Two Projection Theorems and Symbol Calculus for Operators with Massive Local Spectra

Math. Nachr. 162 (1993) 167-185

Two Projection Theorems and Symbol Calculus for Operators with Massive Local Spectra

By T. FINCK, S. ROCH, and B. SILBERMANN of Chemnitz

(Received October 29, 1992)

Abstract. In this paper we construct a symbol calculus for Banach algebras generated by two idempotents and a coefficient algebra. This, combined with local principles for “embedding algebras”, leads to a symbol calculus for singular integral operators on spaces with Muckenhoupt weight and for singular integral operators with measurable coefficients.

1. Introduction

In the last decade notable advances in understanding the structure of concrete Banach algebras of operators had been made. Most of these new insights are essentially based on only two simple observations which are characteristic for a large variety of operator algebras, and on two closely related technical ingredients. The first observation is that the Calkin image of many operator algebras contains a rich center which offers the applicability of local techniques such as the local principles of ALLAN (see below) or of GOHBERG/KRUPNIK. In its consequence this associates with each operator algebra a whole family of small or local algebras which are labeled by the points of a topological space, viz. the maximal ideal space of the center. In other words, operator algebras can often be viewed as Banach algebra bundles. Now the second observation comes to light: the local algebras (or, what is the same, the fibres of the algebra bundle) are of a remarkable simple structure, namely, they are generated by two idempotent elements, and so they are subject to so-called two projection theorems.

The idea of combining local principles with two projection theorems has been successfully employed for algebras generated by one-dimensional singular integral operators with piecewise continuous coefficients on different function spaces (see, e.g., [9], [lo], [ll]), for algebras of Wiener-Hopf and multiplication operators (see [ll]), for algebras of Toeplitz operators with piecewise continuous or piecewise quasicontinuous generating functions (see [8], [l], [13]), for algebras of Toeplitz and Hankel operators or of singular integral operators with Carleman shift (see [11]), and so on.

Nowadays, in all of these situations, effective symbol calculi for Fredholmness are available (for instance, the famous symbol calculus of Gohberg and Krupnik for

168 Math. Nachr. 162 (1993)

singular integral operators [2] can be derived in this way), one can describe algebraic invariants of these algebras (their centers, commutator and quasicommutator ideals, . . .), and one knows such things as their inverse closedness.

However, these results base heavily on the fact that all algebras mentioned above possess dense subalgebras the elements of which have thin spectra, i.e. spectra without inner points, whereas, on the other hand, there are simple examples leading in a natural way to elements with thick or massive spectra. Consider, for instance, the smallest closed unital subalgebra alg( p , q) of the algebra Y(L'(T)) of all bounded linear operators on the Lebesgue space Lr(T) ( I # 2) over the unit circle T, which contains the operator p of multiplication by the function

1 I m t 2 0 , 0 I m t < O ,

1 2

and the operator q = -(I + S ) where

Notice that p and q are idempotents and that the spectrum of p q p contains inner points. So, the theory of [lo] applies, and gives an effective criterion for invertibility in alg( p , q). But, since nobody knows whether alg( p , q) is inverse closed in 9(Lr(T)) , an invertibility criterion for elements of alg( p , q) in 9 (Lr (T) ) remains unknown.

Analogous questions arise also on a much higher level. I. SPITKOVSKI had investigated Fredholmness of singular integral operators with piecewise continuous coefficients on Lr-spaces with general Muckenhoupt weights. He pointed out that each dense subalgebra of the Banach algebra generated by all of these operators contains elements with massive essential spectra, which involves serious difficulties in establishing a symbol calculus or the inverse closedness.

It is the aim of the present paper to explain how to overcome these problems. The crucial point is to embed the algebra one is primarily interested in into a larger Banach algebra which is, on the one hand, large enough to be inverse closed in the algebra of all operators and, on the other hand, small enough to be accessible to local principles. A prominent example for an embedding algebra is the algebra of all operators of local type, say on L'(T). This algebra is defined as the collection of all bounded operators on Lr(T) which commute modulo compact operators with each operator of multiplication by a continuous function on T. The Calkin image of this algebra has both a rich center (viz. the algebra of all continuous functions on T) and it is clearly inverse closed in the Calkin algebra.

Before establishing a symbol calculus in SPITKOVSKI'S context and in some others, we shall formulate and prove a generalization of the two projection theorem which is not only our starting point but certainly of its own interest. Suppose we are given a unital Banach algebra 9, idempotents p and q in 9(i.e. p z = p , q2 = q), and a unital subalgebra '3 of 9 the elements of which commute with both p and q. In what follows, we refer to 9 as embedding algebra and to B as coefficient algebra. Now consider the smallest

Finck/Roch/Silbermann, Two Projection Theorems 169

closed subalgebra d of 9 which contains p , q, and 9. Our goal is to find criteria for the invertibility of elements of d in 9. Notice that the case where d = 9 is a C*-algebra, Y = C, and p and q are self-adjoint, was subject of investigation by many people (see [3], [7], [8], [16] and others). The nice result reads as follows:

If the spectrum of p q p is the closed interval [0, 11, then the algebra d = alg( p , q) is isometrically isomorphic to the algebra of all continuous 2 x 2-matrix valued functions

with

flZ(0) =f21(0) = f d ) =f21(1) = 0 .

