, one possible choice for L(cp) is the function:
t
L(cp) =-iEJe-i(t-S)O Po cp(s)ds. o
Hence, in this case the general solution to equation (3.1) is given by
where the functions p (A.) and k (t) and the vector x are as in Theorem 3.1.
We conclude this section with the specification of Theorem 3.1 for IR-regular pencils.
COROLLARY 3.2. Let A.G -A be an n xn regular matrix pencil. Then equation (3.1)
158 Kuijper
has a unique solution for every right hand side <jl E (::I n )' if and only if 'AG - A is IR -regular,
and in this case the solwion is given by
p=l * <jl+p(i!!")<jl, dt
where the functions p and I are as in Theorem 3.1.
(3.9)
PROOF. From the proof of Theorem 3.1 it follows that the general solution of the homo
geneous equation
in (::I n )' is given by
p(t) = Ee -i/O. x, -00 < t < 00,
where x runs over all vectors in 1m Po, the image of the left spectral projection of 'AG - A
corresponding to the real axis. Thus, equation (3.2) has a unique solution in (Y n)' for every
right hand side <jl E (::I n)' if and only if Po = 0, or in other words if 'AG - A is IR-regular. In this
case, the formula (3.9) for the solution immediately follows from (3.2). D
The solution (3.9) can also be obtained via the frequency domain by applying the Fourier
transform to (3.1).
4. SINGULAR DIFFERENTIAL EQUATIONS ON THE HALF-LINE
Let A.G - A be a regular n x n matrix pencil. In this section we consider the equation
i!!..G[P] =A [p] + [<jl] dt
(4.1 )
in the context of the space (Y n)' [IR + I, the quotient space (::I n)' / (Y n)' (IR - ) of tempered dis
tributions in (0,00). The corresponding quotient mapping we denote by q +, and equivalence
classes in (::In)'[IR+J are denoted by square brackets, like in [pl. Given an element [<jl] in
(Y n)' [IR+ I, an element [p] E (Y n)' [IR+ J is called a solution of equation (4.1) if
-i([p],C T a')=([p],A T a) + ([<jl],a), aE Y n, supp (a) c (0,00),
where a' indicates the classical derivative of a E ::I n •
THEOREM 4.1. Let 'AG - A be an n x n regular matrix pencil, and let
(E, n, P; P _, P +, Po, Po<» be the data for the decomposition of A.G - A. Let [<jl] be an arbi
trary element of (Y n)' [IR +]. Then the general solution of equation (4.1) in the space
(Y n)' [IR +1 is given by
Kuijper 159
(4.2)
where x is an arbitrary vector in 1m P _ + 1m Po. the element \jI E (:I n )' is any distribution
such that q + (\jI) = [<1>]. the functions I and p are given by
{ iEe -iiO P + , t < 0,
l(t) = -'E -ilOp ° I e _, t>,
N , ' pCA) = - 'LEQ} p ",,/,..J,
j=O
L(\jI) = Ee -iQl P 0 FE (:I n)' with F a primitive in (:! n)' of -ie iOl P 0 \jI. and N is the order of
the pole at infinity of CAG - A) -I ,
PROOF. From Theorem 3.1 we know that
PI =1 * \jI+p(i~)\jI+L(\jI) dt
is a solution of the equation i...'!.G P = A P + \jI in the space (:I n )'. Hence, q + (p 1 ) is a solution of dl
equation (4.1). Also, lEe a!lx] is an element of(:! 11)' [JR.+] and satisfies
Thus, every [p] of the form (4.2) is an element of (:! 11 )' [JR. + ] and a solution of (4.1).
To prove the converse, it suffices to show that every solution of the homogeneous equa-
tion
(4.3)
is of the form lEe-ali x I. with x as in the theorem. Let [p 1 be a solution of (4.3). Put [p] =E[p 1 ] + E [p 2], with [p 1 1 = PE-1 [p] and [P2] = P ""E- 1 [p]. By straightforward verifica
tion one shows that [p 1 ] and [p 21 satisfy the following equations:
(4.4)
The key remark to finish the proof now is to repeat the arguments in steps (iii) and (iv) in the
proof of Theorem 3.1. Firstly, note that [p I] and [p 2], being tempered distributions in (0,00),
may be viewed as clements of :J) /I (IR +)' , the space of distributions in (0,00) (see [H], where the
space :J) n (IR T)' is denolcd by UD /I ) , (lR +». Moreover. the equations in (4.4) are satisfied in
160 Kuijper
j)n(JR+)' -sense. Since this space has exactly the same structure as (j)n)' itself, the arguments
in the proof of Theorem 3.1, Step (iii) and (iv), still hold. Therefore [p 2] = O. Furthermore, [p 1 ]
will be of the form
[Pl]=P[Pl]=[e-itOx],
where x is a vector in 1m P. However, [pJlE (.'I'n)'[JR+1, which implies that X=XE 1m
P _ + 1m Po. Thus [p] =E[p 1] is of the desired form. [J
To conclude this section, we specify the above result for the case when $ is in L ~ (JR + ),
the space of cr n -valued L2 -functions with support in [0,00). For such a function, we indicate by
[$] the corresponding element in (.'I' n)' [JR +].
COROLLARY 4.2. Let A.G - A be an regular n x n matrix pencil. Let $ be a given function in L ~ (JR +). Then the general solution of equation (4.1) in (.'I' n)' [JR + I is given by
[p] = [jZ (t-s)$(s)ds] +p(i~ )[$] + o dt
I
- [iEJe-i(I-S)O Po$(s)ds] + [Ee-iIO x], x E 1m P _ + 1m Po. o
Here /, p, P _, Po, E, and Q are as in Theorem 4.1.
5. PRELIMINARIES ON REALIZATIONS
This section deals with a special representation for rational matrix functions. A triple
0= (A.G - A,B,C) is called a realization of a given rational m x m matrix function W if
W(A.) =1 +C(A.G _A)-l B. (5.1)
Here A.G - A is a regular matrix pencil of order n , say, and Band C are matrices of sizes n x m and m x n, respectively. In [OK1] an explicit procedure to build a realization for any rational
m x m matrix function is described. This procedure is based on the Laurent series development
of a function W(A) at its poles, and yields a realization 0=(AG -A,B,C) with the extra pro
perty that the pencil AG - A has a zero at 1..0 E cr if and only if the function W has a pole at 1.. 0 ,
To be precise, if the function W has no poles in a certain subset V of cr, then one may choose
realizations 0=(A.G -A,B,C) with a V-regular pencil A.G -A.
In the sequel of this manuscript we shall need the following result for rational matrix
functions of the form (5.1). Recall that the Fourier transformation f of and L1-function f is
defined by 00
j(A)= J eiNf(t)dt, A.E JR.
Kuijper 161
LEMMA 5.1. Let W be a rational m x m matrix function with a realization
E>=(AG -A,B,C). Assume that the pencil AG -A IS JR.-regular, and let
(E, 0, P; P _ , P + ' Po, P",,) denote the data for the decomposition of AG - A. Then Po =0, and ~ ~
W has a unique decomposition W (A) = q (A) + k(A), where q is an m x m matrix polynomial and k
is the Fourier trall.lform of all L 11unction k. The functions q and k are are given by:
N. . q(A)=I- I,AJCEOJP""B,
)=0
k(t)=l iCEe-itQ P +B,
-iCEe-itQ P _B,
t < 0,
1>0.
Here N is the order o{the pole at infinity of(AG -A)-I.
(5.2)
(5.3)
PROOF. Since W is a rational matrix function, we can write W in a unique way as
W = q + V, with q a matrix polynomial and V a rational matrix function which is analytic and
zero at infinity. To find the above decomposition of W in terms of the chosen realization E>, write
W(A) =1 +C(AG -A)-I B =1 +CE«AG -A)E)-I B =
= 1 + CE«AG -A)E)-l PB + CE«A.G -A)E)-I (I-P)B.
Using the identities in (2.1) we have
W (A) = 1 + CE(AO(l- P) -I) -I (I-P)B + CE(A-Q.) -I PB. (5.4)
By Lemma 2.1, the matrix O(J -P) is nilpotent of order N+1. Hence, the polynomial part ofW
is given by the first two terms in the right hand side of (5.4), whereas the third term represents
the part of W which is analytic and zero at infinity. Since 1- P = P"", we conclude that q is
given by formula (5.2). Furthermore, Lemma 2.2 and the assumption on the pencil A.G-A
imply that the function k defined by (5.3) is an L 1 -function, and that P + +P _ =P. Applying
Fourier transformation, it follows that V = k, which proves the lemma. 0 ~
Henceforth, we shall sometimes refer to the functions q and k in (5.2) and (5.3) as the
polynomial part and the strictly proper part of W, respectively. Also, we shall say that a rational
matrix function V is (strictly) proper if it is analytic at infinity (and V (00) = 0).
Another important feature of the representation (5.\) is the connection between (5.1) and
the invertibility of the function W. Let W <A) be given as in (5.\), and put A x =A - Be. Then,
by Theorem 4.2 in [GK \], the following identity holds:
det W (A) = det (AG - A x) . det(AG-A)
(5.5)
162 Kuijper
In particular, the function det W (A.) is not identically equal to zero if and only if the pencil
A.G - A x is regular. Furthermore, if this condition is satisfied, then the inverse function W (1..)-1
admits a realization ex =(A.G -A X,B, -C):
(5.6)
From the above remarks it follows that the conditions (i) and (ii) in Theorem 1.3 are
equivalent to the conditions (i') and (ii'), respectively, stated just below Theorem 1.3. Indeed,
the equiValence of (i) and (i') is an immediate consequence of identity (5.5). To see that the
condition (ii) implies (ii') and vice versa, apply Lemma 5.1 to the realization (5.6).
In the sequel we shall often need information about the interplay between the spectral
data of the pencils W - A and A.G - A x appearing in the realizations (5.1) and (5.6). The next
three lemmas present a number of auxiliary results in this direction. In the statements and in the
proofs of these lemmas,
denote the data for the decomposition of A.G - A and A.G - A x , respectively.
LEMMA 5.2. The space Ker P _ II 1m p~ II (lKer CEX(O,X)j =(0). j?o
PROOF. Take a vector x E Ker P _ II 1m p~ II (lKer CEx(O,X)j. Then x E 1m p~, j~O
and hence the identities in (2.1)-(2.3) imply that
C(A.G -A X)-I x=CEX«W -A x)Ex)-1 p~x =CEX(A.-O,X)-I X.
Using the Neumann series expansion for the right hand side in the above equality, it follows that
C(A.G _Ax)-I x= 1:CEX(O,X)V xl.. -v-I =0. (5.7) v=o
Now, use the identity BC =A -A x =(W -A x)_(W -A) to verify that
(5.8)
By equation (5.7), the above formula applied to the vector x yields
(A.G -A x )-1 x =(A.G -A)-I x.
PremuItiply this identity by G, and integrate the result over a contour r _ which encloses all
zeros of the pencils A.G - A and A.G - A x in the open lower half plane but excludes all other
zeros. This yields x =p~x =P _x =0. 0
The above lemma may be viewed as a generalization of [BGK3], Lemma m.1.1.
Kuijper 163
Indeed, take G =1 and assume that both A and A x have no spectrum on the real line. Then
Po = P 00 = 0, and hence Ker P _ = 1m P +. Thus, Lemma 5.2 reduces to Lemma III.I.1 in
[BGK3].
LEMMA 5.3. For j ~ 1, the space
PROOF. The proof relies on the formulas
and
The above identities are based on the relations (2.1 )-(2.3), applied to the pencils A.G - A and
A.G -A x. By combining the above idenities, one obtains
which implies the desired inclusion in the lemma for j = 1. To pass to the case j > 1, we apply
induction to j. Assume that the claim holds for j = I. Then. for j = 1 + I we have
Thus. using the induction hypotheses
which proves the inclusion for j = / + I. 0
LEMMA 5.4. Let r be a nonnegative integer, and assume that the function W(A.)-I A.-r
is proper. Then
(5.9)
PROOF. The assumption on W(A)-I implies that W(A)-I is of th.: form
where W j. j = 1,2 ...... are certain m x m matrices. In particular. the polynomial part of W (A.)-I
has degree at most r. Now, recall formula (5.8) and consider the identity
(5.10)
164 Kuijper
(5.10)
As PG()"G -A)-I =P(A-O)-I by the identities in Section 2. the right hand side in (5.10) has a
pole at infinity of order at most r. Hence. the same has to hold for the left hand side of (5.10).
So. the function
is analytic at infinity. Using the nilpotency of the matrix OXp~ (cf. Lemma 2.1). this proves
the lemma. D
The converse of Lemma 5.4 is not true. To see this. let W be the function
[1 A) [1 0) [-1 0) [0 1) [1 0) _I [0 0) W (A) = 0 1 = 0 1 + 0 0 (A 0 0 - 0 1) 0 1 .
The corresponding left separating projection P =0. So. the equation (5.9) is certainly satisfied
for any r. However. the function W (A) -I,) is not analytic at infinity for j = 0.1. thereby con
tradicting the assumptions in Lemma 5.4.
6. PROOF OF THEOREM 1.1
In this section we prove Theorem 1.1. For all notations we refer to Section 1.
PROOF OF THEOREM 1.1. Take an element ell E H ~ (IR + ). Since the pencil W - A is
JR.-regular. Corollary 3.2 states that the first equation in 0:] with u = ell has a unique solution in
(:In)' given by
p=p(i.E..)Bell+l * B<I>. dt
where the functions p and I are given by
N . . p(A.)=-~EOJ p 00 A.1 •
j=O
I (I) = { 'E -itOp I e +.
-iEe -itO P _ •
(6.1)
(6.2)
t <0.
1>0, (6.3)
Here N is the order of the pole at infinity of (W _A)-I. A comparison of the formulas (6.2)
and (6.3) with those for the functions q (A) and k (t) in Lemma 5.1 yields that
Cl(t)B =k(t). I +Cp(A)B =q(A).
So. we have. <l>E D(Tw) and Tw(<I»=[f] if and only if C([p])+[<I>]=[f]. which proves the
theorem. D
Kuijper 165
As a preparation to Section 7, we remark that the unique solution p E (::/ n)' to the first
equation in (1:] is a distribution in the Sobolev space H~ (IR) for 11 sufficiently small. Indeed,
this fact is an immediate consequence of the formula in (6.1).
The restriction to the numbers sand r in this section is not essential. With the appropri
ate modification of the definition of T w, Theorem 1.1 also holds when sand r are arbitrary real
numbers.
7. PROOFS OF THEOREMS 1.2 AND 1.3
In this section we prove Theorems 1.2 and 1.3. We shall freely use the notations intro
duced in Section 1. In particular, throughout this section -s and r + s are nonnegative integers,
the rational function W is given by the realization (1.9), with associated decomposition data
(1.12)-(1.13), and Tw (H~(IR+)-7H~+r(IR+]) is the maximal half-line integro-differential
operator with symbol W. We also assume that the pencil "AG -A is IR-regular, and AX =A -Be.
