Feynman’s operational calculi for noncommuting operators: The monogenic calculus

FEYNMAN'S OPERATIONAL CALCULI FOR NONCOMMUTING OPERATORS: THE MONOGENIC CALCULUS

B. Jefferies School of Mathematies The University of New South Wales N S W 2052 A USTRALIA e-mail: b.jefferies~unsw.edu.au

and

G .W. Johnson Department of Mathematics and Statistics 810 Old]ather Hall The University of Nebraska, Lineoln Lincoln, NE 68588-0323 U.S.A. e-mail: [email protected], edu

(Received: October 09, 2001; Accepted: December 11, 2001)

A b s t r a c t . In recent papers the authors presented their approach to Feynman 's operational calculi f o r a system of not necessaxily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutat ive Banach algebra, and the disentangling map which carries this commutat ive s tructure into the noncommutat ive algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distr ibution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.

1991 Mathematics Subject Classi]iccation. Primary 47A60 46H30 ; Secondary 47A25, 30G35

K e y w o r d s : functional calculus, disentangling

Advances in Applied Clifford Algebras 11 No. 2, 239-264 (2001)

240 Feynman's monogenic calculus . . . B. Jefferies and G. W. Johnson

O. I n t r o d u e t i o n

In three earlier papers [5], [6], [7] the authors studied properties of a class of operational calculi for systems of noncommuting bounded operators Al , . �9 �9 A,~ acting on a Banach space X. The basic ideas are as follows: time Ÿ are at tached to keep track of the order of operations in products. Operators with smaller time indices are to act before operators with larger time indices no mat ter how they are placed on the page. With time indices attached, functions of the operators are formed j u s t a s if the operators were commuting. Finally, the operator expressions must be restored to their natural order or "disentangled". More detail regarding Feynman's heuristic ideas on which the operational calculi are based can be found in [5] and in Chapter 14 of [11].

Implementing these basic ideas, we assign a measure on the set of time indices to each of the operators. These measures determine a particular operational calculus depending on the set of measures. In the case that the measures are all equal, we obtain the equally weighted functional calculus, which, in the case of self-adjoint operators acting on Hilbert space, coincides with the Weyl functional calculus [15], [14], [1].

The purpose of the present work is to identify the support of the operator valued distribution defined by the just mentioned functional calculus, with the set of singularities of a certain function which serves as the "Cauchy kernel" of a higher-dimensional analogue of the Riesz-Dunford calculus. The existence of the operator valued distribution depends on certain exponential estimates, usually satisfied, for example, when the bounded operators A l , . . . , Ah act on a Hilbert space and are self-adjoint. If the exponential estimates fail, then we have recourse to the higher dimensional analogue of the Riesz-Dunford calculus--it is important to know that the two approaches agree. At least in the case of commuting operators, we obtain an effective method of computing the support of the associated distribution [13].

Under the assumption that the exponential bound (1.11) holds, we defined the operator f ,1 ..... ~~(A1 , . . . ,An) in [7, Section 3] whenever f is smooth in ah open neighborhood of a compact subset 7t,(A) on l~n, called the lz-joint spectrum of A. In the case n = 1, we know that there is also a representation

/ (A) = ~ (AI - A)-~f(A) dA

Advances in Applied Clifford Algebras 11, No. 2 (2001) 241

via the Riesz-Dunford functional calculus for functions f which are analytic in a neighborhood of the spectrum a(A) of A and where C is a suitable closed curve about a(A) in the complex plane.

By analogy with the situation for n -- 1, any real analy_tic function f defined on an open subset U of ~n has a monogenic extension f to an open subset of lt~ ~+1 , that is, ] satisfies a higher dimensional analogue of the Cauchy-Riemann equations and the restriction of ] to U is equal to ] . The function / takes values in a 2'~-dimensional Clifford algebra and is represented by a higher-dimensional Cauchy integral formula.

Now suppose that A l , . . . , Ah ate bounded linear operators and #1,-- �9 #n ate continuous probability measures on [0, 1]. Just under the assumption that real linear combinations (~, A), ~ E ~n, have spectra lying on the real line, it has recently been established [9] that if A l , . . . , A~ commute with each other of if/~1,...,/~,~ are all equal, then ]~~ ..... ~~ ( A l , . . . , Ah) may be defined whenever f i s real analytic in a neighborhood of a suitable subset "),(A) of ~'~. The set -),(A) is in a sense minimal, and serves a s a joint spectrum for the n-tuple A = ( A l , . . . , A~). We emphasize that no exponential bound is assumed and f,~ ..... ~~(A1,. . . ,A,~) is defined from the higher-dimensional Cauchy integral formula.