Thereby, the elements p , q, e go over into the functions

respectively. Only a few years ago, an analogous result for Banach algebras generated by two

idempotents was achieved in [lo]. The methods used there are quite different from the C*-setting. They result from a combination of Allan’s local principle and of Krupnik’s theory of algebras with polynomial identity (see [6]). Here we shall generalize these results into three directions: the hypotheses for the spectrum of p q p are less restrictive than in [lo], the coefficient algebra must not be trivial, and we consider all things in an embedding algebra to handle massive spectra.

2. Spectra, symbols, inverse closedness

Let us start with fixing some notations. If d is a Banach algebra with identity e and a E d, then the spectrum aJa) of a in d is the set of all complex numbers A such that a - l e is not invertible in d. If, in particular, X is a Banach space, Y ( X ) the Banach algebra of all linear bounded operators on X , and X ( X ) the ideal of all compact operators in 5?(X), then the essential spectrum of A is the spectrum of the coset .n(A):=A + X ( X ) in the Calkin algebra 5 ? ( X ) / X ( X ) . It is well known that the spectrum of an element of a unital Banach algebra is a non-empty and compact subset of the complex plane C.

Now let d, 39, %, 9 be unital Banach algebras with d c 39, %? c 9, and let W : 39 + 9 be a bounded linear mapping. This mapping is called a symbol map for d if

(i) W ( d ) z W .

(ii) W(e) = e .

(iv) invertibility of W(a) in 9 implies invertibility of a in 39 for all a E d.

170 Math. Nachr. 162 (1993)

W(a) is then referred to as the symbol of a. If Wis a symbol map and a E d is invertible in 98, then, by (ii) and (iii), the symbol W(a) of a is invertible in 9. Since the reverse implication holds by (iv), we have

(1) o,(a) = o,(W(a)) for all a E d.

Let us introduce the notation smb(d, W; V, 9) for the set of all symbol maps in sense of (iHiv), and abbreviate smb(d, d; V, 9) to smb(d; V, 9), smb(d, d; 23, 9) to smb(d; 9). The elements of the latter set are clearly Banach algebra homomorphisms which are usually referred to as symbols. The necessity to generalize this standard notion of a symbol comes from the idea of embedding d into a larger algebra, W. Finally, the subalgebra d of 98 is said to be inverse closed if the embedding operator from d into 98 belongs to smb(d; B) or, in other words, if invertibility of a in &implies invertibility of a in d for all U E ~ or, again equivalently, if

(2) oJa) = a,(a) for all a E d .

It is obvious that

(3)

for all V c 9 and that

smb(d, 98; V, 9) c smb(d, 9; 9)

(4) srnb(d, 98; V) E smb(d, 97; V, 9)

whenever W is inverse closed in 9. The latter inclusion is also a consequence of the general implication

( 5 ) W, E smb(d, 98; V, 9), W, E smb(V, 9; 8, 9) ==- W, 0 W, E smb(d, W; 8,9).

3. Allan’s local principle

In this section we are going to summarize some facts around Allan’s local principle. Let d be an arbitrary Banach algebra with unit element e and V be a central closed subalgebra of d containing e (recall that W is central in sit if ac = ca whenever a E d and c E V). For every maximal ideal t of the commutative Banach algebra W, we let I , stand for the smallest closed two-sided ideal of d which contains t , and we write d, for the quotient algebra &/Iz and a, for the coset a + I , containing a E d. Possibly, I, = d. In this case, d, is an algebra whose only element is simultaneously the zero and the identity, and we make the conventions that ((a,(/ = 0 and that a, is invertible for every a ~ d .

Finally, let M ( V ) refer to the maximal ideal space of V. Then Allan’s local principle reads as follows.

Theorem 1. (a) The element a E d is invertible in d if and only if the cosets a, are invertible in A, for all t E M(V). (b ) The mapping t H ((atl( assigning to each maximal ideal a non-negative number is upper semi-continuous for anyfixed a E d. In particular, ifa E d and a,, is invertible in d,,, then a, is invertible in d, for all t belonging to a neighborhood of to.


Let Mo(W) stand for the collection of those t E M ( W ) for which I,# d. The set Mo(V) is closed in M(V). Indeed, if to~M(V)\Mo(%'), then O,, is invertible in dt0 by definition, whence by Theorem l(b) it follows that 0, is invertible in d, for all t in a neighborhood of to. But 0, is the zero element in d,, thus, d, = (0,) or, equivalently, t E M(V)\M,(W). This shows that the complement of Mo(W) is open, hence Mo(W) is closed.

It is evident that Theorem 1 remains valid if M(W) is replaced by its compact subset Mo(%) or by any other subset containing Mo(V). Next we describe a suitable subset. Let MG(W) stand for the collection of all maximals ideals t of V which contain in d invertible elements.

Proposition 1. Mo(W) c M(%)\MG(%').

Proof.Let usshow that MG(%) E M(W)\M,(W).Let tEMG(W)andcetbeinvertiblein d. Then, since t E I,, c is an invertible element in the ideal I, whence I , = d follows. Thus

In what follows we shall employ this simple assertion for singly generated algebras V, that is, we suppose that there is a c E d such that W is the smallest closed subalgebra of d containing c and the identity e. It is well known that M(W) is homeomorphic to the spectrum oe(c) of c in W, and we shall always identify M(W) and a&$.

t E MG(%)\Mo(%).