Furthermore, we shall freely use the equivalence of conditions (i) and (ii) with conditions (i')
and (ii') in Section I (see Section 5). We start with three lemmas and a proposition.
LEMMA 7.1. Assume that the conditions (i) and (ii) in Theorem J.3 are fulfilled. Then
the kernel of T w is given by
KerT w = {(Cee-iIQx x)+ - -£IVj(i.E..)j 0 I j=o dt
-5-1 x E 1m P ~, x + i L n) Bv j E Ker P _ },
j=O
dim Ker Tw = dim (1m P~ n (Ker P _ + 1m .:8» +dim (Ker P _ n 1m .:8) +dim Ker.:8.
(7.1)
(7.2)
PROOF. Let <I> hI' an element of Ker T w. Then, by Theorem 6.1, there exists a distribu
tion p in some Sobolev space H~ (IR) such that
i.E..Cp =A p +B<I>, [0] =C [p] + [<1>], dt
By applying the mapping q+ to the first equation in (7.3) we have
i~G[P]-A[p] =B[<I>], [O]=C [p] +[<1>]. dt
(7.3)
Next, subtract the second equality, premultiplied by B, from the first of the above identities.
Thus, one obtains that [p] E H~ [IR + I satisfies the equation
i~C[Pl-A x [pl = [0], dt
166 Kuijper
and hence, by Corollary 4.2 and assumption (i) of Theorem 1.3, we have
for some vector x E 1m p~. As for the distribution p itself, this implies the existence of a tem
pered distribution T _ with support in (-00,0] such that
p=(EXe-itilX x)+ +T_.
Furthermore, we have [I/>] =-C [pl = [-CE x e-itilX xl, and hence
1/>=_(CEXe-itilX x)+ +To
(7.4)
(7.5)
for some tempered distribution To with support in (-00,0]. In fact, as I/> and the first term in the
right hand side of (7.5) are elements of H 'J' (IR + ), the distribution To has support at the origin
and is an element of H'J' (iR-:- J. In other words, using a well-known result from the theory of dis
tributions (cf., e.g., [Tr], Theorem 24.6), To is of the form
T l: (.d)l: (.d)-s-Il: o =vov+vl 1- v+",+V-s-1 1- v, dt (It
(7.6)
where \' j' j = 0, 1, .. , -s-I, are certain vectors in cr m. Now, insert the formulas (7.4)-(7.6) into
the first equation in (7.3), and use that GExpx =px and AXExpx =Qxpx (cf. Section 2).
This yields
The above equality is equivalent to
d -s-I d· i-GL -AL =-ixo+B ~ l'j(i-)'O.
dt j=o dt (7.7)
As the pencil 'AG -A is lR-regular, Corollary 3.2 states that the distribution L is given by
I -.1-1 I· -s-i d· L=p(i.!:.-)(-ixO+B ~ Vj(i.!:.-)10)+I*(-ixO+B 1: Vj(i-)'O),
dt j=O dt j=O dt (7.8)
where the polynomial p and the integrable function I are as in (6.2) and (6.3). But T _ has its
support in (-00,0]. The same holds for the first term in the right hand side in (7.8). As for the
last term in (7.8), observe that
Kuijper 167
. I 1* (~}j &= IV) (t) +'i: IV-I-v) (0)( ~} v &.
dt v=o dt
Hence, it follows that the third term in (7.8) has its support on the negative halfline if and only if
or, equivalently, if
-itQp (. B OD o-s-IB )-0 e _ -IX+ Vo +uDvl + .. +u V-s-I = ,
-s-I X + i 1: gj Bv j e Ker P _.
j=!J (7.9)
Combining this observation with (7.5) and (7.6), the distribution cp is an element of the right
hand side in (7.1). Conversely, assume that cp is of the form (7.5) with x in the image of P ~ and
To given by (7.6) and (7.9). Define a distribution T _ by (7.8), and let p be as in (7.4). Then T_ has its support in (-00,0] and satisries (7.7), the equality C[p] + [cp] =0 holds and a computation
as above shows that the first equation in (7.3) is satisfied. But then, Theorem 6.1 shows that
cpe Ker Tw.
Next, let us prove (7.2). Introduce an 2n x(n-ms) matrix r by
and denote by v the column vector containing as elements the vectors Vo up to V -s-I • More pre
cisely, v = (Vb v r ... V:s-I ) T. In terms of the matrix r the kernel of Tw consists of those
Ii> E H:' (JR + ) of the form (7.5) for which the vectors x and v satisfy the condition
(7.10)
We claim that the correspondence between the vectors x and v satisfying (7.10) and elements
li>eKerTw is one-one. Indeed,takevectorsxEImP~ and Vj,j=O,l, .. ,-s-l,incr m satisfy
ing (7.1 0), and assume that the distribution cp in (7.5)-(7.6) is the zero distribution. Then it is
immediately clear that v j =0 for j =0, l, .. ,-s-l, and that CEXe-itQx x =0 for t > O. Hence v=O,
and thus x is contained in 1m P~ n Ker P _ because of (7.10). Now apply Lemma 5.2 to con
clude that x =0. This proves the claim.
By the results of the previous paragraph we have
dim Ker Tw = dim r -I [1m p~ xKer P _]=dim r -I [(1m p~ xKer P _}n 1m r]=
=dim Ker r+dim «1m p~ xKer P _}n 1m r).
Consider the second term in the above expression. If the vector r [ :) is contained in the space
168 Kuijper
1m p~ xKer P _, we have
X e 1m P~ II (Ker P _ + 1m jJ) =: 1J.
Conversely, if x e 1J, then there exist vectors v 0 up to v -s-\ such that r [ :) is an element of
1m P ~ x Ker P _, and the number of possible linearly independent vectors r [ :) equals the
dimension of 1m jJ II Ker P _. Thus, we derived that
dim (1m P~ xKer P _)11 1m r=dim 1m P~ II (Ker P _ +Im jJ)+dim (Ker P _ II 1m jJ).
As Ker r = (0) x Ker jJ, this establishes formula (7.2). [J
LEMMA 7.2. Assume that the conditions (i) and (ii) in Theorem 1.3 are fulfilled. Then
the image ofT w is given by
1m Tw = {q+(f) If eH':'.tr(JR), R,.s(Bj)e 1m P~ +Ker P _ +lm jJ}, (7.11)
codim 1m T w = codim (1m P ~ + Ker P _ + 1m jJ). (7.12)
Furthermore,for a function [f]e 1m Tw the general solution to the equation Tw(cp)=[f] in
the space Hr;' (IR + ) is given by
r~ . . cp =f++I.CEx(inX)}p~B(fU»++
j=O
+ ± CEx(inX)jp~B(.!!..-)j-r-S«(f('+S»+)+ j=r+s+1 dt
ooJ -s-I d +( C(t-s)f+(s)ds-CExe-itO,x x)+ + I. Vj(i-)jo+
o j=O dt
(7.13)
+ -f ,+s:i:-1CE x (inx) v+l p~Bf(V-j) (O)(.!!..-)j 0, j=O v=j dt
where f e H':'.t,(IR) is any representative of [f], and x e 1m p~ and v j e «: m , j =0, l, ... ,-s-l,
are vectors such that
R"s(Bj)+x +i(Bvo +nBv] +···+n-s- I BLs-l) e Ker P _. (7.14)
Before proving Lemma 7.2 we need the following result. Recall that:J m is the space of
rapidly decreasing functions with values in cr m .
LEMMA 7.3. Define the mapping Roo,s::J n ~ cr n by
Roo,s(f)=in-S(nX)-S ~(inX)j+1 P~fU)(O)+ijeiSo'x P~f(s)ds, j=O 0
Kuijper 169
Then Ker P _ + 1m P ~ + 1m P ~ + 1m Roo,s B + 1m :JJ = ([ n .
PROOF. Recall from Lemma 2.1 that the matrix QX P~ is nilpotent. Hence, the map
ping Roo,s is well-defined. Let v be a vector in the image of P _, and choose a function / e :J m
such that f (t) = CEe -ilQ v for t > O. First, we prove the following two identities:
00
iJe iSQx P~Bf(s)ds=P~v, o
i ~(iQX)j+1 P~ BfU)(O)=P~v. j={)
To verify (7.15), use the identities AEP = QP and P x A x E x = Q x P x to check that
00 00
iJe iSQx P~Bf (s)ds = iJe isQx P~ (A -A x )Ee-isQ vds = o 0
00 00
=iJei.l'Qx P~Qe-isQvds-iJeiSQx P~QX(Ex)-IEe-isQvds. o 0
Rewrite the first integral in the previous formula as
00
=P~ I' +iJe iSQx P~ QX e-isQ vds. o
(7.15)
(7.16)
Since pXGEx =p x and GEP=P (cf. the formulas (2.1)-(2.3», we have PX(EX)-IEP=pxP,
which proves (7.15). Next. to verify (7.16), recall the formulas AEP =QP and P~A x EX =P~,
and check that
i i (iQx )j+l p~BfU) (0) =iI:. (iQx )j+l P~ (A -A x )E(-in)j v = j~ j={)
=-1: (iQx )j+l P~ (_iQ)j+l V - i 1: (iQx )j+l P~ (Ex )-1 E(-iQ)j v. j~ j={)
To complete the proof of (7.16), it now suffices to compute the right hand side in the previous
expression by employing the identities
(cf. Section 2). which yields the desired identity. Now. to conclude the proof. combine the iden
tities in (7.15) and (7.16) and verify that Roo,s(Bf)=P~v+Q-S(Qx)-SP~v. Hence, we may
170 Kuijper
write the vector v as
v =P~v +Pgv +R .. ,s(Bf)+(P~ v _Q-s (QX)-S p~ v).
By Lemma 5.3, the last term in the previous expression is an element of Ker P _ + hn 91. As the
same holds for vectors v in Ker P _, this finishes the proof. [J
The above lemma may be viewed as a generalization of Lemma II.l.l in [BGK3].
Indeed, take G =1, and assume that both A and A x have no spectrum on the real line. Then
Po =P .. =0, and hence Ker P _ =Im P +. Furthermore, we have P~ =O=Pg and p~ +P~ =1,
and so
ImR .. ,s B=lm [P~B P~QxB ... p~(Qx)nB]chnP~ +hn [B QXB ... (Qx)nB].
It follows that 1m p~ +Im P++hn [B QXB ... (Qx)nB]=([n, which is [BGK3], Lemma
11.1.1.
PROOF OF LEMMA 7.2. The proof is divided into three parts.
Part (A). Let [f] E 1m T w. In this part we prove that [f] is contained in the right hand
side of (7.11), and that any solution $ E H ': (IR +) of the equation T w ($) = [f] is of the form
(7.13)-(7.14). Throughout Part (A), the expression f denotes a representative of the class
[f] E H':+r [IR + I in the space H':+r (IR).
To start the proof, let [f], f and $ be as above. Then, by Theorem 6.1, there exists a dis
tribution p satisfying the equalities.
i!!.Gp-Ap=B$, [f]=C[p]+[$]. dt
By passing to the inverse system, one obtains
;!!'G[P] =A x [p] +B [f], [$] =-C [p] + [f]. dt
(7.17)
Now, apply Theorem 4.1 to the first equation above. Using assumption (i) of Theorem 1.3, it
follows that there exists a vector x E 1m P ~ such that
(7.18)
where
NX
pXO ... )=-l:Ex(ilx)j p:, ')). j=O
Kuijper
t < 0,
t > 0,
171
and N x is the order of the pole at infinity of CA.G - A x ) -I . Hence, the second equality in (7.17)
implies that the distribution <I> belongs to the coset
[<1>1 = [qx(i~)! + + fe(t-s)! +(s)ds] - [Cee- itQx x], dt 0
(7.19)
with e the function given in Theorelll 1.2 and qX the polynomial
r .. qXO .. )=/ + I.CEx(nx)J P>;;'B'AJ.
)=0
Here the summation stops at the level j = r because of assumption (ii) of Theorem 1.3.
Next, we establish the formulas (7.13) and (7.14). This part of the proof is divided into
three steps.
Step (i). As a preparation we first prove that qX(i3...)!+ =H +R for distributions dt
H E H ': (IR + ) and R E (J III )' (IR - ) gi ven by
r+s H= f++ I.ce(inx)Jp':x,B(j(j»++
j~O
+ ± Ce(inX»)p>;;'B(~»)-r-S«j(rH)+)+ J=r +s+ 1 dt -1-1 r+l+jl J
+ L L ce(inX)jp:Bf(j-l-jl)(O)(~)jlo, 11=0 j =jl+ 1 dt
To establish (7.20) and (7.21), first observe that
For integers j ~ r +s we have
Combining these formulas, a straightforward computation shows that
(7.20)
(7.21)
172
r+s =/ + + I.CExUO x ») P":"B(jU»+ +
)=0
+ ± CEx(inX)jp~B(!!..)j-r-s«j(r+s»+)+ j=r+s+! dt
Kuijper
(7.22)
(7.23)
(7.24)
The tenns in (7.22) belong to H':(IR +). To analyse the tenns in (7.23) and (7.24), perform the
substitution 11 := j -1-v, recall that -s $ r by the assumption on the integers r and s, and separate
the terms containing derivatives of 0 of order smaller than -s from those with derivatives of
order greater than or equal to -so Thus, the sum of the tenns in (7.23) and (7.24) may be rewrit
ten as
The second double sum in the previous formula equals R, while the first double sum and (7.22)
together equal H. It follows that q x (i ~)/ + has the desired representation. dt
Step (ii). Let the function e and the vector x be as in (7.19), and introduce a distribu
tion <I>! by
00
<I> I =H +(fe (t-s)/ + (s)ds)+ - (ce e-itO.x x)+. o
In this step we show that
(7.25)
(7.26)
for certain vectors \' j , .i = 0, 1 , ... , -s -1, in ~ m. Indeed, from (7.20) and (7.25) we see that <I> 1 is
an element of H ~ (IR + ). Furthennore, because the distribution R in (7.21) has its support in 1R - ,
Kuijper 173
we have
and hence the distrihution ~ 1 satisfies [~I ] = [~] (cf. (7.25) and (7.19». But then, the support of
~-~l is contained in (-00,0]. On the other hand ~-~I is contained in H':(JR+). Combining
these observations, equality (7.26) follows.