In the case that f i s replaced by a distribution T, there exists a monogenic representation of T off ~n [2, Theorem 27.7] and, in Section 3, we use this to obtain a representation of f~~,:. ,,~ ( A l , . . . , Ah) a s a surface integral of functions with values in a Clifford module under the assumption that (1.11) holds. At the end of Section 3, we briefly discuss how the exponential estimate (1.11) can be replaced by a weaker assumption along the lines considered in [9] for the special cases of commuting operators and the Weyl calculus.

In Section 1, we briefly review the definition of Feynman' functional calculus considered in [5], [6], [7]. Section 2 is devoted to a brief outline of the relevant parts of Clifford analysis [2] needed to obtain our representation of Feynman's calculus.

The representation of Feynman's functional calculus a s a surface integral is given in Corollary 3.6 and the identification of the support of the functional calculus with the set of singularities of the Cauchy kernel is given in Theorem 3.8.

242 Feynman's monogenic calculus... B. Jefferies and G. W. Johnson

1. T h e B a n a c h A l g e b r a s A and D a n d t h e D i s e n t a n g l i n g M a p s .

We begin by reviewing the two commutat ive Banach algebras A and D and noting the close relationship between them.

Given a positive integer n and n positive numbers r l , �9 �9 �9 rn, let A(r l , �9 �9 �9 r,~) or, more briefly A, be the space of complex-valued functions ( Z l , . . . , z,~) f ( z l , . . . , zn) of n complex variables, which are analytic at ( 0 , . . . , 0), and are such that their power series expansion

o o

f ( z l , . . . , zn )= E cm, ..... m,,Z'~ l ' ' ' z ¡ (1.1) r n , , . . . , r n n ~ 0

converges absolutely, at least on the closed polydisk ]Zl[ _< r x , . . . , [zn[ _< rn. Note that any entire function of n complex variables belongs to A( r l , . �9 rn) for all positive numbers r l , . . . , rn. For f E A ( r l , . . . , rn) given by (1.1), we let

IIfl] : IIfllAr ..... ~~) : = ~ Icm, ..... m~lrŸ ~ ' ' ' ' r~ ~. 7"~1J,..(f'/ ' �91 = 0

(1.2)

The function on A ( r l , . . . , r n ) defined by (1.2) makes A ( r l , . . . , rn) into a commutative Banach algebra under pointwise operations.

We turn next to the Banach algebra D. Let X be a Banach space and let A l , . . . , Ah be nonzero operators from s the space of bounded linear operators acting on X. Except for the numbers I[A11[,..., I[An[[, which will serve as weights, we ignore for the present the nature of A l , . . . , Ah as operators and introduce a commutative Banach algebra consisting of "analytic functions" f(A1 . . . . ,,4n), where A l , . . . , A,~ are t reated as purely formal commuting objects.

Consider the collection D = D ( A 1 , . . . , Ah) of all expressions of the form

f ( A , , . . . , A n ) = ~ cm, ..... ,,,,A'~' . . .A~o (1.3) r n l , . . . , r n n = 0

where cm, ..... m. E C for all m l , . . . , m n = O, 1 , . . . , and

I I f ( A 1 , . . - , - ~ . n ) l l = I I / ( -~~ , . - . , -An) I ID(A1 ..... A~)

c,o

E m 1 , . . . , r o n ~-.~-0

(1.4) : = q ..... ~ . I I I A l l l m , . . . l l A n l l m~ < oo.


The function on D ( A 1 , . . . , Ah) defined by (1.4) makes D(A1, . . . ,Ah) into a commutat ive Banach algebra under pointwise operations. In fact, if we take rj = [[Aj[[ for j = 1 , . . . , n , then D ( A 1 , . . . , A n ) is obtained from & ( r l , . . . , r ~ ) simply by renaming the indeterminates; hence, ID and A are isometrically isomorphic in a natural way as Banach algebras.

We refer to lI}(A1,.. . , A , ) as the disentangling algebra associated with the n-tuple ( A l , . . . , A~) of bounded linear operators acting on X.