Proposition 2. If % is a singly generated ( b y c) central subalgebra of d, then M O W S ) = %&).

Proof. Let t E Mo(W), i.e. I, # d. Then (c - te), = 0, is not invertible in d, and via Theorem 1 it follows that c - te is not invertible in d. That means, t E ad(c).

Let now t E ad(c), i.e. c - te be not invertible in d. Theorem 1 yields that there exists a z E CT%(C) for which (c - te), is not invertible in d,. But since for all z # t the element (c - te), = ((7 - t)e), is invertible in dz, the element (c - te), = 0, can't be invertible in d,. Thus I, # d, i.e. t E Mo(W). H

Consequently, if %? is singly generated, then the set M ( B ) in Theorem 1 can always be replaced by ad(c).

4. The two projections theorem

As in the introduction, let 9 be a unital Banach algebra with identity e, p and q idempotents in 9, 9 a unital subalgebra of 9 the elements of which commute with p and q, and denote by d the smallest closed subalgebra alg,(9, p , q) of 9 containing 8, p and q.

First of all one observes that the element ( p - q)' commutes with both p and q, and therefore with all elements of d. Hence the smallest closed subalgebra alg(p - q)' of d, which contains ( p - q)' and the identity e, is a central subalgebra of d. This simple fact will prove to be extremely useful in what follows.

(6)

Let us suppose here and hereafter that

0, 1 E C are not isolated points of oF(( p - 4)').

172 Math. Nachr. 162 (1993)

Before stating our version of the two-projections theorem we have to introduce some more notations. We let B stand for the set

{ f E 9 : ( P - 4 I 2 f = f ( P - d 2 } ’ Obviously, A9 forms a closed subalgebra of 9, W is inverse closed in 9, and W contains

d. Thus, our problem of investigating invertibility of elements of d in 9 can be reduced to study invertibility of elements of d in W.

Further, it is immediate from the definition of W that alg( p - 4)’ is a singly generated central subalgebra of 93. For every maximal ideal t E M(alg( p - 4)’) = aa,g(p-qp(( p - 4)’) (in fact it will suffice to consider t E a& - 4)’)) we let I , stand for the smallest closed subalgebra of A9 which contains t, and we write W f for the quotient algebra W/I, , b, for the coset b + I , containing b E W, and df, Y, for the images of d and Y under the canonical homomorphism b -+ b,.

In case the intersection as(( p - q)’)n (0, 1) is not empty, we further need the following hypothesis:

(7) df is inverse closed in Wt for t E as(( p - q)2)f l (0, l} . If 0 E a*(( p - q)’), then we let f, refer to the smallest closed two-sided ideal of do which

contains ( p - q),. Possibly, this ideal consists of the zero element only, but we shall see later on that f, is always proper. Further, we let I , , and I , , stand for the smallest closed

ideals of the quotient algebra do/fo which contain the cosets e, - ~ + 4o + f, and

+ 4o + f,, respectively. We write sdo0 and do, for the algebras (do / fo ) / Ioo and 2

(d,/f,)/I,,, and a,, and a,, for the cosets a, + f, + I , , and a, + f, + I , , , respectively. Analogously, if 1 E O ~ ( ( p - q)’), then f, stands for the smallest closed two-sided ideal of

d, containing (e - p - q),, and I , , and I , , denote the smallest closed two-sided ideals of

2

+ fl and 2 2 + p1 - “ + f,, respectively. The quotient el - P 1 + 41 d/ f , which contain ~ ~

algebras (dl/~,)/I1, and ( d , / f , ) / I , , will be abbreviated to d,, and dl and the cosets a , + fl + I , , and a , + f, + I , , to a , , and a , , .

for the algebra of all 2 x 2 matrices with entries in a given Banach algebra W and, if Wl and W2 are Banach algebras, then we denote by Wl x W2 the Banach algebra of all ordered pairs (cl, c2) with c, EW,, c ~ E W ~ , provided with componentwise operations. For convenience, we identify each ordered pair (c,, c2) with

the diagonal matrix

Now we have:

Finally we write W2

Theorem 2. Let d, B, 9, B and p , q be as above and suppose (6 ) and (7 ) to be fulfilled. Then there exists a symbol map

4 E smb(d, A9; 9),

where 9 stands for the algebra of all functions on osF((p -4)’) taking at


tEaF((p-q)2)\{0, l } a v a l u e i n 9 ~ x 2 a n d a t t E a s ( ( p - q ) 2 ) n { 0 7 l } a v a l u e i n 4 , x dl. This symbol map sends g E g7 p , and q into the functions

respectively. Herein, 1/ refers to an arbitrary branch of the square root function. Taking into account the inverse closedness of W in 9, one can reformulate this assertion as follows:

(8) An element a E d is invertible in 9 i f and only if its symbol &(a) is invertible in 9.