Step ( iii). In the third step we show that the vectors \' j in (7.26) have to satisfy the con
dition in (7.14). To prove this, we first define a distribution P 1 hy
(7.27)
Here the functions p x and I x and the vector x E 1m P ~ are as in (7.18). Then, as [p] = [p 1 ], the
distribution p is of the form p = p 1 + T _ for some tempered distribution T _ with support in
(-00,0]. Insert this formula into the first equality in (7.17). This yields
Next, recall from Corollary 3.2 that
is the unique solution to the equation i~Gp -A x P =Bf +. Using this fact as well as the formula dl
in (7.27), we obtain the identity
Furthermore, observe that <p 1 + R = f + - C PI' Hence, employing (7.26), we arrive at the fol
lowing equality for the distribution T _ :
d cof -.1'-1 d i-GL -AL = (-ix + e isOx P~Bf(s)ds)O+ I Bl'j(i-)jo-BR.
dt 0 ]=0 dt (7.28)
Now, recall that T _ has its support in (-00,0], and repeat a n:asoning in the proof of Lemma 7.1
(see the formulas in (7.7)-(7.9». It follows that
-ix + -rQj Bl' j + fe iSOX P~Bf (s)ds - r~ (-iQ)~ Bw~ E Ker P _, (7.29) j=O () ~=-s
174 Kuijper
where
r wI! = L ce(iQX)j p;;"BjU-I-I!) (0).
j=I!+1
Premultiply the last tenn in (7.29) by P _. This yields, using the equality A x EX P;;" =P~,
,--] ,-] , P _( L (-iQ)~BwJ.l)= L L P _(-iQ)J.lBCe(i!y)j P~Bf(j+~)(O)=
J.l~-s J.l~-s j ~J.l+]
= ri i P _(-iQ)l!A£x(iQx)j p;;"BjU-I-I!) (0) + I!=-S j =I!+ I
- ri i P _(-iQ)I!(iQx)j p;;"BjU-I-I!) (0).
1!=-sj=I!+1
(7.30)
(7.31)
As PA = QPE-I =QPG and GExP~ =QxP;;" (cf. the identities (2.l)-(2.3», the expression in
(7.30) equals
ril i iP_(-iQ)I!+IGe(iQX)jp;;"BjU-l-I!)(O)= 1!=-sj=I!+1
r-I r = L L P _(-iQ)I!+1 (iQX)j+l p':.,BjU-l-l!) (0).
1!=-sj=I!+1
Hence, the expressions in (7.30) and (7.31) add up to
r r L P _(_iQ)I!(iQx)r+1 P':., Bj (r-I!) (0)- L P _(_iQ)-s(iQX)i p':.,BjU-I+s) (0). 1!=-s+1 j=-s+1
Because of the second assumption of Theorem 1.3, the first sum vanishes. Indeed, by Lemma
5.4, the matrix (-iQ) I! P _ P ':., (0. x) r+ I B = O. The second sum equals
r+s-I - L P _(_iQ)-s(iQX)-s+j+l P':.,BjU) (0). j=D
Thus, as a condition on the vectors v j' we derive from (7.29) that
which is condition (7.11).
From the results obtained in Step (i) up to Step (iii), and in particular the identities in
(7.20) and (7.25)-(7.27), it follows that the solution q, is of the fonn (7.13)-(7.14). Furthennore,
we proved that the representative f of U'] E 1m T w (and hence any representative of [f]) satis
fies condition (7.14), and thus [f 1 is contained in the right hand side of (7.11). This concludes
Kuijper 175
Part (A).
Part (B). Assume that the function J E H~+r (JR) and the vectors x E 1m P ~ and
v j E ([ m • j = O. 1 •...• -s -1. satisfy the condition in (7.14). In this part of the proof we show that.
under this assumption. the distribution $ in (7.13) solves the equation T w ($) = q + (j). In partic
ular. we prove that any distribution q + (j) contained in the right hand side of (7.11) is an ele
ment of 1m T w .
To prove our claim. let J. x. and v j be as above. Define a distribution $ by (7.26) with
$ 1 as in (7.20) and (7.25). Then $ E H~ OR + ). $ is given by (7.13). and [$] = [$ d. Next. define
distributions Rand PI as in (7.21) and (7.27). let L be the unique solution to equation (7.28)
(which exists since AG-A is lR-regular; see Corollary 3.2). and put P=Pl +L. Then. since
supp R c (-00.0]. the calculations in Step (i) show that [~] = [~d is given by (7.19). Further
more. using the expression for the solution of (7.28) given in Corollary 3.2 and condition (7.14).
it follows that L E (:In)' has its support in (-00.0]. Hence. [p]=[pd is as in (7.18). and a
straightforward computation shows that
[J ]-C[p] =[J + ]-C[p] = [~].
Moreover. employing the calculations in Step (iii).
In other words. the distributions $, P and [J] satisfy the equations in (7.17). But then, Theorem
6.1 shows that Tw (~) = [J J.
Part (C). In this part we verify equality (7.12). Introduce a mapping
by
By equality (7.11) the above mapping is well-defined and injective. We claim that it is surjec
tive as well, or. equivalently. we claim that
1m R r,s B + Ker P _ + 1m P ~ + 1m .:8 = ([ n .
To establish the above identity, combine assumption (ii) of Theorem 1.3 and Lemma 5.4 to
176 Kuijper
check that
An application of Lemma 7.3 now completes the proof. D
PROPOSITION 7.4. Let X be a Banach space, and let M be a closed subspace of X of
finite codimension. If D is a dense subspace of X that contains M, then D =X.
PROOF. This lemma immediately follows from (and, in fact, is equivalent to) the well
known fact that any subspace D that contains a closed subspace of finite codimension, is closed
itself. D
PROOF OF THEOREM 1.3. The proof is divided in three steps. In Step (i) we show
that T w cannot be Fredholm if det W (A) is identically equal to zero. In the second step we
assume the operator T w to be Fredholm and derive the properties (i) and (ii) mentioned in the
theorem. Step (iii) concerns the reverse implication and the statements in (1.16) up to 0.22).
Step (i). Suppose that det W (A) == O. Apply the Smith-McMillan form and write the func
tion W as W (A) = E (A)D (A)F (A), where E and Fare m x m matrix polynomials with constant
nonzero determinants and D is a rational matrix function of the form
with D 1 a diagonal matrix. By the assumption on det W, the size of D 1 is I x I for some I < m. Now, recall that F- 1 is a matrix polynomial as well, and define a subspace '/Jo by
Here Y (IR +) stands for the rapidly decreasing functions in Y with support in 1R +. Then, using
the relations between differentiation, convolution and rourier transformation in Y " a straight
forward computation shows that '/J o c Ker T w. Furthermore, the subspace '/J o is infinite
dimensional. Indeed, as
F(·d)F-1(.d) - pE(y nJ )', ldi I dt p-p,
the mapping p F-1 (i!!"-)P acting from (Y m)' to (Y m)' is injective. But then the space Ker Tw dt
is infinite dimensional, and thus T w cannot be a Fredholm operator.
Step (iiJ. Assume that the operator T w is Fredholm. Then, by Step (i), the determinant
of W (A) does not vanish identically. Therefore, using identity (5.5), the pencil AG - A x is
Kuijper 177
regular. In particular, the decomposition data (1.13) for AG -A x are well-defined. -
Next, we reduce the problem to the case r = s = O. Define a function W by
WOe) = (A+i)'H W (A)(A- i) -J,
and let T w (L ~ (IR + ) ~ L ~ [IR +]) be the maximal half-line integro-differential operator with
symbol W. Here the expressions L ~ (IR +) and L ~ [IR + 1 stand for the spaces H ~ (IR +) and
H~ [IR+J, respectively. The operators Tw and Ti' are linked by the following relations. A
function <I> E L~I (IR +) is an element of D(T w) iLand only if (i~ - i) -J <I> E D(Tw), and in that dt
case
Observe that the operators
are invertible bounded linear mappings. Hence, along with the operator T w , the operator T w is Fredholm. Once it is proved that this implies that det W (A) :;t 0 for A E IR and that the function
W- I (.) is analytic at infinity, the properties (i') and (ii'), and hence (i) and (ii), for W(A) follow
immediately.
From now on we assume the integers r and s to be zero. First, we introduce two linear
subspaces D 1 and D 2 of L ~ fIR +], namely:
t
D 1 = (q+(f) II E L;n (IR), 3x E 1m Po : (e-itnX (x _iksnx PoBI (s)ds»)+ E q (lR +)},
o
Here N X is the pole at infinity of (AG - A x) -I. We claim that both D I and D 2 are dense sub
spaces of L;n [IR + I. Indeed, all functions q + (f) with f E L;n (IR) and supp f compact belong to
D 1 . In fact, for such a function q + (f) one may take for x the vector
~
x=iIe iSnX PoBI(s)ds. o
As the set of functions with compact suppon is dense in L f (IR), it follows that D 1 is dense in
178 Kuijper
L ~ [IR +]. As for the space D 2, note that D 2 contains the subspace (:J ) In [IR +] of restrictions of
functions in:J In to (0,00). Since :J In is a dense subspace of L ~ (IR), the subspace D 2 is dense in
L~ [IR+], as well.
Next, we prove that 1m Tit cD I. Take an element [f] E 1m T w. Let f be any
representative of [f] in L ~l (IR), and let lJ> E D (T It) be such that T w (lJ» = [f]. Then, by
Theorem 6.1, there exists a distribution p E (:J n)' satisfying the equations in (7.17). By a
remark just after the proof of Theorem 6.1, we know that P E fI ~ (IR) for some integer 11. Now,
repeat the argument in the second paragraph of Part (A) in the proof of Lemma 7.2 and observe
that [p] E H ~ fIR + I satisfies the equations
i~G[P]=AX[P]+B[f], [lJ>]=-C[p]+[f]. dt
Hence, Theorem 4.1 states that [p] is of the form
I
+[EXe-ilnx (x~ _ijei,ln x P~Bf(s)ds)], o
(7.32)
where pX is some n Xn matrix polynomial, IX is an n Xn matrix function with components in
L I (IR) and x~ and Xo are vectors in 1m P~ and 1m Po, respectively. The terms in the first line
of formula (7.32) are elements of the space H ~ [IR + I for V sufficiently small. As the same holds
for [p] itself, we must have [X(Bf)] E fI ~ [IR + j, where
t
X(Bf):=(e-itnX (xo _ijeisnx PoB!(s)ds»+. o
We claim that, in fact, [X(Bf)] E L~ [IR + j. Indeed, for v ~O this is clear. For negative v employ
the identity
i~[X(Bf)]- i[X(Bf)] =(Qx - i)[X(Bf)] + [P~ Bf]. dt
(7.33)
As [P~Bf]E L~ IIR+lcH~ [IR+], equation (7.33) implies that, together with [X(Bf)], the distri
bution (i..!!..-i)(X(Bf)]EH~fIR+]. Now, introduce an operator D_:(:Jn)'-7(:Jn)' by dt
D_(f)=i"!!"f -if. It is easy to check that the operator D_ is invertible and maps H~+I (IR) onto dl
H~ (IR). A closer look reveals that D _ maps H~+I (IR) onto H~ (IR -). Here fI~ (IR -) denotes
the space of distributions in H~+I (IR) with support in (-00,0]. As a consequence, the operator
D_ :H~+I [IR+I-7H~ [IR+j, given by [p]-7i"!!"[p]-i[p], IS invertible, and hence, dl
Kuijper 179
[X(Bf)] E H ~+I [IR + ,. Repeating this argument -v-I times, we obtain that [X(Bf)] E L ~ [IR +1.
which proves the claim. But then, we have X(Bf) E L ~ (IR +), and so [f] E D I .
From the results in the preceding paragraph and Proposition 7.4, it follows that
D I =L~ [IR + J. Using this equality, we now prove that P'OB =0. Indeed, suppose that P'OB ;to.
Then there exist a number a E IR and vectors 0 = v -I , Vo, V I, ... , v I-I such that
vjB=O (j=0,I, ... ,1-2), v/_IB;tO.
Now, let 1 -:; j -:; m be an integer such that c = (v H B) j ;t 0, and let I be the cr m -valued function
with all components zero, except for I j, which is given by
. I I J(t)=e-lIa --, t >0, IJ(t)=O, t <0.
1+ t
The function [f J E L ~ [IR + J = D I, and hence there exists a vector x ~ E 1m P'O such that
(x(BI» + E L ~ (IR + ). However, a premultiplication of the expression (x(BI» + by v /-1 yields
( . )/-1 I
-ita(( ( .) -It) x .J isa( B)/()d ) e l'/_I + -It V/-2 + ... + Vo Xo -I e V/_I S S + = (I-I)! 0
( .)H = e-iltJ (l' /-1 + (-it)v /-2 + ... + -It vol xo -iclog(l + t)) + e: L2 (IR +).
(I-1)!
Thus, we arrive at a contradiction, proving that P'OB =0.
To establish the lR-regularity of AG -A x, or, equivalently condition (i'), it now suffices
to recall formula (5.6), which states that eX = (Ae - A X ,8, -C) is a realization of W- I (.).
Hence, we have
and it follows that the function W- I (.) is analytic at all points AE lR, indeed.
It remains to prove (ii). Again, we first show that D 2 =LT [IR + J. To verify this, let
[f] E 1m T w, and let f be any representative of [f J in L ~n (IR). Note that P a = 0 because of the
IR -regularity of 'AG - A x. Hence, employing (7.17) and (7.32), there exists a vector x~ in the
image of P ~ such that
As the last two temlS in the above formula are functions in L T IIR + \, the same has to be true for
Cp x (i~)8 [f J. But then, since <II
180 Kuijper
NX
pXO,,)=-l:EX(QX)j P::"B').) , j={)
the function [f] E D 2. Thus 1m T weD 2, and hence by Proposition 7.4 D 2 =L'{' [IR +]. Next,
take an arbitrary vector x E cr m , and define a function f by f (t) =x for 0 < t < 1 and f (1) = 0 for
other values of t. Then, because [f] E D2,
q+(CEX P>;"Bf) -q+( f CEx(iQx)j P>;"B(!!.. )j-1 ()1 x) E L2 [IR +]. (7.34) j=1 dt
Here () 1 is the distribution () 1 E :I ' , given by «) 1 ,a.) = 0.(1) for a. E :I. Since the first term in
(7.34) is an element of L'{' [IR +], it follows that
N" . d. l:CEx(iQx)J P>;"B( _)J-l ()I X + L =g E L'{' (IR) j=1 dt
for some T _ E (:I m)' with support in (-00,0]. Now, decompose the function g as g = g + + g _,
where g± E L'{' (IR) and g± vanishes for ±t < O. Thus, one obtains
In particular, there exists a tempered distribution To with support in {OJ such that
'fCEx(iQX)j P>;"B(.!!..)j-1 ()lx+To e L,? (IR t )cL,? (IR). j=1 dt
Applying Fourier transformation to the above expression yields
eo.· 'fiCEX(QX)j P>;"BXA,j-1 +:J(To)e L,? (IR),
j=1
where :J(T 0) is a polynomial (see [Tr], Theorem 24.6). But then, a straightforward reasoning
implies that CEx(QX)j P>;"Bx =0 for j '2.1. Since x is an arbitrary vector in cr m, this proves
condition (ii).