Let Al , . �9 A,~ be nonzero operators from s and let # 1 , . . - , #,~ be continuous Borel probabil i ty measures on [0, 1]. The idea is to replace the opera- tots A l , . . . , A,~ with the elements Al , . �9 Ah from ID and then form the desired function of Al , . �9 �9 A~- Still working in D, we time order the expression for the function and then pass back to s simply by removing the tildes. We now present formally the ideas which we have just expressed.

Given nonnegative integers m i , . . . ,ron, we let m = rol + . . . + ron and

�9 = . m . ( 1 . 5 ) pm, ..... m " ( Z l , . . , z , ~ ) zŸ nl- "z n .

We are now ready to define the map T~I ..... ,~ which will return us from our commutat ive framework to the noncommutat ive setting of s For j = 1 . . . . ,n and all s 6 [0, 1], we take As(s) = Aj and, for i = 1 , . . . , m , we define

{ Al(s) i f i 6 { 1 , . . . , m i } , A2(s) i f i 6 {mi + 1 , . . . , m i + m 2 } ,

Ci(s) := . . (1.6)

Ah(s) if i 6 {mi + - - . + m,~-i + 1 , . . . , m } .

For each m = 0, 1 , . . . , let Sin denote the set of all permutat ions of the integers { 1 , . . . , m } , and given 7r 6 Sin, we let

Am(;r) = { (S l , . . . , Sm) 6 [0,1]m: 0 < S~(1) < . . . < S~(m) < 1}.

"-]-~1 ..... lt,~ (pro, ..... m . ( 2 ~ l , . . . , ~ n ) ) :__~ D e f i n i t i o n 1.1

f C,~(m)(s~(m)) '"C~(1)(s~(1))(#? x• .. xp ' f f " ) (ds l , . . . , d s ,~ ) . (1 .7 ) ~6Sm JA~(~r)

Then, Ÿ t i ( A l , . . . , 2~n) E I]~(AI, . . . , Ah) given by

o o

. f ( : , , , . . . , ~ , , ) = ~ ~,~1 ..... ,,, A Ÿ (1.8) m l , . . . , m n ~ 0

244 F e y n m a n ' s m o n o g e n i c c a l c u l u s . . . B. Jeffer ies a n d G. W . J o h n s o n

~e set % ..... , . ( / ( ~ i , , . . . , ~ ~ ) ) equal to

o o

Z Cm1 ..... m,T~l ..... •,, (pro, ..... m , (2~1, . . . , Z~n)) . ( 1 . 9 )

r~�91 , . . . , r o n ----0

In the commutat ive setting, the right-hand side of (1.7) gives us what we would expect [5, Proposition 2.2].

As is customary, we shall write the operator "F~l ..... , . x in place of T~I ..... ,n (x) for an element x of I ) (A1 , . . . , Ah). The mapping

/ , > T,1 ..... , ~ ( / ( . 4 ~ , . . . , A n ) ) , I e A(HAI[I,...,[IA,dl)

into Z:(X) is a / unc t i ona l calculus studied further in Section 3. Here we are identifying A(IIA~II,..., IIA~II) with D(A1 , . . . , A~). We shall sometimes write the bounded linear operator T,1 ..... ~,= (/(ei ,1, . . . , .3.n)) a s / , 1 ..... ,~ ( A 1 , . . . , A ~ ) . In particular,

P2: ,::(A) = T.1 ..... , . ( e = l ..... = . ( ~ ~ , . . . , ~ o ) ) (1.10)

The following exponential estimate was used in [7] to obtain a functional calculus. The bound will also be the underlying assumption of the present work.

D e f i n i t i o n 1.2 Let A l , . . . , Ah be bounded linear operators acting on a Banach space X . Let q = (#1 , . . . , /~n) be an n-tuple o/ continuous probability measures on B[0, 1] and let

T,, ..... ~. : D ( A 1 , . . . , A n ) - ~ / : ( X )

be the disentangling map defined in Definition 1.1. I / there exists C, r, s >_ 0 such that

IIT,, ..... ,~(ei<r162 , f o r a l l ~ e C '~, (1.11)

then the n-tuple A = ( A l , . . . , Ah) o/operators is said to be o/Paley- Wiener type ( s, r, Iz ) .