Proof. We shall verify the existence of symbol maps

(9) 1 0 1)g.7 (o 1 0 0)et7 (p- y e t t (1- t ) t

1 7 4 Math. Nachr. 162 (1993)

if t E oF((p - q)')\{o, I}, into

if tEo9((p-q)')n{O}, and into

if t E o,((p - q)')n {I}.

(12) t - 4i(4

Then Theorem 1 entails that the mapping 4 sending bE98 into the function

is the desired symbol map, and we are done. Let us introduce the elements N : = e - p - q and S:=p - q. It is easily checked that

(13) S 2 + N 2 = e and S N + N S = O .

We first suppose that t E a,(S2)\{0, l}. Then te, - S: = 0, that is S: = te,, and from (13) we conclude that N : = (1 - t ) e,. Let c1 and P denote complex numbers such that a2 = t

and P' = 1 - t, and set s, = - S , and n, = -N, . Then we have

(14) s," = n," = e, and s,n, + n,s, = 0.

1 1 c1 P

Consider the mapping

The matrices standing left and right from (2 l) in (15) are inverse to each other.

Thus, q, is a Banach algebra homomorphism, and we claime that qt is even a symbol map, precisely

q, E smb(98,; 9?; ') .

Indeed, if qt(bt) is invertible in W : x 2 , then (:, :) is invertible in and

this clearly implies invertibility of b, in 98,. Let us emphasize that qt maps g, E $,, s,, and n, into the matrices


respectively. For the next step, we abbreviate by p + , p - , a n d j the matrices

respectively, and define the mapping p, by

P, : %-@ ”, b,Hp+v,(b,) P+ + p-jvl,(b,)jp- . We are going to explain that p, is a symbol map, more exactly,

(17)

It is evident that pt is a linear bounded operator from at into a:”, and taking into account that p : = p + , p ? = p - , j ’ = e, and p + + p - = e, it is also obvious that p, is unital. Let us verify condition (iii) from section 1; that is, given a, E d, and b, E a, we have to show that

p, E smb(d,, W,; W,” ’) .

(18) Pt(a,)Pt(b,) = P t ( 4 d and Pt(bt)Pt(4 = Pt(bt4 *

Explicitly written, the left hand side of the first identity in (18) equals

(19) (~+vl,(a,)p+ + p-jrl ,(a,)jp-)(p+v,(b,) P+ + P-jn,(bJjp-)

= ~+v, (a , )p+v , (b , )~+ + p-jv , (a , )p-vAWp_- . Now use that p+ and p- commute with each element of q,(d,) since the cosets g, E Q,, s,,

and n, span the whole algebra dp Thus, (19) is nothing else than

P+ vt(a3 v,(b,)p+ + p-jvt(a,) r,(bJjp- = rur(arbt).

The second identity in (18) follows analogously. It remains to show that invertibility of ’ implies that of a, in 99, for all a, E d, or, since qt is a symbol map, that of q&) p,(u,) in in B,” ’. Let m E 93: be the inverse of p,(a,). A straightforward computation gives

P+Pt(%)P+ + P-jP,(at)jP- = v t ( 4 for all a, E dr Thus,

vXa3 ( P+ mp+ + P- jmjp- )

= ( P+ ~, (a , ) P+ + p-juXa3jp- 1 ( P+ mp+ + p-jmjp- 1

whence it follows that qt(ut) is invertible in B; ’ and that

v t W = P+ mp+ + p- jmjp -

This proves (17) To finish the proof in case t E o#’)\{O, l}, we choose complex numbers y, 6 satisfy-

176 Math. Nachr. 162 (1993)

ing y z = 1 + a, ti2 = 1 - a, and define a matrix by

Since d 2 = rt O), it is obvious that the mapping 0 et

4, : 9YZw99; ’, b p d p , ( b , ) d

belongs to smb(d,, a,; &I: ’), and direct computation yields that 4,(gt), 4,( p,) and 4,(qt) are just the matrices (9) as desired.

Suppose now t = OE a,(S2). Then one can argue as above to see that Si = 0, and thus So and N o are subject to the identities

(20) Si = 0, N ; = e,, S O N , + N O S , = 0.

claim is that the canonical homomorphism If a, E do is invertible in go, then it is also invertible in do by hypothesis (7). Our first

v, : do + do/fo, U O H U , + r, is a symbol map, precisely

(21) v, E smb(d,; do/fo).

It clearly suffices to show that the ideal f, belongs to the radical Rad do of do or, since To is generated by So = po - q,, that So E Rad do. Equivalently (see [5]) , we have to show that e, + a,S, is left invertible in do for all a, E do. But this is evident since

(e, - a,S,)(e, + a,S,) = e, - a,S,a,S, = e,

(observe that each element a , ~ d , can be approximated by elements of the form ghe, + giS, + g;N, + g$,N, with gb E B,, and then use (20) to find that a,S,a,S, = 0).

Let p+ and p- stand for the elements v, t o _____ and v, ( ~ e~ 1 in do/r0. Since

N ; = e,, these elements are obviously idempotents, moreover, they are non-trivial idempotents, that is, neither p+ nor p- is invertible. Indeed, suppose for example that p +

is invertible. Then ~ eo + N o is invertible in do by (21), and since it is an idempotent, we

have - = e , or, what is just the same, N o = e,. On the other hand, SE = 0, and these

two identities give p o = qo = 0. Consequently, we would get that e, - po( = e,) is invertible in do, and now the second part of Allan’s local principle entails that e, - pr is invertible in gt for all t E a,(S2) belonging to a neighborhood of 0 (such points exist by hypothesis (6)). But, as we have already seen,

2 eo +No

2


which fails to be invertible. This contradiction shows that p+ is not invertible, and analogously the non-invertibility of p- follows.