Step (iii). Assume the conditions (i) and (ii) hold. By Lemma 7.1, the kernel of Tw then
is given by 0.16) and has a finite dimension given by (1.17). Furthermore, Lemma 7.2 states
that the image of T w is the closed subspace given in (1.18) with finite codimension as in (1.19).
Since T w is a closed operator, this proves that T w is Fredholm. To compute its index, note that
codim 1m Tw =dim (1m p~ n (Ker P _ +Im ~»+n -dim (Ker P _ +Im ~)-dim 1m P~.
Thus, the index is given by
ind Tw =dim 1m p~ -n+dim (Ker P _ +Im ~)+dim Ker JH
Kuijper 181
+ dim (Ker P _ n 1m :13) = dim 1m P ~ - dim 1m P _ - ms,
which proves (1.20). It remains to prove that (1.21) satisfies 0.22), or in other words to prove
that (1.21) is a solution to the equation T w (<1» = q + (f), with f a function in H~, (JR.) satisfying
the condition in (1.18). Define the vectors x and v j, j = 0, I, .. , -s-I, as in Theorem 1.3. Then
x E 1m P ~ . Furthermore, as
1m A! elm:l3,
the definition of :13 + implies that
-s-1 i L n)Bv)=-A!P_At(l-P~)R"s(Bf). )=0
Therefore, by the definitions of A! and At, we have
-s-1 P - (i L n) Bv j) = -P - At (I - P ~ )R ',S (Bf).
)=0
(7.35)
Next, employ (1.18) to conclude that (I - P ~ )R ',S (Bf) = (I - P ~)u for some vector u in the
space Ker P _ + 1m (B QB ... n -s-1 B). Hence,
A t (I - P ~ )R ',S (Bf) = (I - P ~ + P ~ )A t (I - P ~ )u =
=(I-P~)R "s(Bf)+P~ At(l-P~ )R"s(Bf).
Combining (7.35) and (7.36) we arrive at
-s-1 R ",(Bf) +x + i L n j Bv j =(I-P~ )R ',S (Bj) +
j=o
-s-1 . + P ~ A t (I - P ~ )R ',S (Bf) + (I - P _ + P - )( i L nl Bv ) ) =
j=O
-s-1 (I-P _)At(l-P~ )R"s<Bf)+(I-P _)(i L BnjVj)E Ker P_,
j=O
An application of Lemma 7,2 now finishes the proof, D
(7.36)
Under the condition that 'A.G -A x is IR-regular, the space D2 introduced in Step (ii) of
the above proof can be identified with the domain of the maximal integro-differential operator
T w-1 (L '{' (lR + ) ~ L'{' [IR + [). Using this observation it is possible to give an alternative proof of
the inclusion 1m Tw e D 2 , and hence, proceeding as in Step (ii), of the necessity of condition
(ii) for T w to be Fredholm.
PROOF OF THEOREM 1.2. If the operator T w is invertible, then the operator T w is Fredholm, injective and surjective. Thus, by Theorem 1.3, the conditions (i) and (ii) hold, and
182 Kuijper
the right hand sides in (1.17) and (1.19) are zero. In particular. the first two terms in (1.17) are
zero. which together with (1.19) implies condition (iv). Finally. the matrix :B is injective
because the third term in (1.17) is zero. This implies condition (iii). Conversely. if the condi
tions (i) up to (iv) hold. then the operator T w is Fredholm and. using the formulas in Theorem
1.3. we have
codim 1m Tw =O=dim Ker Tw.
So. Tw is invertible. To prove (1.14). it suffices to check that the vectors x and v j. j=O.I •..• -s-l. defined in (1.15). satisfy the condition in (7.14). Indeed. since 1m ll2 elm:B.
we have "2R"s(Bj)=:Bu for some ue cr-ms . Hence. it follows that
-s-I R "s(Bj)+x +i l: oj Bv j =R"s(Bj)+x -:B:B (-I) :Bu=
j=o
=R"s(Bj)+x-:Bu=(I-lll -ll2)R"s(Bj)e Ker P_.
This finishes the proof. [J
8. AN EXAMPLE
Let us illustrate the results of the previous sections by considering the system of half-line
equations
{ q+(~k*l\Il +1\12) =[fIl.
q+(iL *1\11 +i~1\I2 -iI\l2) =[f2]· dt
(8.1)
Here k(t)=e- 111 and L(t)=k(t)-k+(t). The equations (8.1) have as their symbol the func
tion
The determinant of the function Wequals
det W (A) =-2i(A 2 + 1) -I .
In particular. W (A) is invertible for all real A. and a straightforward computation shows that Wo.rl is the polynomial
-I 1 .[(A2 +1)(A.-i) -(A.2 +1)j W(A.) = Y21 -(A. + i) 1 .
Kuijper 183
So, it follows, e.g. hy taking the Fourier transformation, that the operator i w : (;! 2)' ~ (;! 2)' is
invertible. Also, we see that the conditions (i ') and (ii ') are fulfilled for r ~ 3, which implies
that the maximal half-line operator Tw (H~(IR+)~H;+r[1R +[) with symbol W is Fredholm for
r ~ 3. We shall compute the Fredholm characteristics of the operator T w for r = 3 and
s =0,-1,-2 and -3. respectively, and give formulas for possible generalized inverses.
To analyse the Fredholm properties of the operator T w, we first write W in realized
form, i.e., in the form (1.9). Put
o 1 000 1 000 0 u I+i
o 0 000 o 1 00 0 o -I _ [0 0 1 -'hi 'hi)
G= 00 000 A= 00 1 0 0 , B= -I C- 100 10' 000 1 0 000 i 0 0 o 0 0 0 1 o 0 0 0 -i 0
Then a straightforward calculation verifies that, for the above matrices, the identity (1.9) holds.
The next step is to compute the data for the decompositions of the pencils A.G - A and
/..G - A x. As for the pencil /..G - A, Theorem 1.2 shows that we need only the projection P _
and the matrix Q. These matrices are given by
o 0 0 0 0 o 1 00 0 00000 0000 0
P._= 00000 , Q= 0000 0 o 0 0 0 0 000 i 0 0000 1 00 0 0 -i
In particular, we have
Ker P _ = span [e \ , e 2, e 3, e 4 },
where e j is the /h unit vector of ([ 5. Next, we direct our attention to the pencil A.G - A x ,
where
-i 0 0 -1-i 0 1 1 0 0
AX=A-BC= 1 0 0 'hi +1 -'hi
0 o -1 1 'hi -'hi
0 o -\ 'hi -\ 'hi
The pencil A.G -A x has determinant -2i, and hence is regular and has no zeros in the complex
plane. Therefore, we immediately have
p~ =P~ =Po =p x =0, p~ =1 5 ,
To compute the matrices EX and QX, we first invert the matrix /..G -A x. To do so, we use the
184 Kuijper
fonnula manipulation program Maple, which yields
-O .. +i -I) -(A 2 +(i -1)1.) 1.. 2 -1.+ J-j V2iA-V2i +V2 -V2A+V2i-V2 -I -A-2i A+i V2i -V2i
-1.2-2 -1.3 -21. A 3-iA 2+2A-2i V2iA 2+A+2i -V2iA 2_1..
A+i A2+iA -A 2 -I -V2iA-V2 V2iA+V2
A-i A 2-iA -A 2 +2iA+ 1 -V2iA-I V2 V2iA+ 1 V2
Using this formula. a straightforward calculation involving contour integration shows that
!hi +v:! 0 V2i +V2 'l4i +'14 -1/4-'I4i -'I2i 1 -V2 -'14 '14 -V2i -V2 -'14 '14 0 0 0 0 0
EX: -i 0 -I 0 QX: 0 0 0 0 0 -V2 0 -V2i -'I4i 1;"i -'12 o -V2i -'I4i 'l4i !h 0 V2i -3Jii o/4i V2 0 V2i o/4i -0/4i
We are now in a position to compute kernel and image of T w. Firstly, verify the identi-
ties
p ~ = 0, codim Ker P _ = I, Ker P _ + 1m B = ([ 5 .
Inserting these formulas into Theorem 1.3, the Fredholm indices of T w are given by the follow
ing table:
s dim KerTw codim 1m Tw indTw
0 0 1 -I
-1 1 0 1
-2 3 0 3
-3 5 0 5
In particular, for s =0 the operator Tw is injective. As for the image, note that P~ =0. Hence,
by the formula in Theorem 1.3, the image 1m T w equals
{q + (f) I ! E H ~ (IR), -lIzi! 1 (0) -liz! 2 (0) + ill (0) + liz! ~ (0) -liz! F) (0) =O}.
Here! = (f I ! 2) T. For s < 0 the mapping T w is surjective. and it remains to compute the ker
nel. Observe that the vector x in Theorem 1.3 must be the zero vector because P~ =0. As for -.1'-1 .
the vectors v j' the condition that L n J Bv j E Ker P _ comes down to requiring the last comj=O
ponent in the summation to be zero. Thus. an easy calculation leads to
Ker T w = { [~~ 1 /) I VI = O} = ( [~) /) I a E ([ }
in case s = -1.
Kuijper 185
in case s =-2, and
in case s =-3.
Finally, let us compute formulas for a possible generalized inverse of Tw. To deal with
the case s = 0, recall that P ~ = P ~ = O. Hence, by Theorem 1.3 it remains to verify that
x x x [lhi 0] CE Q P ooB = -lhi 0 '
Observe that the above four matrices are just the coefficients of ,.) for j = 0, 1,2,3 of the polyno
mial W(A)-l. From the above results one immediately obtains the following formula for a gen
eralized inverse of T w for s = 0:
+ _ [lhf 1 - 'hih -lh/1 -lhnJ + 'hif~2l +lhf\3l 1 T w (q+ (j» - 'hf 1 +lhih +YZ/1 '
+
where f = (j 1 f 2) T E H; (lR). Note that, in this case, the operator Ttv is a left inverse of T w . To compute a general izcd inverse in case s < 0 we need the operators Ai, A!, :R + and R 3,s'
Recall that Ker P _ + 1m B = «: 5 , and that p~ = O. Hence, for any negative s one may take the
identity matrix of «: 5 for Ai. For A! one may choose, independent of the parameter s, the
operator mapping the vector n.e5 in 1m P _ to the vector (O,O,n,n,n)T in the image of B. It
remains to find :R + and R 3,s' For s = -\, the operator :R + may be chosen as a left inverse of B,
e.g.
[0 0 0 0 1] :il+ = 0 -1 0 0 0 .
Furthermore, a straightforward calculation shows that
R H (Bn = [* * * * -if J{O)-1f2!2 (O)+1f2i(1 (0) ] T,
with * an irrelevant element. Thus one derives the following generalized inverse of
Tw (H:l (IR+)~H~ IIR+\):
+ . -l' ,4 I - 'hif 2 -lh/1 -lhf?l + Y2inl )
T w(q+ (j» - lhf 1 +lhih + lh/1 + +
186 Kuijper
with f = (j 1 f 2) T E H ~ elR.). In case s = -2, the matrix 53 + may be chosen as a left inverse of
(B QB), say
0 0 0 lh lh
0 -1 0 0 0 :Jj+=
0 0 o -lhi lhi -1 -l-i 0 0 0
The vector R 3,-2 (Bf) has as its last component -Ifzf 1 (0). Thus, a generalized inverse of
T w (H:2 (IR +) -7 H ~ [IR + I ) is given by
~«-lh/l +lhi/;)+)+lh(~)2«(r'1 )+) dt dt [0]
Ttv(q+(j» = 0 + lh/l(O) 0+
[lh/l-lhif2-lh/l] [-lh/l(O)+lhih(O)] d
+ ' + -0 lh/l + lhih + lhl 1 . 0 dl . +
Here f = (j 1 f 2) T E H ~ (IR). To compute a generalized inverse for T w when s = -3, observe
that (Q x) 4 = 0, and hence R 3,-3 = 0. Thus, the vectors v j' j = 0, 1,2 are all zero, and as general
ized inverse of Twin case s = -3 we find
-lh~«(f I )+)+lh(~)2 «(f 2 - f I )+)+ lh(~)3 «(f 1)+) dl dt dl
lh~«(fl)+) dt
where f=(jl f2)T E L~(IR), In all three cases 5=-1,-2,-3, the formulas for Ttv define a
right inverse for the operator T w .
RE}t'ERENCES
[BGK I] Bart, H., Gohberg, I. and Kaashoek, M. A: Wiener-Hopf integral equations, Toeplitz matrices and linear systems, in "Toeplitz Centennial" (Ed. I. Gohberg), Operator Theory: Advances and Applications, Vol. 4, Birkhiiuser Verlag, Basel, 1982, pp. 85-135.
[BGK2] Bart. H., Gohberg, I. and Kaashoek, M. A.: Convolution equations and linear systems, Integral Equations and Operator Theory 5 (1982),283-340.
[BGK3] Bart, H., Gohberg, I. and Kaashoek, M. A.: Explicit Wiener-Hopf factorization and realization, in "Constructive methods for Wiener-Hopf factorization" (Eds. I. Gohberg and M. A. Kaashoek), Operator Theory: Advances and Applications, Vol. 21, Birkhauser Verlag, Basel, 1986, pp. 235-316.
Kuijper 187
[BGK4] Bart, H., Gohberg, I. and Kaashoek, M. A.: The state space method in problems of analysis, Proc.ICIAM 87, Paris-La Villette, 1987.
[GGK] Gohberg, I., Goldberg, S. and Kaashoek, M. A.: Classes of Linear Operators Vol. 1, Operator Theory: Advances and Applications, Vol. 49, Birkhauser Verlag, Basel 1990.
[GK1] Gohberg, I. and Kaashoek, M. A.: Block Toeplitz operators with rational symbols, in "Contributions in Operator Theory and its Applications" (Eds. I. Gohberg, 1.W. Helton, L. Rodman), Operator Theory: Advances and Applications, Vol. 35, Birkhliuser Verlag, Basel, 1988, pp. 385-440.
[GK2] Gohberg, I. and Kaashoek, M. A.: The state space method for solving singular integral equations, in "Mathematical system theory, the influence of R. E. Kalman" (Ed. A. C. Antoulas), Springer Verlag, Berlin, etc., 1991.
[GKLR] Gohberg, I., Kaashoek, M. A., Lerer, L. and Rodman, L.: Common multiples and common divisors of matrix polynomials I: spectral method, Ind. J. Math. 30, no. 3 (1981),321-356.
[GKr] Gohberg, I. and Krein, M. G.: Systems of integral equations on a half line with kernels depending on the difference of arguments, Uspehi Mat. Nauk 13 (1958), no. 2 (80),3-72, (Russian) = Amer. Math. Soc. Transl. (2) 14 (1960),217-287.