2. C l i f fo rd A n a l y s i s

CLIFFORD ALGEBRAS

Let F be either the field ~ of real numbers or the field C of complex numbers. The Clifford algebra F(n) over F is a 2n-dimensional algebra with unit defined as


follows. Given the s tandard basis vectors e0, el, . . . , e,~ of the vector space F ~+1 , the basis vectors e s of F(,~) ate indexed by all finite subsets S of {1, 2 , . . . , n}. The basis vectors are determined by the following rules for multiplication on F(~):

eo = 1,

2 - 1 , for l < j < n e j ~- _ _

e j e k = - e k e j = e { j , k } , for 1 _< j < k _< n

e j l e j 2 . . . e j j = e s , if l _ < j l < j 2 < . . . < j , _ < n and S = { j l , . . . , j s } .

Here the identifications eo = e0 and e j = e{j} for 1 < j _< n have been made. The p roduc t of two elements u = ~-~~s u s e s , u s E F and v = E s v s e s , v s E

F is u v = ~ 8 , R U S V R e s e R . According to the rules for multiplication, e s e R is ~=1 times a basis vector of F(n). The sca lar p a r t of u = ~'-~~s u s e s , u s E F is the term u0, also denoted as u0.

The Clifford algebras ~(o), 1~(1) and ~(2) are the real, complex numbers and the quaternions, respectively.

The conjugate ~ of a basis element e s is defined so that e s e s = e s e s = 1.

Denote the complex conjugate of a number c G F by ~. Then the operat ion of c o n j u g a t i o n u ~-4 ~ defined by ~ = ~-'~~s u s e s for every u = ~-~~s u s e s , u s E F

is an involution of the Clifford algebra F(n). Then ~ = ~ F for all elements u and v of F(n). An inner product is defined on F(~) by the formula (u ,v) = [UF]o = ~ u s g - g for every u = E s u s e s and v = E s v s e s belonging to F(n). The corresponding norm is writ ten as I-[ .

Suppose tha t m < n are positive integers. The vector space i~m is identified m with a subspace of F(,~) by virtue of the embedding ( x i , . �9 �9 x m ) ~+ ~ j = l x j e j .

On writing the coordinates of x E ]~~+1 as x = (x0, x i , . . . , x~), the space i~~+1 is identified with a subspace of F(n) with the embedding ( x o , x l , . . . , x , ~ ) ~-4

7�91

~"~~ j =0 x j e j .

BANACH MODULES

A Banach space X with norm [[ �9 [[ over F with an operat ion of multiplication by elements of F(n) turning it into a two-sided module over F(n) is called a B a n a c h m o d u l e over F(n), if there exists a C _> 1 such tha t


II=ull < Cl~l I1=11 ana Ilu=ll < Clul It=11 for all u E F(n) and x E X. The vector space of all continuous right module homomorphisms from a Banach module X to a Banach module Y is denoted by s Y). Thus, a bounded linear map A : X --+ Y belongs to s Y) if (Ax)u = A(xu) for all x E X and u e F(n). Both s and the space s Y) of continuous linear operators from X to Y are considered as Banach spaces over IF with the uniform operator norm I1" [I.

The algebraic tensor product X(~) = X | F(~) of a Banach space X over F with F(n ) is a Banach module. Elements of X(~) may be viewed as finite sums u = ~-~~s x s | es of tensor products of elements xs of X with basis vectors es of IF(n). Multiplication in X(n) by elements A of the Clifford algebra F(~) is defined by uA = ~'~~s x s | (esA) and Au = E s x s | (Aes). The tensor product notation x s | es is written simply as xse s . The norm on X(n) is taken to be

I1~'11 = ( E s Ilxsll~-) '/=. The analogous procedure applies to a locally convex space E to define the

module E(~) with its induced locally convex topology. If E and F are two locally convex spaces, then the spaces ( s and s F(n)) ate

identified by defining the operation of T = E s Tses on u = E s u s e s as T(u) = E s , s , Ts (us , ) e ses , . In the case that E and F ate equal to a Banach

space X, the norm of T is given by HT[I = ( ~ , s IITs[l~(x)) 1/2" In particular, for n bounded operators 2 '1 , . . . , Ta acting on X, we have

~2 r�91

= I I T j l l L ( x ) ) II Erje~ll~,.,Ix,.,~ ( ~ 2 ,/2 j=1 j=a

Given x 6 E and ~ 6 F', the element (Tx,~) 6 F(n) is defined for each T = E s T s e s belonging to s F(.)) by (Tx, ~) = ~'~~s(Tsx, ~)es.