As a little thought shows, each element of do/fo can be approximated by elements of the form

(22) ( g , ' + f , ) p + +(go +f,)p- with go+, go€%,.

This yields that the collection of all linear combinations ap+ + P p - , a, f l E C, forms a central subalgebra of do/fo which suggest to study invertibility of elements (22) by invoking Allan's local principle again. Since the central subalgebra is singly generated by p + , its maximal ideal space is homeomorphic to the spectrum of p + , and the latter equals (0, l} because of the non-triviality of p + . The corresponding local ideals are just the ideals I,, and IOLintroduced above. Hence, by Theorem 1, a, E do is invertible in do if and only if (a, + I,) + I,, =: a,, is invertible in do, and (a, + f,) + I,, =:aol is invertible in do, or, equivalently, if the matrix

is invertible in do, x dol. The simple observation that the homomorphism assigning to a , ~ d , the matrix (23) sends the elements g 0 E 9 , , po and qo into the matrices (10) completes the proof for t = O E & S ~ ) . In case t = l ~ o ~ ( S ~ ) the proof runs analogously.

Let us complete and comment this result by a few results and corollaries. The first remark concerns the repeated factorization at the points 0 and 1 which seems to make the things rather non-transparent. This point of view is in a sense inadequate since each factorization evidently simplifies invertibility criteria (the coset containing a given element is rather invertible than the element itself). But, on the other hand, cosets often give the impression of somewhat mysterious objects. We want to moderate this impression by showing that the quotient algebras by the ideals f, are always of a simple structure.

Remark 1. (a) The algebra do is the direct sum of its subalgebra which contains 9, and N o and of the ideal f, that is,

( b ) The quotient algebra &,Ifo is isomorphic to alg(%,, No) .

, respectively, and eo + N o anA eo - N o 2

Proof. Let p + and p - stand for the idempotents ~

2 define a mapping o : dowd0 by

4ao) = P + a o P + + p-aop-.

Straightforward computation shows that o is a Banach algebra homomorphism and, since p + and p - are complementary idempotents, that o(o(a,)) = ~(a,). The identity o2 = o implies that do decomposes into the direct sum

do = I m o + Kero. 12 Math. Nachr., Bd. 162

178 Math. Nachr. 162 (1993)

Observing that w(go) = g o for g o E go, @(No) = No, and w(S,) = 0, we find that Imw = alg(g0, N o ) and Kerw 3 ro. Now remember that each element of do can be approximated by elements of the form gAeo + g i S 0 + g:No + g$,N_,, g b E 9,. These elements are evidently in alg@,, N o ) + f o which shows that Kerw E I , and finishes the proof of part (a). Part (b) is an immediate consequence of (a).

There is a nice situation where actually no further factorization at 0 and 1 is needed:

Corollary 1. Let the hypothesis of Theorem 2 be in force and suppose moreover that

(24) 9, is simple for tEoF(S2)n(0 , 1 ) .

Then there exists a symbol map $I E smb(d, 9; 9), where now 9 stands for the algebra of allfunctions on oF(S2) taking at t E aF(S2) a value in 99: '. This symbol map sends g E 9, p , and q into the functions

respectively. The proof is based on the simple fact that, if Vl and V2 are Banach algebras and

H : Vl --+ V2 is a homomorphism, then Im H is isomorphic to WS,/Ker H . If, moreover, V, is simple, then either Ker H = (0) or Ker H = Vl, whence it follows that either Im H is isomorphic to %, or Im H = (0). Thus, ifgo is simple, then goo is either isomorphic to go or consists of the zero element only, with the latter being impossible since p + and p - generate proper ideals as we have already seen. Applying this isomorphism to the symbol map quoted in Theorem 2, we obtain our corollary.

In case 9 is simple, then the very same arguments show that (24) is satisfied. In particular, Corollary 1 holds with 9 = Ck '. For the special case k = 1 this has been the subject of [ l o ] by two of the authors (but under stronger conditions imposed on the spectrum of S'). We are greatful to N. L. VASILEVSKI, who pointed out to the authors that some part of the proof given there can be modified. We employed this modification to prove (9). To avoid misunderstandings, let us mention a formal difference between [ lo ] and Theorem 2 above. The central role in [RS 11 is played by the element p q p + (e - p ) (e - q) (q - p ) instead of S '. Since evidently

p q p + (e - p)(e - q)(e - p ) = e - S 2 ,

we have

4 P 4 P + (e - P I @ - -PI) = 1 -

(the algebraic difference!) whence it follows that the symbol maps of [lo] and of Theorem 2 can be transformed to each other by substituting t t t 1 - t. The advantage of choosing p q p + (e - p ) ( e - 4) (e - p ) in place of S is that the spectra of p q p + (e - p ) (e - q) ( e - p ) and p q p coincide under suitable conditions and that, in many applications, p q p proves to be a local Toeplitz operator which is a well studied object with well known spectrum.