[GLR] Gohberg, I., Lancaster, P. and Rodman, L.: Matrix polynomials, Academic Press, New York, etc., 1982.
[H] Hormander, L.: Linear partial differential operators, Springer Verlag, Berlin, etc., 1963.
[Ka] Kailath, T.: Linear systems, Prentice-Hall, Englewood Cliffs, NI, 1980.
[Kr] Krein, M. G.: Integral equations on a half-line with kernel depending upon the difference of the arguments, Uspehi Mat. Nauk 13 (1958), no. 5 (83), 3-120, (Russian) = Amer. Math. Soc. Transl. (2) 22 (1962), 163-288.
[Ku] Kuijper, A. B.: The state space method for integro-differential equations of convolution type with rational matrix symbols, Ph. D. thesis, Vrije Universiteit Amsterdam, Amsterdam, 1992.
[Rl] Ramm, A. G.: Theory and Applications of some new classes of integral equations, Springer Verlag, New York, etc., 1980.
[R2] Ramm, A. G.: Random fields estimation theory, Mathl. Comput. Modelling 13, No.4 (1990),87-100.
[Ro] Roozemond, L.: Systems of non-normal and Wiener-Hopf equations, Ph. D. thesis, Vrije Universiteit Amsterdam, Amsterdam, 1987.
[S] Schwartz, L.: Theorie des distributions, Hermann, Paris, 1966.
[St] Stummel, F.: Diskrete Konvergenz linearer Operatoren II, Math. Z. 120 (1971), 231-264.
[Ta] Talenti, G.: SulIe equazioni integrali di Wiener-Hopf, Boll. Un. Mat. Ital. (4) 7, suppl. fasc. 1 (1973),18-118.
[Tr] Treves, F.: Topological vector spaces, distributions and kernels, Academic Press, New York, 1967.
,[VGl] Volevich, L. R., and Gindikin, S. G.: The Cauchy problem and other related
188
[VG2]
[VG3]
[VG4]
[VG5]
Kuijper
problems for convolution equations, Uspehi Mat. Nauk 27 (1972), no. 4 (166), 65-143 (Russian) = Russian Math. Surveys 27 (1972), 71-160.
Volevich, L. R., and Gindikin, S. G.: The Wiener-Hopf equation in generalized functions, Trudy Moskov. Mat. Ob.~('. 35 (1976), 165-214 (Russian) = Trans. Moscow Math. Soc. (1979), no. 1 (35),167-216.
Volevich, L. R., and Gindikin, S. G.: The Wiener-Hopf equation in generalized functions that are smooth on the half line, Trudy MoskoI'. Mat. Ob§('. 38 (1979), 29-74 (Russian) = Trans. Moscow Math. Soc. (1980), no. 2 (38),25-70.
Volevich, L. R., and Gindikin, S. G.: The Wiener-Hopf equation for smooth and generalized functions in a half-space, Trudy MoskoI'. Mat. Ob.~('. 40 (1979), 241-295 (Russian) = Trans. Moscow Math. Soc. (1981), no. 2 (40),231-284.
Volevich, L. R., and Gindikin, S. G.: Boundary value problems for the Wiener-Hopf equation in a half-space, Trudy Moskov. Mat. ObH. 42 (1981) (Russian) = Trans. Moscow Math. Soc. (1982), no. 2 (42), 175-198.
Department of Mathematics and Computer Science Vrije Universiteit de Boelelaan \081a 1081 HV Amsterdam The Netherlands
MSC 1991 Primary: 45J05, 45ElO Secondary: 15A22, 93C05
Operator Theory: Advances and Applications, Vol. 58 © 1992 Birkhauser Verlag Basel
SYMBOLS AND ASYMPfOTIC EXPANSIONS
Harold Widoml)
189
A general principle is proposed that in all the usual asymptotic expansions for tr I(T) where I is a "general" function and T is a Toeplitz, Wiener-Hopf, or pseudodifferential operator, each term of the expansion is an integral of (or, more generally, some distribution applied to)/(c1*), where 0'. is a "Symbol" associated with that term of the expansion. These Symbols are defined in terms of 0', the symbol of the given operator, but have larger domain. Although nothing so general is proved we consider several operators with smooth or nonsmooth symbol, evaluate early terms of asymptotic expansions associated with them, and show that they can all be cast into the form described.
O. INTRODUCTION
Since its earliest years the study of Toeplitz and Wiener-Hopf operators has focused on
questions of invertibility and Fredholmness on the one hand and the asymptotics of [mite sections on the other. One of the fundamental early results is that the Toeplitz operator T(cp) with (scalar
valued) continuous symbol cp is Fredholm if and only if cp is nonzero on the circle ']I', and is
invertible if and only if the winding numbers of cp equals zero. For their study of operators with
piecewise continuous symbols, Oohberg and Krupnik [OK] introduced a function cp* which has
domain lr x [0, 1] and is defined by
(0.1) cp*(9, t) = (1 - t) cp(9-) + t q>(9+), 0 S t S 1.
(As usual we identify 9 E 1R with ei9 E lr.) It turns out that T(cp) is Fredholm on H2 if and only if
cp* is nonzero, and is invertible if and only if a suitably defined winding number of cp* is zero.
This object cp* Oohberg and Krupnik called the symbol of the operator but we prefer to call it the
Symbol. Thus, the symbol cp is the function defining the operator whereas the Symbol cp* arises in
the study of its qualitative behavior.
1) Research supported by National Science Foundation Grant DMS 8822906.
190 Widom
The basic limit theorem is that, denoting by T,.( cp) the usual n x n section of the mauix
for T(cp), one has for quite general symbols cp and functions/an asymptotic formula as n -+ 00
(0.2) trfl..TII(cp» =1i Jf(<J(O» dO+ o(n).
(If cp is real-valued then/ need only be continuous; otherwise/is required to be analytic in a convex
set containing the essential range of cpo In the sequel we always take our functions/to be analytic
in an appropriate domain.) If cp is smooth then the term o(n) in (0.2) can be sharpened to: constant
+ 0(1). We will give an expression for the constant in a moment but first we tell what happens when cp is only piecewise smooth with jump discontinuities. The sharpening of o(n) in (0.2) in this
case is: constant· log n + o(log n). The constant here is equal to
(0.3) ~ L U(cp(Oj-), CP(Oi +),j) 411: j
where the OJ are the points of discontinuity of cp and where U is defined by
(0.4) U(a, b,j) = r fl..(l- t) a + t b) - [(1 - t)fl..a) + tfl..b)] tit. Jo t(1- t)
(The Wiener-Hopf version of this is in [W2]. A sharp form of the stated result, wherein cp is only
required to be of bounded variation, is in [F]. Of course, the sum (0.3) might then be infinite, but
still convergent)
Basor [B] pointed out that (0.3) could be interpreted as an integral over the domain of the Gohberg-Krupnik Symbol cp* ofT(cp). In fact this constant is the integral ofj{cp*), adjusted by
subtracting the function which is linear in t and has the same values at t= 0 and 1 (we call this the "linear part" of fl..CP*», with respect to the measure on T x [0, 1] which is the product of counting
measure on T and the measure
(O.S)
on [0, 1]. (Thus we have a linear functional applied tof(cp*) which is a regularized integral like
those familiar in distribution theory.)
We see, therefore, that although the Symbol was introduced for the study of qualitative properties of T( cp) it arises also in the answer to a quantitative question about these operators.
If cp is smooth then of course (0.3) vanishes and the term o(n) in (0.2) takes the form:
constant + 0(1). The constant here can be written in several ways. (It was obtained first, in a different form, in [L].) We prefer, for the context of this paper,
(0.6) ~ J J U(cp(O), cp(V),j) dO dl¥ . 1611: sin2 HO - cp)
Widom 191
This also can be written as a regularized integral ofl(cp*) for an appropriately defmed Symbol cp*.
Its domain is TxT x [0, 1] and it is given by
(0.7) <p*(9, 'II, t) = (1 - t) <p(9) + t <p('II),
and (0.6) is the regularized integral of I(<p*) with respect to the product of the measure
1 cosec2 l. (9 - 'II) de d\jI 4 2
on TxT and the same measure (0.5) on [0, 1]. Why it is just these measures which occur is, as
of this writing, a mystery. More such mysteries are to come.
There seems to be reason to think that the following speculation has some merit: in all
the usual asymptotic expansions for tr /(T), where T is a Toeplitz, Wiener-Hopf, or
pseudodifferential operator, each term of the expansion is the integral of (or another distribution applied to) 1(0'*) where 0'* is some Symbol of T associated with this term of the expansion. The
bulk of this paper consists of discussion of four other examples where this is the case. We
describe these now.
I. Let T be the integral operator on ~(lR "), depending on the large parameter a, with
kernel given by
(0.8) (f,tr f eia~.(x-y)O'(x, ~) ~ where, to simplify matters, we assume 0' is a Schwartz function on m II x mil. (Thus T is a
pseudodifferential operator with symbol, in the sense of that theory, equal to O'(x, ~/a).) Then
there is a complete asymptotic expansion as a -+ 00
(0.9) trf(n - L Ck, aft-Ie 1=0
where each ck is given by an integral formula involving derivatives of/applied to 0', times products
of derivatives of 0' itself. (We require of /, in addition to analyticity in the convex hull of the range
of 0', that it vanish at 0 so thatf(n is trace class. This expansion was fust derived in [WI], and
can be found in [W4] also. In both cases the context was somewhat different: in the fust the setting was a compact manifold and in the second 0' vanished for x outside a compact set in mil but
was only required to be a symbol of negative order rather than a Schwarz function. The condition on the x-support of 0' could easily have been weakened to obtain the result as stated above.) We
have
(0.10) Co =bir J J f(O'(x,~) dx d~, as one expects in all expansions like this. We shall show here that if we define a Symbol 0'* with domain (mil x m ")2 x [0, 1] by
192 Widom
(0.11) O'*(x, ~,J,11, t) = (1 - t) O'(x, ~) + t O'(y, 11)
then we have for k = 1 and k = 2 the formula
(0.12) Ck= (;'k~k(t-r f f r (V"VTlf(O'*) 1,=" ~dxdf" 7t J IRa J l1.a Jo f\~ t(1 - t)
where now the integral does not have to be regularized. Here V x V TJ. denotes the operator II a2
L ax. ::w. .• i=1 ,VIII
This formula for k = 1 was not hard to discover; that is holds also for k = 2 we found utterly
surprising. For k > 2 the fonnula is incorrect and the domain of the Symbol very likely has to be
enlarged.
II. In [B] Basor found a fonnula for the third tenn in the asymptotic fonnula (0.2) when q> is piecewise smooth. If we write the expansion in the fonn
(0.13) trf(Tn(q>)) = aln + a210g n + a3 + 0(1)
then of course al = (27t)-1 I f(q>(e» de and a2 is given by (0.3). The constant a3 was itself
written as a sum a3(1) + a3(2). The fonnula for a3(1) in [B] can be stated in Symbolic tenns by
defining a Symbol q>* with domain '][' x '][' x [0, 1] by
q>*(e, 'II, t) = (1- t) q>(0-) + t q>(",+).
(In the smooth case, of course, this is the same as (0.7). The restriction of this q>* to the diagonal of'][' x '][' is exactly the Gohberg-Krupnik Symbol (0.1).) In these tenns the formula for Q3(1)
can be written
(0.14) -Li: r [kf f eik(8-1I')f(q>*(e,,,,,t» dO dv-£kL f(q>*(e,e,t»J~ 2n-21=1 Jo JT Jy 9 t(1- t)
where the integral over t is regularized as usual. The sum in the integrand is taken over the points of discontinuity of q>, the tenns corresponding to other points contributing zero.
The second constituent, a3(2), of a3 is given in [B] by an infinite series of integrals of
increasing order. Here we shall show that it is also representable as the regularized integral
-L2 (1 L f(fp*(e, e, t» u(t) dt (0.15) 2n- Jo 9 where
r ' (1 + -L log l.::.l) u(t) = ~ 2 27ti t / t(1 - t).
r (1 + ~ log l=.1) 2 2m t
Widom 193
So once again the measure (0.5) appears, but this time with a mysterious factor involving the
gamma function. (Stirling's formula shows that this factor behaves like log log 1/t(1 - t) near
t = 0,1.)
Notice that if <p is smooth then the diagonal part of <p* does not contribute at all and it is
easily seen that (0.14) reduces to (0.6).
III. The finite Wiener-Hopf operator on L2(0, a) with symbol O'(~) is unitarily
equivalent to the operator on ~(O, 1) with kernel given by (0.8) with n = 1 and O'(x, ~) replaced by
O'(~). In [W4] we considered the generalization of this to operators on L 2(0), where 0 is a
bounded subset of lR n with smooth boundary, with 0' in (0.8) allowed to depend on x also. Here,
also, there was shown to be an expansion (0.9), but with coefficients given by much more
complicated formulas than in the situation described in I. In [W5] we considered the even more
general case of operators on L2(lR n ) with kernel given by (0.8) but where 0' jumps across a
hypersurface 'J1l. in x-space. (For the operators on ~(O) described just above we would have a
symbol which is smooth for x E 0 and vanishes for x f 0; thus 'J1l. in this case is ao.) Again it
was shown-but this time only for polynomial/-that there is an expansion (0.9). Of course Co is
given by (0.10) but now cl has two constituents. The first is just (0.12) with k = 1, the x
integration taken over the complement of 'J1l.. For the second we shall here define a Symbol 0'*
with domain T*('J1l.) x lR x lR x [0, 1], where T*('J1l.) is the cotangent bundle of 'J1l., and show
that this constituent is obtained by applying a certain distribution (essentially another regularized
integral) toj(O'*). In the finite Wiener-Hopf case this reduces to (the continuous analogue of)
formula (0.6).
N. Also in [W5] we considered the case where in (0.8) the symbol O'(x,~) jumps
across a hypeIplane 'J1l. in (x, ~)-space, and is otherwise smooth and rapidly decreasing. It was
shown, again only for polynomial/, and for generic 'J1l., that there is then an expansion
(0.16) trj(n-L c/can - kI2• /c=o
Of course Co is given by (0.10). We shall show that if we defme 0'* on 'J1l. x [0,1] by
(0.17) O'*(x, ~, t) = (1 - t) 0' +(x, ~) + t 0' -(x, ~)
where O'±(x, ~) are the limits of 0' from the two sides of 'J1l., then cl is equal to the regularized
integral
(0.18) Cl = -~- ( 11 j(O'*(x,~, t» wet) dt d(x, ~) (2x)n Jm
194 Widom
where P is a constant depending on the orientation of 111., d(x, ~) denotes Lebesgue measure on
111., and
wet) = dE-let) dl
where £-1 is the inverse of the incomplete error functions given by
(0.19) E(s) = [ e...JffT2 dr.