CLIFFORD ANALYSIS

What is usually called Clifford analysis is the study of functions of finitely many real variables, which take values in a Clifford algebra, and which satisfy higher dimensional analogues of the Cauchy-Riemann equations.

A function f : U --+ F(n) defined in an open subset U of i~~+1 has a unique representation f -- ~-~~s f s e s in terms of F-valued functions f s , S C_ { 1 , . . . , n} in the sense that f ( x ) = ~-]s fS (x )es for all x E U. Then f is continuous, differentiable and so on, in the normed space F(~), ir and only if for


all finite subsets S of { 1 , . . . , n}, its scalar component functions f s have the corresponding property. Let Oj be the operator of differentiation of a scalar function in the j ' t h coordinate in I~n+l--the coordinates of x E /l~ '~+1 are written as x = ( xO ,X l , . . . , xn ) . For a continuously differentiable function f : Ii~ '~+1 -+ F(n) with f = E s f s e s , the function D f is defined by set-

D f = ~-~~s ( (Oofs)es + ~jn=l(Oj fs )e jes ) and f D is specified by f D = %

ting

Now suppose that f is an F(,~)-valued, continuously differentiable function

defined in an open subset U of II~ n+l. Then f is said to be left monogenic in U if D l ( x ) = 0 for all x E U and right monogenic in U if f D ( x ) = 0 for all x E U .

For each x E ~n+l , the function G ( . , x) defined by

1 w - x G(w, x) - (2.1)

~ n I~ - x i n §

for every w ~ x is both left and right monogenic as a function of w. Here the volume of the unit n-sphere in ~n+l has been denoted by ah and we have used the identification of 11~ '~+1 with a subspace of 11~(~) mentioned earlier.

The function G ( . , x ) , x E II~ n+l plays a special role in Clifford analysis. Suppose that f~ C ll~ n+l is a bounded open set with smooth boundary cOf~ and exterior unit normal n(w) defined for all function f defined in a neighborhood U of f~,

lo G ( w , x ) n ( w ) f ( w ) d # ( w ) = ( f ( x ) , if 0, if

w E c0s For any left monogenic the Cauchy integral formula

x E f~; _ (2.2) x e-U \ f~.

is valid. Here # is the surface measure of c3fL The result is proved in [2, Corol- lary 9.6]. If g is right monogenic in U then fo~ g(w)n(w)f(w)dl~(w) = 0 [2, Corollary 9.3].

E x a m p l e 2.1 For the case n = 1, the Clifford algebra ~(1) is identified with C. A continuously differentiable ]unction f : U -+ ~(1) defined in ah open subset U of R 2 satisfies D f = 0 in U ir and only ir ir satisfies the Cauchy Riemann equations in U. For each x, w E ~2, x ~ w, we have

1 1 G ( w , x ) - 2Ÿ w - x

The inverse is taken in C. The tangent at the point ~(t) of the portion {((s) : a < s < b} of a positively oriented recti]iable curve C is i times the normal


n(r at r so the equality d~ = i.n(r dl~ ] shows that (2.2) is the Cauchy integral formula ]or a simple closed contour C bounding a region f~.

VECTOR VALUED FUNCTIONS

It is a simple mat ter to check from the definition of a Bochner integrable function, tha t for a Banach module X over F(n), the integral fE f dp of an X-valued Bochner p-integrable function f has the property that

= f j ( ~ ) dp(a),

( f E f ( a ) d P ) ~ = f E f ( a ) u d # ( a ) �9

for all u E F(n). �9 k o o Let X be a Banach space. A sequence { Ÿ }k=l of X-valued functions

fk : f~ --~ X is normally summable in X if there exists a summable sequence {Mk}k~=l of nonnegative real numbers Mi such that Ilfk(w)l I < Mk, for all w E 12 and all k = 1 , 2 , . . . . Thus, a normally summable sequence {fk}k=l of X-valued functions on 12 is absolutely and uniformly summable on f~. In the case that X is a Banach module over F(n), we have u ~ k fk = ~-~~k ufk and

( Z k fk)u = ~-~~k ufku for all u e F(n). The definition of monogenicity extends readily to other vector and operator

valued functions. In particular, if g is a left monogenic F(n)-valued function, then the tensor product g| : w ~-~ g(w) | of g with an element x of a Banach space X is left monogenic in X(n). If {gj | }~=1 is normally summable in X(n)

o o and each function gj is left monogenic, then ~i=1 gY | xi is left monogenic in X(n).