Next we discuss the inverse closedness of d in 99.


Corollary 2. Let all hypotheses of Theorem 2 be satisfied and suppose moreover that

(25) o-(SZ) = Od(S2)

and

(26)

Then d is inverse closed in 9, and the mapping &from Theorem 2 is even a symbol map for invertibility in d, i.e. & E smb(d; 9).

Proof. Let & refer to the symbol map established in Theorem 2 and &‘ to the symbol map resulting from this theorem by specifying 9 = d. Both mappings coincide on d because of (25), and it is easy to see that &,(a) and &:(a) belong to Q: for all tEo9(S2)\{0, l } and a e d . Thus, if a E d is invertible in 9, then +Jt(a) is invertible in 9:” and, by (26), even in 9:”. This involves invertibility of &;(a) in Q:x’ and, consequently, that of a in d.

The following remark contents a simple but often very effective criterion for deciding the inverse closedness of given algebras.

8: is inverse closed in g: ’for t E osF(S2)\{0, l } .

w

Remark 2. Let 2’1 and 2’’ be unital Banach algebras and 2’1 E z2. I f contains a dense subalgebra LY0 the elements of which have thin spectra in Y1 then Y1 is inverse closed in 2’z. (Recall that a subset of the complex plane is called thin if it does not contain inner points.)

Proof. Let c E 2’o. Then the spectra 09,(c) and a9,(c) can only differ by a connected component of C\o,,(c) (see [12]). Since a9,(c) is thin, we have necessarily 09,(c) = 09,(c), whence it follows that the elements in are invertible in Yl if and only if they are invertible in g2. Now let ~ € 2 ’ ~ be invertible in gZ. Approximate c by a sequence (c,)E 2’o. For large n, the c, are invertible in 2’’ too, and it is easy to check that c,-’-+c-l as n + m . But C , - ’ E ~ ’ ~ , and, since S1 is a closed subalgebra, we conclude that c- E g1.

For example, if $2 is isomorphic to Ck k, then Y is simple, and each element of Q and Q2” has a discrete (and thus thin) spectrum. Combining Corollaries 1 and 2 and Remark 2 yields:

Corollary 3. Let d, 9Y, 9, Q and p, q as above, suppose that (6) and (25) are jiulljXed, and let 92 be isomorphic to Ckxk (for simplicity we shall identifv Q with Ckxk in what follows). Then there exists a symbol map & E smb(d, 9; 9) where now 9 stands for the algebra of all functions on oJ(p - 4)’) taking values in CZk 2k. This symbol map sends g E Ck = 59, p , and q into the matrix functions

where e denotes the k x k unit matrix. Moreover, d is inverse closed in 9, and & even belongs to smb(d; 9).

12.

180 Math. Nachr. 162 (1993)

5. Singular integral operators on spaces with Muckenhoupt weight

Let T denote the complex unit circle and let p be a non-negative function on T which does not vanish almost everywhere. For 1 < p < 00 consider the space Lp( p)

with dm referring to the Lebesgue measure. The function p is called a Muckenhoupt

weight, and we write p E A, in this case, if p E Lp( l), p - E Lq( 1) with - + - = 1, and if 1 1 P 4

where the supremum is taken over all subarcs Z of T, and ) Z I denotes the arc length of Z. Further, we let S stand for the singular integral operator on T,

, tET. s - t

T

The problem of describing all weights p with the property that S maps LP(l)flLP(p) into itself and extends from LP(l)nLP(p) to a bounded operator on all of Lp(p) was solved by HUNT, MUCKENHOUPT and WHEEDEN [4]: they showed that S is boundedly extendable to Lp(p) if and only if p E A,. Examples for elements in A, are given by the well known Khvedelidze weights. These functions are of the form

(27) n

p(t)= n I t - tk lQk with t, tkET, CtkER, k = 1

1 P

and they belong to A, if and only if 0 < - + Ctk < 1 for all k = 1,. . ., n.

Throughout what follows let p E A,. Later on we shall need a result of SPITKOVSKI concerning essential spectra of Toeplitz operators on weighted Hardy spaces which we are going to explain now.

Let Pi :=-(I k S). These operators are bounded projections (i.e. P: = Pi) on Lp(p). The 1 2

Hardy space H P ( p ) is defined as the image of P+ in Lp(p), that is,

HP(p):=P+LP(p) .

Further, given a function a E L"(T), the Toeplitz operator T(a) is the operator on HP(p) which sendsfe H p ( p ) to P+(af). Since p E A , this operator is clearly bounded. For p EA, and S E T we set

ZJp, p ) = { r ~ R : I t - s l ' p ( t ) ~ A , ) .