(This is slightly different from the usual incomplete error function Erf.) Thus the measure wet) dl
is just the push-forward to [0, 1] of Lebesgue measure on IR via the map E : IR -+ [0, 1]. The
weight function wet) is more singular than 1/t(1- t) by a factor of order [log 1/t(I- 1)]1/2. Why it
arises is another mystery.
Although the results just described fit the same general pattern their proofs are
completely different.
For I we take the existing formulas for cl and c2 and check that (0.12) holds. The
check for k = 1 is trivial, for k = 2 not completely so.
In n, the derivation of formula (0.15) for the second constituent of a3, we modify
Basor's argument by using the functional equation for the Barnes G-function to evaluate a certain integral rather than its infinite product representation.
Section ill is by far the longest of the paper and we derive our formula for the second constituent of cl (when a in (0.8) jumps across a hypersurface in x-space) by a very circuitous
route. Using the result in [W5] we show that it suffices to prove the correctness of the formula in the case n = 1. This case is done from the very beginning and requires two steps. First we show that Cl has two constituents, that the first is what it ought to be, and that the second is equal to the
trace of a certain operator associated with f and a singular integral operator on ~(IR). Second, we
find a formula for this trace with the help of the inversion of singular integral operators via Wiener
Hopf factorization. The reader may wonder why we cannot use the result of [W5] directly. The problem is that what is derived there is a formula for cl in the case where f(A) = Am with positive
integer m and this formula involves an m-fold integral. Thus the formula increases in complexity
as m increases and what we need is one in which m appears in a more benign way. There ought to be a more direct route from one to the other, but we do not see it.
Widom 195
For IV, the case where a in (0.8) jumps across a hyperplane in (X, ~-space, we do use
the formulas found in [W5] for the traces of the powers. They also increase in complexity as the
power increases and involve constants given by integrals I (q'th smallest component of (0) doo
where, for the m'th power, the integrals with q = 1, ... , m are taken over the unit sphere in the subspace of JRm defmed by Uo; = O. We shall show here that these integrals can be represented
nicely in terms of the error function E(s) and this representation will lead to the expression (0.18)
for cl'
The four parts of the paper, which follow, can be read independently of each other.
I. SMOOTII SYMBOLS ON JRn
Here is how (0.9) is obtained. The trace of a pseudodifferential operator (under, as usual, suitable conditions) equals (2x)-n times the integral of its symbol over JR n x JR n• Our
operator T with kernel (0.8) has symbol o(x, !;fa) andj{n is a pseudodifferential operator whose
symbol has an asymptotic expansion
(1.1) 1=0
with each 'Yk a Schwartz function. Integrating (1.1) term-by-term, introducing the variable change
; ~ a;, gives (0.9). The first few 'Yk are given by
'YO = j{o)
Y1 = -f VxeJ V~f"(o) and 'Y2 = 'Y2,2 + 'Y2,3 + 'Y2,4 where
'Y2,2 = - ~ V;o. V~f"(O)
(1.2) 'Y2,3 = - M V;o. (V~O)2 + V xeJ (VxV~o) V~o + (V xeJ)2 vto],"'(O)
'Y2,4 = - i (VxO)2 (V~o)2 ,(iv)(o).
Here we have used tensor notation so that, for example, 2 2 ~ cPa aa aa
Vxo (V;o) = ~ ----ij dXi dXj d;i d~j ~ d20 ~ dO dO
Vxo(VxV~o)V~o=~ --. ~ --. i dXi d~i j dXj d~j
(These formulas, in the context of compact manifolds, can be found in [WI, Sec. 5]. It is simpler in JRn.)
196 Widom
It follows immediately that Co is given by (0.10) and cl by
-f (2if J J V xcr V ~cr 1"( cr) dx df,.
This clearly equals the right side of (0.12) when k = 1. As for k = 2, the right side of (0.12) in this case is easily computed to be - (21t)-n/4 times the integral over 1R n x 1R n of
(1.3) V;cr V~cr f"(cr) + M V;cr (V~cr)2 + (V xcr)2V~crJt "'( cr) + ~ V xcr V~cr fiv)(cr).
The tenns here are in general like those in (1.2) except for the tenn
(1.4) Vxcr (VxV~cr) V~crf"'(cr)
in the expression for 12,3 which does not appear in (1.3). However since we are eventually going
to integrate over 1R n x 1R n , integration by parts allows us to make a replacement for this tenn.
Removing the V x from the second factor in (1.4) and distributing it among the others (and
remembering to change sign) shows that (1.4) can be replaced by
- V;cr (V ~cr)2 ! "'( cr) - V xcr V ~cr V x V ~cr! "'( cr) - (V xCJ)2 (V ~cr)2 ! (iv)( cr).
Noting that the middle tenn here equals minus (1.4) itself, we see that (1.4) can also be replaced by
- Mv;cr (V~cr)2 f"'(cr) + (Vxcr)2 (V~cr)2 !(iV)(cr)].
By symmetry in x and S, (1.4) can also be replaced by the symmetrization of this over x and S, which is
- ~ V;cr (V~cr)2 + (V xcr)2 v~cr]! "'(cr) - t (V xcr)2 (V~cr)2 !(iv)(cr).
Making this replacement for the tenn (1.4) appearing in the expression for 12,3 we find that the
integral of 12 is equal to -1/4 times the integral of (1.3) and this establishes (0.12) when k = 2. A
miracle?
II. PIECEWISE SMOOTII SYMBOLS ON 1r
A word about the derivation of (0.13). With integration over an appropriate contour,
we have
tr!(Tn(<P» = 2~i f !C)..) tr (A - Tn(<p)t1 tA
= -21. f !(A) JL log det (A - Tn(<P» tA = - -21. f f'(A) log det Tn(<P - A) tAo m tA m
One inserts the known asymptotics of Toeplitz determinants with piecewise smooth symbol into the
integral and evaluates the result. It is here that the Barnes G-function arises. (We refer the reader
to [BS, Chap. 10] for a discussion of these things.)
Widom 197
By this method the formula for a3 was derived in [BJ. With a3 = a3(l) + a(2) the
expression for a3(l) was expressible as (0.14). We shall not state the final formula for a3(2)
derived in [BJ but only that it is equal to
-~J f'CA) L log J~IOgA-P(9+»)tA 2m 9 5t2lt1 A - <p(9-)
where the sum is taken over the points of discontinuity of <p and where g is related to the Barnes G
function by
g(A) = GO + A) G(l - A).
The logarithm appearing in the argument of g is determined by saying that it vanishes when A = 00.
Consider a term corresponding to a single discontinuity e, and for convenience
set a = <p(e-), b = <p(9+). Thus we consider
(2.1) --21 . J J'(A) log J-21.10g~)tA. m 5l m A - a
Now G is an entire function with zeros at the nonpositive integers, and so g is entire with zeros at
the nonzero integers. The argument of g in the integral can never be an integer since a "# b. Moreover at A = 00 this argument equals 0, and g(O) "# O. It follows that the log g term in the
integral is single-valued and analytic in the A-plane cut along the line segment joining A = a to
A = b. The path of integration can be replaced by this cut, transversed twice in opposite directions,
with the limiting values of log g from the two sides of the cut in the integrand.
If we face in the direction from a to b then the limiting value of the argument of g as A
approaches the cut from the left is
1 + -.L log ~ = 1 + -.L log ll..:=.A . 2 27ti ~ 2 27ti A - a '
for the limiting value from the right of the cut we change the summand 1/2 to -1/2. It follows that
(2.1) equals
J1 + -.L log leA) f'(A) log 512 27ti A-a tAo
J_1 + ~ log l2=:..A) 5l 2 2m A - a
Now the functional equation for the G-function is G(z + 1) = r(z)G(z) and it follows
that
g(z) r(z)
g(z-l) r(1-z)
Hence the logarithm of the quotient in the integral equals
198 Widom
r (1 + ~ log il=A) log 2 2m A - a
r(l- ~IOg12=A) 2 2m A-a
This equals
2i gm log r[t+ 2~ log t~l since f(z) takes conjugate values at conjugate points and (b - 1..)/(1.. - a) is positive real for A on
the cut Hence if we parametrize the cut by 1..= (1 - t) a + t b O~t~ 1
our integral becomes
~11 () ~f«(1- t) a + t b) {]m log r t + ~ log 1 ~ t dt.
Before integrating by parts we observe that if the factor involving/ were replaced by a constant the result would be zero. This is shown by making the variable change t --+ 1 - t and
using once again the fact that f(z) takes conjugate values at conjugate points. So we can replace
f«(1-t)a+tb)by
(2.2) f«1 - t) a + t b) - [(1 - t)f(a) + tf(b)]
in the integral and then integrate by parts. (This replacement is important since (2.2) vanishes at t= 0, 1 and the r factor in the integral is mildly singular there.) What results is
il r' (l+LIOg1=1) _1_ V{(1- t) a + t b) - [(1- t)lla) + tllb)] 11« (2 2xi t) ~. 21t2 0 r 1 + L log .l..=.1 t(1 - t)
2 21ti t
Recalling that we are to replace a by cp(9-) and b by cp(9+) and then sum over the discontinuities e we see that the representation (0.15) for a3(2) has been established.
1lI. PIECEWISE SMOOrn SYMBOLS ON JRn
In this section, for later convenience of notation, we replace ~ in (0.8) by ~ and reserve
~ for a variable running over JR. We are given a hypersurface minx-space and assume that
cr(x, ~) is Coo in the complement of m x JR n. with all derivatives bounded and rapidly
decreasing as Ixl + I~I --+ 00. It was shown in [W5] that under a uniform smoothness condition on
'JT1. and for polynomial/vanishing at 0 there is an expansion (0.9) with Co given by (0.10). The
formula for cl had two parts, and so we write here cl = cl(I) + cl(2). The fIrst summand cl(1) is just (0.12), the integral being taken over the complement of 'JT1. x JR n. The formula for cl (2), in
Widom 199
the casej{A) = Am, involves a multiple integral whose order increases with m. We here derive a
completely different expression for cl (2), and we begin by inttoducing some notation.
(3.1)
For x e 'JIl. we denote by v x a unit normal vector to 'JIl. at x and defme, fll'st
O'i(x, ~) = lim O'(x +svx, ~), s-+Oi
and then for ~ e vt, ~ e JR.
The pair (x, ~) is a point of the cotangent bundle T*('JIl.) and we integrate over it by the formula
I'JIl. L; .. . d~ dx , with d~ denoting Lebesgue measure on vt and dx surface measure on 'JIl..
We associate with an ordered pair of functions (0'1,0'2) on R a 2 x 2 matrix of
functions (at. 0'2)~j on JR x JR x [0,1] as follows:
(0'1, 0'2)~i~' 11, t) = (1 - t) O',~~) + t O'J~) if i, j ~ 2, I,
(3.2) (aI, 0'2);,1 = (at. 0'2);,2 .
Here of course i and j take the values 1 and 2. The matrix is symmetric but depends on the order of
the pair of functions.
Our Symbol 0'* is defmed to be the matrix-valued function on T*('JIl.) x JR x JR x
[0, 1] whose value at the point (x, ~), ~, 11, t is obtained from (3.2) by setting + -
0'1 = O'(%'~)' 0'2 = O'(%,~) •
(This Symbol, appropriate for the representation of cl (2), is of course different from the 0'* defined
by (0.11) which is appropriate for the representation of cl(1).) With these defmitions the formula
we shall establish for the second constituent of cl is
(3.3) cl(2) = M...L2 r1 ( (1 (1 ± (-I)i+j j{O'~j) --ilL drt tJ; d~ dx 1r JT*( m) JR R Jo iJ= 1 t(1 - t) (~- 11 + 002
where, as usual, the integral is regularized by subtracting from f( O'~j) its linear part. The integral
over 11 is interpreted as the limit as E ~ 0+ of the corresponding integral with Oi replaced by Ei.
Here is why the result of [W5] allows us to reduce the proof of this to the case n = 1.
The expression for Cl (2) obtained there can be cast into the form
Ll..r1 f Ft (O'(X'~)' O'(x,~» d~ dx \21r Jr.(71/,)
200 Widom
where Ffis a function, depending on the polynomial/but not the dimension n, on ~(JR) x ~(JR).
(Here ~(JR) denotes the Schwartz class on JR.) If we can prove (3.3) when n = 1 and 'IT\. = {O}
then we will have enough infonnation to identify Ff (crI' cr:z) as equal to
_1_11 f f ± (-I)i+j f( crl, cr2)~) ---11L- drt d~ (3.4) 87t2 .1m. .1m. ij= 1 t(1 - t)(~ -11 + Ozi
and so (3.3) will follow for general n.
From now on we take n = 1, 'IT\. = {OJ, VX = 1 and, since there is only the single point
(0, 0) in T*('IT\.), we drop the subscript (x, ~). Thus
cr±(~) = lim cr(x, ~). x~o±
We shall prove two lemmas which, when combined, give the desired result. In them we use the notation P ± for multiplication by the characteristic function of JR ± and op cr(x, ~) for the
pseudodifferential operator with symbol cr(x, ~), the operator whose kernel is given by (0.8) with
a = 1. Also, we denote by cr±(x,~) any functions on ~(JR x JR) which agree with cr(x, ~) for
x E JR±. (Of course these have no connection with the functions in (3.1).) Thus cr±(~) =
cr±(O, ~). In these terms we can write
T = P + op cr+(x, ~/a) + P _ op c:r(x, ~a)
and we shall prove the following.
LEMMA A. 1// is a polynomial vanishing at 0 we have as a ~ 00
tr/(1) = Co a + cI(1) + cI(2) + 0(1)
where
Co = f,; J J f( cr(x, ~» dx d~ cl(1) = -f 2~ J J Vxcr V~crf"(cr) dx d~
(3.5) Cl(2) = tr~P+op cr+ +P_ op en -P+opf(cr+) -P_opf(cr-) J. (In the last, crt denote the functions cr±@, which are independent of x.)
LEMMA B. For any/which is analytic on a convex set containing the ranges o/the Schwartz junctions cr I (~) and cr2@ the trace
tr [ftp+ op crl + P _ op cr2) - P + op f(crl) - P _ opf(cr2) ]
is equal to (3.4).
Clearly Lemmas A and B yield (3.3) in the case n = 1, 'IT\. = {O}. For the proof of Lemma A, which will require some sublemmas, we use the fact that the operators op cr±(x, ~/a)
Widom 201
and op Cf±(x/a, ~) are unitarily equivalent (via the unitary map cp(x) -+ a 11l cp(ax) of ~(lR»
and so for each power m we have (3.6) tr rm = tr([P + op w(x/a., ~) + P _ op cr(xla, ~)]m).
In what follows we denote by 1:, 1:1"" any functions in ~(1R x JR.).