As in the case of vector valued analytic functions [4, Section 3.10], there is a choice of possible topologies in which to take limits. The proof of the following assertion follows the case of vector valued analytic functions [4, Theorem 3.10.1].

P r o p o s i t i o n 2.2 A function is monogenic for the weak topology of a locally convex module E(n) ir and only ir it is monogenic for the original topology. Moreover, for a Banaeh space E, ir g : U -+ E(n) is right monogenic and f : U -+ F(n) is lefl monogenic and ~ is ah open set with smooth boundary OO such that f~ C u, Then the function w ~-4 g(w)n(w)f(w), w E 0~, is Bochner p-integrable in E~n) and fo~ g(w)n(w)f(w) dp(w) = O.


In particular, this is valid in the case that X is a Banach space and E = /:(X) with the uniform operator norm. It follows from the principle of uniform boundedness and the Cauchy integral formula that an L(X)-valued function is norm monogenic when it is monogenic for the weak or strong operator topologies.

MONOGENIC EXPANSIONS

For any subset A of ~n+l , let M(A, F(~)) be the collection of all F(,~)-valued functions which ate left monogenic in an open neighourhood of A in i~n+l, if A is open in i~n+l, then M(A, F(~)) is given the compact-open topology (uniform convergence on every compact subset of A). If K is a compact subset of ]~n, then M(K,F(,~)) is the union of all spaces M(U,F(~)), as U ranges over the open sets in i~n+l containing K. Equipped with the inductive limit topology, M(K, ~'(n))becomes ah LF-space [2, Theorem 9.11]. Then M (K, F(n)) becomes a topological algebra under the C-K product [2, Theorem 9.11], and the closed subspace M(K, F) of M(K, F(n)) consisting of left monogenic extensions of F- valued functions on K is a commutative topological algebra. The topological algebra M ( K , F ) is isomorphic, via monogenic extension, to the topological algebra H(K, F) of F-valued functions analytic in an open neighborhood of K with pointwise multiplication.

Suppose that f is an analytic F-valued function defined on an open neighborhood of zero in II~ n and the Taylor series of f is given by

~ 1 ~ • Y(~) = Z y , "'" a,...,~~,~ . . . ~ ,~ ,

k=O 11=1 / k = l

(2.3)

for all x E ~'~ in a neighborhood of zero. The coefficients atx...t~ are assumed to be symmetric in l l , . - - lk. Then the unique monogenic extension ] of f i s

](X) = ~ Z a'l*[k V'l""lk(x) k=O (t~ ..... lk)

(2.4)

for all x belonging to some neighborhood of zero in R n+l. Here, the sum ~ ( h ..... tk )"" is over the set {1 _ ll _< . . . _< lk _< n•, and for ( /1 , . . - , lk) E

{ 1 , 2 , . . . , n } k, the function V tl'''t~ : ll~ n+l ~ F(n) is defined as follows. For


each j = 1 , . . . , n, the monogenic extension of the function xj : x ~ xj, x E i~n is given by zj : x ~ xjeo - xoe j , x E R~+I. Then Vo(x) = eo,x E ~n+l and

1 v"t~=~. ~ ~~,...~j~, (2.5) j~ ..... jk

where the sum is over all distinguishable permutations of all of ( /1 , . . . , lk ) , and products are in the sense of pointwise multiplication in F(,~). If ] is left monogenic in the open ball BR(0) of radius R about zero in 1~~+I, then (2.4) converges normally in Bn(O) [2, p82].

MONOGENIC REPRESENTATION OF DISTRIBUTIONS

Let T E s176176 be an F(n)-valued distribution with compact support. Then T is interpreted as a right module homomorphism from C~ into F(n). Thus, we may represent T a s a finite sum T = ~-~~s Tses with Ts E s176 F). The function T(w) = T(G(w, �9 )) for all w E II~ '~+1 \ i~n is called the left monogenic representation of ~. The following result is proved in [2, Theorem 27.7].

T h e o r e m 2.3 Let T E l:(n)(C~176 Then f~ may be extended to a right monogenic function, still denoted by T, in ~n+l \ supp T.

Furthermore, liml~l_~oo Ÿ 0 and for any r E C c (~ )(n), we have

T(r = ~-~0+lim fR- [T(w + ce0) - T(w - eeo)]r dw.