One can prove (see [15]) that I , (p , p ) is an open interval of length not greater than 1 containing the origin and, conversely, for any prescribed interval (-a, p) of length o! + p < 1 containing 0 there exists a weight p EA, such that (-a, 8) = I J p , p ) . If, for example, p is the weight (27), then

and

Now define numbers v , f ( p , p) by

Finally, given two real numbers y, 6 such that 0 < y I 6 -= 1 and two complex numbers z, w, we introduce the (y, d)-horn joining z to w as the set

H(z, w; y, a):= u w, w, P I , l 4 y . a

where B(z, w, p) stands for the circular arc

5-z 5 - w

<EC\{Z, w } : arg- = 27cp

Note that 0 E H(z , w; y, 6) if and only if z # 0, w # 0, and arg $[27cy, 2n4. Now the

main result of [15] can be restated as follows:

Theorem 3. Let p E A , and a be a piecewise continuous function (that is, a function having one-sided limits a(t f 0) at each point of the counterclockwisely oriented unit circle). Then the essential spectrum of T(a) on Hp(p) is the set

a p , p : = u H M t - 01, a(t + 0); v; (P, PI, v:(P, P)) * RT

Let PC stand for the algebra of all piecewise continuous functions on C. Our goal in this section is a symbol calculus for Fredholmness of operators in the smallest closed subalgebra alg(S, PC) of 9(LP(p) ) containing the singular integral operator S and all operators of multiplication by a function in PC. In other words, we look for a symbol map for invertibility of elements of the quotient algebra alg"(S, PC):= alg(S, PC)/X(Lp(p)) in the Calkin algebra 9(Lp(p ) ) / X(LP(p)). A basic observation is that, for each continuous function f; the coset f + X(Lp(p)) belongs to the center of alg"(S, PC), and thus,

182 Math. Nachr. 162 (1993)

forms a central subalgebra of alg"(S, PC). Since it is by no means clear whether alg"(S, PC) is inverse closed in the Calkin algebra, we have to work in a larger subalgebra of 9(LP(p))/X(LP(p)) viz. in the algebra OLT":= OLT/Y(Lp(p)), where OLTstands for the algebra of all operators of local type (an operator A is of local type if fA - AfZ is compact for allfE C(T)). The algebra OLT" is inverse closed in the Calkin algebra, it contains alg"(S, PC) as closed subalgebra, and C"(T) is central in OLT". Therefore, Allan's local principle applies. Since the maximal ideal space of C"(T) is homeomorphic to T, we assign to each point t E T (to each maximal ideal t of C"(T)) the smallest closed two-sided ideal of OLT" which contains t, and we let OLT: and alg:(S, PC) stand for the quotient algebra OLT"/Zt and for the image of the algebra alg"(S, PC) in OLT:, respectively. Let finally 4, stand for the canonical homomorphism from OLT onto OLT:. Theorem 1 involves that an operator A E alg(S, PC) is Fredholm, or equivalently, the coset A + X(Lp(p ) ) is invertible in the Calkin algebra, if and only if the cosets 4,(A) are invertible in OLT: for all t E T .

Now our goal is reduced to the problem of finding a symbol map for the invertibility of elements of alg:(S, PC) in OLT:. It is certainly not very surprising that alg:(S, PC) is just an algebra which is generated by two idempotents. Nevertheless we shall shortly explain this. If zt stands for the characteristic function of the semicircle

{ZET : argt < argz < argt + n},

then p : = 4,(P+) = 4r -(I + S) and q:= 4,(xtZ) are idempotent in alg:(S, PC). In order to

check whether p and q span the whole algebra alg;(S, PC), it is sufficient to show that 4,(aZ) belongs to the algebra generated by p and q for all piecewise continuous functions a. But, as a little thought shows,

G )

= a(t + 0 ) p + a(t - O)(e - p )

and we are done. Before applying Theorem 2 in this situation, we have to determine the spectrum of

( p - q)2 in OLT: or, which is essentially the same problem, the spectrum of the local Toeplitz operator p q p = @,(P+X~P+) in OLT:. To determine this spectrum, we let f be a function on the unit circle having one sided limitsf(t + 0) = 1 andf(t - 0) = 0 at t, being continuous on T\{t} and having the property that

(clearly, such functions exist). The essential spectrum of the Toeplitz operator Tcf) is, by Theorem 3, the union off(T\{t}) with the horn H ( 0 , 1; v;(p, p), v:(p, p)). On the other hand, by Allan's local principle, this essential spectrum equals

Finck/Roch/Silbennann, Two Projection Theorems 183

Let s # t. Then f is continuous at s whence follows that

and that coL~:(+,(T(f)) = { f ( s ) } cf(T\{t)). Comparing this with(28), we conclude that

and since f - zr is continuous at t and vanishes there, we finally obtain

As we have already remarked after Corollary 1, (29) implies that

and this set is clearly subject to our conditions (6) and (7). Applying now Corollary 3 with d, 91y and 9 replaced by alg:(S, PC), OLT:, and C, respectively, we find

Theorem 4. There exists a symbol map Y , E smb(alg:(S, PC), OLT:; 9) with 9 standing for the algebra of all 2 x 2 matrix-valued functions on H(0, 1; v ; ( p , p), v : ( p , p)). This symbol map sends 4t(Z), p = #dP+) and q = +?(x,) into the functions

s(1 -s ) 7 1 - - s .-(o 1). s c ( 0 0). .-(I--- 1 0 1 0

and, moreover, alg:(S, PC) is inverse closed in OLT: and Yt is even in smb(alg:(S, PC); 9). Note that we have the parameter t in Corollary 3 replaced by 1 - s due to (30). A twice application of Allan's local principle both in OLT" and in alg" (S , PC) leads to

local algebras which both have the mapping U, from Theorem 4 as symbol map. Thus, our final result is:

Theorem 5. An operator A E alg(S, PC) is Fredholm i f and only if the matrix function Yt(4?(A)) (with U, referring to the symbol map in Theorem 4 and 4, to the canonical homomorphism involved by Allan's local principle) is invertible for all ~ E T . The homomorphism U , O ~ , sends S and a l into the functions

and, moreover, alg"(S, PC) is inverse closed in the Calkin algebra. For Khvedelidze weights the result conforms exactly with the Gohberg/Krupnik

symbol calculus. Invoking Corollary 3 with k > 1, it is easy to derive an analogous result for matrix-valued piecewise continuous coefficient.