SUBLEMMA 1. (i) As a -+ 00 the operator op 1:(x/a.,~) converges strongly to the operator op 1:(0, ~). (ii) The operator P + op 1:(x/a, ~) P _ - P + op 1:(0, ~) P _ converges to 0 in the HilbertSchmidt norm as a -+ 00, and the same is true if P + and P _ are interchanged.
PROOF. The kernel of op 1:(xla., ~) is k(xla, x - y) where k is the inverse Fourier transform of 1:
in its second variable. This function converges pointwise to k(O, x - y) as a -+ 00 and is bounded
by a constant times (x - y}-N, uniformly in a, for each N. (We use the notation (x) = (1 + x2)1/2.) Assertion (i) follows quickly from this. As for (ii), the kernel of the operator in question is bounded by a constant times Ix 10.1 (x - y}-N, whence the square of its Hilbert-Schmidt
norm is at most a constant times
which is 0(0.-2) if N > 2.
In the next two sublemmas we use the notation T 1 = T 2 for operators Tj depending on the parameter a if tr(T 1 - T z) -+ 0 as a -+ 00.
SUBLEMMA 2. If each operator Rj is either P + or P _, and some Rj is P _, then P + op 1:1(xla, ~) Rl op 1:2(xla,~) R2 ... Rm-l op 1:m(xla, ~) P +
(3.7)
= P + op 1:1 (0, ~) Rl op 1:2(0, ~) R2 ... Rm- 1 op 1:m(O, ~) P +.
The same holds when P + and P _ are interchanged.
PROOF. In the product on the left appear factors P + op 1:j(xla, ~) P _ and P _ op 1:J{x/a, ~) P +.
By Sublemma 1 (ii) these converge in Hilbert-Schmidt norm to P + op 1:1{0, ~) P _ and P _ op 1:iO ~) P +.
By Sublemma 1 (i) the other factors Rk-loP 1:k(xla, ~)Rk converge strongly to
Rk-loP 1:k(O, ~)Rx' It follows by standard operator theory that the operator on the left in (3.7)
converges in trace norm to the operator on the right
In the following recall that Cf±@ = Cf±(O, ~ and that we write op Cf± for op Cf±@.
202 Widom
SUBLEMMA 3. For each positive integer m we have [P + op crt(x/a, ~) + P _ op cr(xla, ~)]m - [P + op cr+(xla, ~)]m - [P _ op cr(x/a, ~)]m
== (P+ op crt +P_ op cr~- (P+ op crt)m- (P_ op cr~.
PROOF. Expand the m'th powers of the sums and apply Sublemma 2, using the general fact
(3.8) tr PiA = tr A P± = tr PiA Pi'
We should remark that, strictly speaking, (3.8) requires that the operator A be trace
class and so we have used, for example, the fact that P + op cr+ P _ op cr is trace class. This is
certainly true-in fact, P + op crt P _ is itself trace class. The issue can be avoided by thinking of
the trace of an operator as the integral of its kernel over the diagonal when it is continuous almost
everywhere on the diagonal.
SUBLEMMA 4. (P ± op cJ±(x/a, ~»m - P ± (op cJ±(xla, ~»m == (P ± op cJ±)m - P ± (op cJ±)m .
PROOF. Take the case of + signs. The left side, after inserting a factor P + to its right (which does
not change its trace) can be written as a sum of expressions of the form of the left side of (3.7), and
the right side as an exactly corresponding sum of expressions of the form of the right side of (3.7).
Thus the conclusion follows from Sublemma 2.
SUBLEMMA 5. There is a complete asymptotic expansion
with
.. tr[p +(op cr+(x/a, ~»'" + P _(op cr(x/a, ~»"'] -L a1 a 1- 1
1=0
ao=...LJ J O'(x,~)mdx~, a1=_im(m-l)..LJ J aaaaam-2dx~ ~ 2 ~ ~~
where the second integration is taken over the region where x ¢ O.
PROOF. We invoke, as we did in I, a fact about pseudodifferential operators, in this case that
there is a representation op tl(x/a, ~) op t2(xla, ~) = op t(x/a, ~)
where t has the asymptotic expansion
~ (iar1--t t(x,~) - ~ -k' Cf~ tl(x/a,~) ~t2(x/a, ~).
1=0 •
This is the standard expansion for the symbol of a product [H, Th. 18.181, modified by the presence of the parameter a. What it means in this case is that the difference between t(x, ~) and
Widom 203
the partial sum ~k<K is of the fonn a-K tK(x/a, ~) with t I( a Schwartz function. By induction
there is an expansion for the symbol of (op t(x/Cl, ~»m beginning
t(x/a, ~)m - a-I f m(m - 1) d~t(x/a, ~) dx t(x/a, ~) t(x/a, ~)m-2 + ....
It follows that there are asymptotic expansions for the (discontinuous) symbols of P +(op cr±(x/a, ~»m beginning
- XR±(x) [ut(X, ~)m - a-I f m(m - 1) d~a~(x/a, ~) dxa±(x/a, ~) a±(x/a, ~)m-2 + ... ] .
From the symbol we obtain the kernel from (0.8) and integrate the kernel over the diagonal to get
the trace.
If we combine Sublemmas 3-5 and use (3.6) and the fact that (op a±)m = op (a±)m
since a± is independent of x, we see that Lemma A is proved.
If a2 = 0 in Lemma B then we are essentially dealing with Wiener-Hopf operators and
the result was proved in [W3]. Our method here is similar to the one there but the ending, the
derivation of the fmal fonnula, is a little different.
We begin with the caseftA.) = A.-I. Of course al and a2 cannot then be Schwartz
functions. We assume they are of the fonn constant (the same for both) plus Schwartz function. It
follows from our assumption that the convex hulls of the ranges of the ai do not contain 0 that the
ai have zero index, and hence have continuous logarithms taking the same values at ~ = ±oo. And
it follows from this that al/a2 has a continuous logarithm, which is in fact a Schwartz function.
We use this logarithm to construct the Wiener-Hopf factorization of al/a2.
We defme the operators t ~ 'tt by
(3.9) t±@=±.L2·f t(T\) dr1 = ± .L2· [PV f t(T\) dr1 + xi a@]. x T\-~±Oi x T\-~
Then (3.10) t = t+ + t_
and t+ (resp. t_) extends to a bounded analytic function in the lower (resp. upper) half-plane. We
define the Wiener-Hopf factors a± (which have no relation to the functions a± earlier in this
section) by
(3.11) a+= exp (IOg~) ,a-= exp (log ( 2) • a2 + a2 -
Thus al/a2 = a +/a - and a + (resp. a -) extends to be analytic, bounded, and bounded away from
zero in the lower (resp. upper) half-plane. The inverse of our operator is given by (P + op al + P _ op (2)-1 = op(al a -)-1 P + op a - + op(a2 a +)-1 P _ op a +.
204 Widom
This is an easy check, using the fact that op (f± leaves invariant the range of P ±. (Or see [OF,
p. 129].)
Next we observe that P + op <Jel + P _ op <J2-1 = P + op(<JI <J -)-1 op <J - + P _ oP(<J2 <J +)-1 op <J +
and so, subtracting,
(3.12) (P+op <JI +p_op <J2)-1- (P+op <Jel +p_op <J2-1)
= [oP(<JI <J -)-1, P +] op <J - + [oP(<J2 <J +)-1, P _] op <J +.
Here we have used the usual bracket notation to denote commutators. It is the trace of the left side
we are interested in.
We pause now for some notations and sublemmas. First we remark that the functions (f± appearing on the right side of (3.12) are no longer of the form constant plus Schwartz function,
so we must enlarge the class of functions we consider. We choose (t + S-. Here, of course, (t
represents the constant functions; S- denotes the class of what are called symbols of negative order, those COO functions 't in lR which satisfy estimates
't(k)(~) = O«~-.r-l) k = 0, 1, ...
for some s > O. The point is that the operators 't -+ 't:t send S- to S- and that, of course, ~(lR) c S-. We extend these operators to (t + S- by defining them to vanish on (t.
The Fourier transform of a function in S- is rapidly decreasing at 00 and O(IxIS-l) near
x= O. The Fourier transform of a function in (t + S- has in addition a &.summand. For 'tl, 't2 E (t
+S-wedefme
(3.1~) ('tlo't2)= J Xil(-x)i2(X)dx,
the integral taken over lR\{ O), where
i (x) = k J ei9 't(~) tt; .
SUBLEMMA 1. If'tl E (t + S- and 't2 E S- then tr [op 'tl, P ±] op 't2 = ± ('tl, 'tV.
PROOF. An easy check, using the fact that [op 'tl, P +1 = p _ op 'tl P + - P + op 'tl P_.
Here, incidentally, there is no guarantee that the operators are trace class. They do, however, have
rapidly decreasing kernels, continuous on the diagonal except at x = 0, and the trace is to be
interpreted as the integral of the kernel over the diagonal.
Widom 205
SUBLEMMA 2. /flog'ti e a: + S- then
('t I> 'tl) = t ('tl 't2, log *) . PROOF. This is left as an exercise for the reader; the proof can be found on pp. 293-4 of [W3].
SUBLEMMA 3. With the same assumptions as Sublemma 2 we have
('tIl, (log 't2>-)- ('t"i, (log't1)+)
= _...L ( ( 11 'tt(~, T\, t)-l ......JJl..- dr1 ~ 4x2 JIR JIR t(1 - t)(~ - T\ + 0i)2
where 'tt(~, T\, t) = (1 - t) 'tl(~) + t't2(T\)
and the integral is regularized by subtracting from 'tt(~, T\, t)-l its linear part.
PROOF. We begin with another representation of our bilinear form,
('tl> 't2) = -fit f 'tl(~) 't~(~) d~, which follows from (3.13) using Parseval's identity. It follows from this, and the defInition (3.9),
that
/'tI1,(IOg't2>-)=...Lf f 't1@-1Iog 't2(T\) drl~ , \ 4x2 (~- T\ + 0i)2
('t"21,(lOg't1)+)=-...Lf f 't2@-1Iog 't1(T\) drl~ . 4x2 (~ - T\ - 002
In the second integral we interchange the variables ~ and T\ and then the order of integration (which
can be justified). We obtain
(3.14)
The trick now is to apply the readily verified identity
11 {«I - t) a + t br1 - [(1- t) tr1 + t b-1]}......JJl..- = (1_1.) (log a -log b) o t(1- t) a b
with a = 'tl(~) and b = 't2(T\). Two of the four terms obtained by multiplying out the right side here
give exactly the negative of expression in brackets in the integrand in (3.14). The analogous
integral with the other two terms
206 Widom
vanishes since, for example,
J dr\ =0 (~-11 + Oi)2
for all~. Thus we see that the sublemma has been established.
We can now quickly finish the proof of Lemma B. First, it follows from (3.12) and
Sublemmas 1 and 2 that
~(P+ op 0'1 + P_ op 0'2r1- (P+ op 0'11 + P_ op 0'21)]
= t\O'l\ log 0'1(0')2)- t\0'21,log 0'2(0'°t·P).
If we use the definition (3.11) of O'± and identity (3.10) we fmd after a little computation that this
equals
\0'11, (log 0'2>-)- \0'1\ (log 0'1)+)'
+ } \0'11, (log 0'1)+ - (log 0'1>-) +} \0'21, (log 0'2)+ - (log 0'2>-).
We apply Sublemma 3 with 'tl = 0'1, 't2 = 0'2 to evaluate the first two terms here, the sublemma
with 'tl = 't2 = 0'1 to evaluate the third term, and the sublemma with 'tl = 't2 = 0'2 to evaluate the fourth. What results is (3.4) withj(A) = A-I.
Thus Lemma B is proved in this special case. To prove it in general we apply this result, with 0' replaced by A - 0', then multiply by f(A), and integrate with respect to A over the
usual contour.
IV. SYMBOLS DISCONTINUOUS ACROSS A HYPERPLANE IN IRnx IRn
Now we assume that 0' in (0.8) is smooth in the complement of a hyperplane 'JI1. in IR n
x IR n, and that all derivatives are bounded and rapidly deceasing as Ixl + I~I -+ 00, and we derive the
formula (0.18) for the coefficient cl in the expansion (0.16) from the formula in [W5]. To state it,
we denote by v a unit vector normal to 'JI1. in 1R~ x 1R! (the subscripts denoting the variables in the
ambient spaces) and by Vx and v~ the components of v in the directions of 1R~ and 1R!. These
components act on each other by the dot product We denote also by Bq(co), for a vector co E IRm,
the q'th smallest component of co. The formula for cI obtained in [W5] in the casej(A) = Am is
(4.1) C1 = tvx'v~ - Oi)1/2 ei7t/4 ~T) 1t(m-1)!2 ~ (~;)~J'lIl. O'+(x, ~)m-q cr(x, ~)q d(x, ~)
Widom 207
where O'±(x,~) denote the limits of 0' from the two sides of em. and where d(x,~) denotes
Lebesgue measure on em.. The coefficients ~m.q are given by
(4.2) ~m.q = L [l>q+I(ro) -l>q(ro)] dro (q = 0, ... , m)
where ;6 denotes the unit sphere in the subspace of R m given by l:roi = 0, and we make the
convention l>Q(ro) = l>m+l(ro) = O.
(The notation here is slightly different than in [W5]. The validity of the expansion (0.16) was proved under the assumption that Vx ·v~ ~ O. There is no doubt that it holds also
when vx'v~ = 0 although a separate argument would have been needed to handle this special case.)
We are going to evaluate fA l>q(ro) dro by relating it to
(4.3) L. e-~2l>q(X) dx
which, in tum, we shall express in tenns of the error function E(s) given by (0.19).
If we write 1\=(XE Rm:Ui=O}
then ;6 is the unit sphere in 1\ and R m is the orthogonal direct sum of 1\ and the space spanned
by the unit vector m-l12 I where I = (1, ... , 1). Thus any vector in R m has the representation
x +s m-j I with x E 1\ and s E R, and (4.3) becomes L L e-1x12 e-S2 l>q(x + s m-t I) ds dx .
If we observe that l)q(x + s m - tl) = l)q(x) + s m -! and that J e-,2 s ds = 0 we see that the above
is equal to
L.L e-1xj2 e-S2 l)q(x) ds dx = ]tIn L e-~2l)q(X) dx.·
Introducing spherical coordinates in the (m - I)-dimensional space 1\ and using the facts that 8q(x)
is homogeneous of degree 1 and that
f" e--rl Tnt-I dr = i r (tt) we deduce that (4.3) equals
208 Widom
Another way to evaluate (4.3) is to observe that it equals m C;ll times the integral over the region
(4.4)
The reason for this is that there are m possibilities for the q'th smallest component of x, and for
each of these there are C;ll possibilities for the q - 1 components which are smaller; by symmetry
the integrals over the regions corresponding to all choices are equal, and the region (4.4)
corresponds to a particular choice. Thus (4.3) equals
m cm- 1 e-Xix dx . II e-x1dx· . II e-x1dx· 100 lX, Loo q-l q q I I •
-- i<Ji -- i>q x,
Making the variable changes Xi --+ xl/2s in all the integrals and recalling (0.18) we see that this
equals
m C;! x(m+l)/2f~ S E'(s) E(s)q-l (1 - E(s»m-q ds .