184 Math. Nachr. 162 (1993)

6. Singular integral operators with measurable coefficients

Let M be a measurable subset of the unit circle T and x its characteristic function,

We let alg(S, C, 2) stand for the smallest closed subalgebra of 9 ( L p ) (with Lp:=Lp(l)) which contains the singular integral operator S, all operators f I of multiplication by continuous functionsf, and the operator X I of multiplication by x. The quotient algebra alg(S, C, x)/X(Lp) will be abbreviated to alg"(S, C, x). As in section 4, all cosets fI + X(Lp), f E C(T) lie in the center of alg"(S, C, x); we can localize via ALLAN over the maximal ideal space T of C(T), and we let alg;(S, C, x) stand for the image of alg(S, C, x) in OLT: under the homomorphism awa,. It is easy to see that the local algebra

alg:(S, C, 2) is generated by its elements p = - ( I + S ) + X ( L p ) + I , and

Q = XI + X(Lp) + I , , and that P and Q are idempotent. Thus, for applying the two- projections theorem, we need the spectrum ooLTn(( P - Q)') or, equivalently, goLT$ PQP). These local spectra were determined by SPITKOVSKI (see [14] or [l], Theorem 5.51). He introduced a subset Y ( M ) of T which consists of those t E T for which there exists a piecewise continuous function f such that X, =A. Roughly speaking, t e Y ( M ) if x is continuous at t or if x has a jump at t. Now SPITKOVSKI succeeded in showing that

1 2

~ O L T , ~ ( P Q P ) =

1 1 with-+-= 1.

P 4 Abbreviating C T O L T , Z ( P Q P ) to o,, one can formulate and prove in complete analogy to

Theorem 5 the following result describing a symbol map for Fredholmness of operators in alg(S, C, x):

Theorem 6. There is a symbol map

4, e smb(alg;(S, C, x), OLT2 9) ,

where 9 stands for the algebra of all 2 x 2 matrix functions on 6,. This symbol map sends

" st, fI,, and X , to the functions 2


respectively. An operator A E alg(S, C , x) i s Fredholm if and only if all functions 4 t (A + .%(Lp) + I,) are invertible. Moreover, i fA is Fredholm, then A possesses a regular- izer in alg(S, C , I), and alg(S, C, x) is inverse closed in the Calkin algebra.

The main results of this paper were presented on the Second International Con- ference in Functional Analysis and Approximation Theory, Acquafredda di Maratea, September 14-19, 1992.

References

[l] A. BOTTCHER, B. SILBERMANN, Analysis of Toeplitz operators. Springer-Verlag, Heidelberg 1989 [2] I. L. GOHBERG, N. YA. KRUPNIK, Singular integral operators with piecewise continuous coefficients.

[3] P. R. HALMOS, Two Subspaces. Transactions A.M.S. 144 (1969), 381-389 [4] R. HUNT, B. MUCKENHOUPT, R. WHEEDEN, Weighted norm inequalities for the conjugate function

and Hilbert transform. Trans. Amer. Math. SOC. 176 (19731, 227-251 [5] A. YA. KHELEMSKI, Banach and polinormed algebras: General Theorie, Representations, Homology.

Moscow, Nauka, Main Editorial Board for Physical and Mathematical Literature, 1989 [6] N. KRUPNIK, Banach algebras with symbol and singular integral operators. Operator Theory 26,

Birkhauser Verlag Basel, 1987 [7] G. K. ~ D E R S E N , Measure theory for C*-algebras, 11. Mathematica Scand. 22 (1968), 63 - 74 [8] S. C. POWER, C*-algebras, generated by Hankel operators and Toeplitz operators. Journal

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Functional Analysis 31 (1979). 52-68 [9] -, Essential spectra of piecewise continuous Fourier integral operators. Proc. Royal Ir. Akad. 81

(1981). 1-7 [lo] S . ROCH, B. SILBERMANN, Algebras generated by idempotents and the symbol calculus for singular

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[ 121 W. RUDIN, Functional analysis. McGraw-Hill 1973 [13] B. SILBERMANN, The C*-algebra generated by Toeplitz and Hankel operators with piecewise

quasicontinuous symbols. IEOT 10 (1987), 73G738 [14] I. M. SPITKOVSKI, On the factorization of matrix functions from the classes An(p) and TL. Ukrain.

Mat. Zh. 35 (1983), 3, 383-388 [15] -, Singular integral operators with PC-symbols on spaces with general weights. Journal Functional

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Two Projection Theorems and Symbol Calculus for Operators with Massive Local Spectra