Comparing our two expressions for (4.3) gives the desired formula
L Oq(CI) dID = 2 ~1-) -lxm12 m C;ll f~ s E'(s) E(s)q-l (1- E(s»m-q ds .
It follows that the constant ~m.q given by (4.2) is equal to 2 ~r.qr xm/2 times
m f~ s E'(s) [Cqn-l E(s)q (1 - E(S»m-q-l - C;: E(s)q-l (1 - E(s»m-q] ds .
An easy computation with the binomial coefficients shows that this is the same as
(4.5) cqn f~ s E'(s) [(m - q) E(s)q (1 - E(S)~-q-l- q E(s)q-l (1- E(S)m-q] ds .
The expression in brackets equals - .JL E(s)q (1 - E(s)m-q
ds and so if q is not equal to 0 or m integration by parts shows that (4.5) is equal to
(4.6) cqn f~ E(s)q (1 - E(s»m-q ds .
Here we used the fact that E(s) vanishes at -- and 1 - E(s) vanishes at +00. If q equals 0 or m
integration by parts like this fails, the last integral being divergent. To see what replaces it observe
that
Widom
[ sE'(s)ds =0
since E '(s) = ~ is even. Hence when q = m (4.5) is equal to
-f~ SE'(S)[mE(S)nt-l_ 1]ds
and integration by parts shows that this equals the convergent integral
f~ (E(s)'" - E(S») ds .
This is to replace (4.6) when q = m. Similarly its replacement when q= 0 is
f~ [(1- E(s»"'- (1- E(s»] ds.
209
Putting all this together with (4.1) and using the binomial theorem we fmd that in the case j(')..,) = ')..,m
Cl = (vx'v~ - 0i)1/2 ei'lt/4 n;/2 ( ( {[(1- E(s» cr(x,~) + E(s) <r(x, ~)r (211:) J'1fL JII.
-[(1 -E(s» cr(x, ~)'" + E(s) <r(x, ~)"']) ds d(x, ~).
If we write w(t) = dE-l (t)/dt , with E-l denoting the inverse function of E, and set E(s) = t the
inner integral becomes
f {[(I - t) cr(x, ~) + t <r(x, ~) r -[(1 - t) a+(x, ~)m + t <r(x, ~)m]} w(t) dt .
Thus, with cr* defined by (0.16) and our integrals regularized as usual, we have shown that for
polynomial f . 1/2 ei'lt/4 n1/2 (( ~ ~ Cl = (vx'V~ - 0,) (211:}1I k h. j(cr*(x, ." t» w(t) dt d(x, .,).
REFERENCES
[B] E. Basor. Trace formulas for Toeplitz matrices with piecewise continuous symbol. J. Math. Anal. Applic. 120 (1986).
[BS] A. Bottcher and B. Silbermann. Analysis of Toeplitz Operators. Akademic-Verlag, Berlin, 1989.
[F] X. Fu. Asymptotics of Toeplitz matrices with symbols of bounded variation. Ph.D. Thesis, Univ. of California, Santa Cruz, 1991.
210 Widom
[GF] I. C. Gohberg and I. A. Feldman. Convolution Equations and Projection Methods for Their Solution. Amer. Math. Soc. Trans!. Math. Monographs 41 (1974).
[GK] I. Gohberg and N. Krupnik. On the algebra generated by Toeplitz matrices. Funcl. Anal. Appl. 3 (1969).
[H] L. Hormander. The Analysis of Linear Partial Differential Operators III. SpringerVerlag, Berlin, 1985.
[L] L. M. Libkind. Asymptotics of the eigenvalues of Toeplitz forms. Math. Notes. 11 (1972).
[WI] H. Widom. Families of Pseudodifferential Operators. In Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York, 1978.
[W2] . On a class of integral operators with discontinuous symbol. J. Int. Eqs. Operator Th. 5 (1982).
[W3] . A traceformulafor Wiener-Hopf operators. J. Operator Th. 8 (1982).
[W4] . Asymptotic Expansion for Pseudodifferential Operators on Bounded Domains. Lecture Notes in Math. 1152, Springer-Verlag, Berlin, 1986.
[W5] . Asymptotic expansions and stationary phase for operators with discontinuous symbol. To appear.
Department of Mathematics University of California Santa Cruz, CA 95064
AMS Classification Numbers: Primary 45M05 Secondary 47B35, 47G05
211
UNIVERSITY OF MARYLAND AT COLLEGE PARK
DEPARTMENT OF MATHEMATICS
WORKSHOP ON CONTINUOUS AND DISCRETE FOURIER
TRANSFORMS AND EXTENSION PROBLEMS
Wednesday, Sept. 25, 1991
11:00 - 11:45 Cora Sadosky
11:50 - 12.35 Misha Cotlar
2:15 - 3:00
3.05 - 3:40
4:00 - 4:35
4:40 - 5:25
5:30 - 6:50
Robert Ellis
Israel Gohberg
Israel Gohberg
Rien Kaashoek
David Lay
Thursday, Sept. 26, 1991
September 25 & 26, 1991
Generalized Toeplitz kernels, weakly positive matrix measures and applications
Two-dimensional Bochner theorem and the theorems of Adamjan-Arov-Krein and Sz.Nagy-Foias
Orthogonal systems related to infinite Hankel matrices
Generalized Toeplitz kernels and the band extension method
Continuous and discrete Fourier transforms, and formulas of Szego-Kac-Achiezer
Connections between continuous and discrete maximum entropy principles
Distribution of zeros of matrix-valued continuous analogues of orthogonal polynomials
3:00 - 3:45 John Benedetto Continuous and discrete Fourier transforms, and sampling formulas
3:50 - 4:35
4:40 - 5:25
Seymour Goldberg Extensions of triangular Hilbert-Schmidt operators
Henry Landau
O~ganizing Committee:
Stability for continuous analogues of orthogonal polynomials
R. Ellis, I. Gohberg, S. Goldberg, and D. Lay
212
Titles previously published in the series
OPERATOR THEORY: ADVANCES AND APPLICATIONS
BIRKHAUSERVERLAG
1. H. Bart, I. Gohberg, M.A. Kaashoek: Minimal Factorization of Matrix and Operator Functions, 1979, (3-7643-1139-8)
2. C. Apostol, R.G. Douglas, B. Sz.-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Topics in Modem Operator Theroy, 1981, (3-7643-1244-0)
3. K. Clancey, I. Gohberg: Factorization of Matrix Functions and Singular Integral Operators, 1981, (3-7643-1297-1)
4. I. Gohberg (Ed.): Toeplitz Centennial, 1982, (3-7643-1333-1) 5. H.G. Kaper, C.G. Lekkerkerker, J. Hejtmanek: Spectral Methods in Linear Transport
Theory, 1982, (3-7643-1372-2)
6. C. Apostol, R.G. Douglas, B. Sz-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Invariant Subspaces and Other Topics, 1982, (3-7643-1360-9)
7. M.G. Krein: Topics in Differential and Integral Equations and Operator Theory, 1983, (3-7643-1517-2)
8. I. Gohberg, P. Lancaster, L. Rodman: Matrices and Indefinite Scalar Products, 1983, (3-7643-1527-X)
9. H. Baumgartel, M. Wollenberg: Mathematical Scattering Theory, 1983, (3-7643-1519-9) 10. D. Xia: Spectral Theory of Hyponormal Operators, 1983, (3-7643-1541-5) 11. C. Apostol, C.M. Pearcy, B. Sz.-Nagy, D. Voiculescu, Gr. Arsene (Eds.): Dilation Theory,
Toeplitz Operators and Other Topics, 1983, (3-7643-1516-4) 12. H. Dym, I. Gohberg (Eds.): Topics in Operator Theory Systems and Networks, 1984,
(3-7643-1550-4) 13. G. Heinig, K. Rost: Algebraic Methods for Toeplitz-like Matrices and Operators, 1984,
(3-7643-1643-8) 14. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu, D.Voiculescu, Gr. Arsene (Eds.): Spectral Theory
of Linear Operators and Related Topics, 1984, (3-7643-1642-X) 15. H. Baumgartel: Analytic Perturbation Theory for Matrices and Operators, 1984,
(3-7643-1664-0) 16. H. Konig: Eigenvalue Distribution of Compact Operators, 1986, (3-7643-1755-8)
17. R.G. Douglas, C.M. Pearcy, B. Sz.-Nagy, F.-H. Vasilescu, D. Voiculescu, Gr. Arsene (Eds.): Advances in Invariant Subspaces and Other Results of Operator Theory, 1986, (3-7643-1763-9)
18. I. Gohberg (Ed.): I. Schur Methods in Operator Theory and Signal Processing, 1986, (3-7643-1776-0)
19. H. Bart, I. Gohberg, M.A. Kaashoek (Eds.): Operator Theory and Systems, 1986, (3-7643-1783-3)
20. D. Amir: Isometric characterization of Inner Product Spaces, 1986, (3-7643-1774-4)
213
21. I. Gohberg, M.A. Kaashoek (Eds.): Constructive Methods of Wiener-Hopf Factorization, 1986, (3-7643-1826-0)
22. V.A. Marchenko: Sturm-Liouville Operators and Applications, 1986, (3-7643-1794-9) 23. W. Greenberg, C. van der Mee, V. Protopopescu: Boundary Value Problems in Abstract
Kinetic Theory, 1987, (3-7643-1765-5) 24. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu, D. Voiculescu, Gr. Arsene (Eds.): Operators in
Indefinite Metric Spaces, Scattering Theory and Other Topics, 1987, (3-7643-1843-0) 25. G.S. Litvinchuk, I.M. Spitkovskii: Factorization of Measurable Matrix Functions, 1987,
(3-7643-1843-X)
26. N.Y. Krupnik: Banach Algebras with Symbol and Singular Integral Operators, 1987, (3-7643-1836-8)
27. A. Bultheel: Laurent Series and their Pade Approximation, 1987, (3-7643-1940-2) 28. H. Helson, C.M. Pearcy, F.-H. Vasilescu, D. Voiculescu, Gr. Arsene (Eds.): Special Classes
of Linear Operators and Other Topics, 1988, (3-7643-1970-4) 29. I. Gohberg (Ed.): Topics in Operator Theory and Interpolation, 1988, (3-7634-1960-7) 30. Yu.I. Lyubich: Introduction to the Theory of Banach Representations of Groups, 1988,
(3-7643-2207-1)
31. E.M. Polishchuk: Continual Means and Boundary Value Problems in Function Spaces, 1988, (3-7643-2217-9)
32. I. Gohberg (Ed.): Topics in Operator Theory. Constantin Apostol Memorial Issue, 1988, (3-7643-2232-2)
33. I. Gohberg (Ed.): Topics in Interplation Theory of Rational Matrix-Valued Functions, 1988, (3-7643-2233-0)
34. I. Gohberg (Ed.): Orthogonal Matrix-Valued Polynomials and Applications, 1988, (3-7643-2242-X)
35. I. Gohberg, J.W. Helton, L. Rodman (Eds.): Contributions to Operator Theory and its Applications, 1988, (3-7643-2221-7)
36. G.R. Belitskii, Yu.I. Lyubich: Matrix Norms and their Applications, 1988, (3-7643-2220-9) 37. K. Schmiidgen: Unbounded Operator Algebras and Representation Theory, 1990,
(3-7643-2321-3) 38. L. Rodman: An Introduction to Operator Polynomials, 1989, (3-7643-2324-8) 39. M. Martin, M. Putinar: Lectures on Hyponormal Operators, 1989, (3-7643-2329-9)
214
40. H. Dym, S. Goldberg, P. Lancaster, M.A. Kaashoek (Eds.): The Gohberg Anniversary Collection, Volume I, 1989, (3-7643-2307-8)
41. H. Dym, S. Goldberg, P. Lancaster, M.A. Kaashoek (Eds.): The Gohberg Anniversary Collection, Volume II, 1989, (3-7643-2308-6)
42. N.K. Nikolskii (Ed.): Toeplitz Operators and Spectral Function Theory, 1989, (3-7643-2344-2)
43. H. Helson, B. Sz.-Nagy, F.-H. Vasilescu, Gr. Arsene (Eds.): Linear Operators in Function Spaces, 1990, (3-7643-2343-4)
44. C. Foias, A. Frazho: The Commutant Lifting Approach to Interpolation Problems, 1990, (3-7643-2461-9)
45. J.A. Ball, I. Gohberg, L. Rodman: Interpolation of Rational Matrix Functions, 1990, (3-7643-2476-7)
46. P. Exner, H. Neidhardt (Eds.): Order, Disorder and Chaos in Quantum Systems, 1990,
(3-7643-2492-9) 47. I. Gohberg (Ed.): Extension and Interpolation of Linear Operators and Matrix Functions,
1990, (3-7643-2530-5) 48. L. de Branges, I. Gohberg, J. Rovnyak (Eds.): Topics in Operator Theory. Ernst D.
Hellinger Memorial Volume, 1990, (3-7643-2532-1)
49. I. Gohberg, S. Goldberg, M.A. Kaashoek: Classes of Linear Operators, Volume I, 1990, (3-7643-2531-3)
SO. H. Bart, I. Gohberg, M.A. Kaashoek (Eds.): Topics in Matrix and Operator Theory, 1991, (3-7643-2570-4)
51. W. Greenberg, J. Polewczak (Eds.): Modern Mathematical Methods in Transport Theory, 1991, (3-7643-2571-2)
52. S. Prossdorf, B. Silbermann: Numerical Analysis for Integral and Related Operator Equations, 1991, (3-7643-2620-4)
53. I. Gohberg, N. Krupnik: One-Dimensional Linear Singular Integral Equations, Volume I, Introduction, 1991, (3-7643-2584-4)
54. I. Gohberg, N. Krupnik (Eds.): One-Dimensional Linear Singular Integral Equations, 1992, (3-7643-2796-0)
55. R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii: Measures of Noncompactness and Condensing Operators, 1992, (3-7643-2716-2)
56. I. Gohberg (Ed.): Time-Variant Systems and Interpolation, 1992, (3-7643-2738-3) 57. M. Demuth, B. Gramsch, B.W. Schulze (Eds.): Operator Calculus and Spectral Theory,
1992, (3-7643-2792-8)
![Politecnico di Torino Porto Institutional Repository · 2017. 12. 15. · Politecnico di Torino Porto Institutional Repository [Book] The Wiener-Hopf method in electromagnetics Original](https://img.dokumen.tips/doc/110x75/5fdf1d7303a41571fa017e84/politecnico-di-torino-porto-institutional-repository-2017-12-15-politecnico.jpg)