3. T h e M o n o g e n i c # - F u n c t i o n a l C a l c ª

The purpose of the remainder of this work is to identify the support "yj,(A) of the/~-functional calculus for an n-tuple A of bounded operators on X satisfying (1.11) with the set of singularities of a certain monogenic function taking values in the Banach module L(n)(X(n))--the Cauchy kernel G , ( . , A) associated with A and ~�91

For a single operator A with real spectrum, the set "~(A) is just the set of singularities on the real axis of the analytic function A ~-~ ( A I - A) -1, that is, the spectrum a(A) of A in /:(X) consisting of all points A E I~ for which AI - A is not invertible in s


INTEGRATION OF VECTOR VALUED FUNCTIONS

One of the arguments of this section requires us to integrate vector valued functions taking values in a locally convex space whose topology is not given by a single norm.

Suppose tha t (~, 8, ~) is a measure space and E is a sequentially complete locally convex space. Let ] : E -+ E be a function for which there exist E-valued #-integrable S-simple functions sn, n = 1, 2 , . . . such tha t su "-+ f ]z-a.e., and for every continuous seminorm p on E, f~. p(sn - sin)d# -~ 0 as n, m -+ oc. Then the integral fA f d# of f wi.th respect to #, over a set A E 8, is defined to be the limit lim,~_,~ fA su d#. The limit is independent of the approximat ing sequence sn, n -- 1, 2 , . . . ; such a function f i s said to be Bochner #-integrable. It follows immediately tha t for a continuous linear map T : E -+ F between sequentially complete locally convex spaces E and F, if f i s Bochner #-integrable, then T o f is Bochner #-integrable and T (fA ] d#) = fA T o fd# for all A E 8.

A bounded continuous function with values in a Fr› space or LF-space is Bochner integrable with respect to any finite regular Borel measure.

If E is a sequentially complete locally convex module over F(n), then the integral fE f d~ of an E-valued Bochner #-integrable function f has the proper ty that

U/Af(a)dP = /Auf(a)d#(a),

for all u E F(n) and A E 8. Moreover, if T : E --+ F is a continuous right module homomorphism and g : E -+ F(n) is a bounded measurable function, then the equalities

T (fA f(a)g(a)d#(a))= fAT(f(a)g(a))d#= fAT(f(a)).g(a)d# (3.1)

hold for all A E 8.

THE ]_�91 KERNEL FOR AN n-TUPLE OF OPERATORS

Let ~ --- (#i,. �9 #n) be ah n-tuple of continuous probabi]ity measures acting on [0, li. Let A = (Al , . . . , Ah) be an n-tuple of bounded operators of Paley- Wiener type (s, r, ~�91 acting on a Banach space X.

252 Feynman's monogenic calculus. . . B. Jefferies and G. W. Johnson

In [7, Section 3], we defined the nonempty compact subset 7~(A) of R n and Feynman's q calculus .TI,.A : f ~ .f~,~ ..... ~,.(A), f E C~176 Here the / : (X)-va lued distribution ~'I,,A is the Fourier transform of ~ ~ (27r)-n

7-~1 ..... ~,. (ei(r ~ E R". The bound (1.11) and the Paley-Wiener theorem shows that ~'~,,A has compact support "y,(A).

The set i~n is identified with the subspace {x G ]IU ~+1 : x0 = 0} of R n+l . The algebraic tensor product ~'t,,A | : C ~ (V)(n) --~ f-(n)(H(n)) of ~'/~,A with the identity operator I(n) on F(n) is also denoted just by ~'z,A. Here V i s an open neighborhood of ?z(A) in R n+l and C~(V)(n) is the locally convex module obtained by tensoring the locally convex space C ~ ( V ) with F(~), as mentioned in Section 2. The mapping ~'z,A : C~176 -'+ s is defined by applying ~z,A to the restriction of functions f E Cr162 to the open subset V Q li( '~ of R n. The map ~'z,A : C~(V)(n) ~ s is a right module homomorphism. The symbols .Tt,,A(f ) and f~l ..... ~, (A) ate used interchangeably.

The support 7z(A) := supp 9Vz,A of the distribution Jrz,A, which is independent of the particular meaning attached to it above, is a nonempty compact subset of R n. Let U be an open neighborhood of 7z(A) := suppgrt,,A in R n and suppose that the function f : U -+ C is analytic. Let f be a left monogenic extension of f to an open neighborhood of U in ~n+l. Then according to the definition of (])z(A) , the equality ( f )~ , (A) = L, , ..... . . (A) | I(n) is valid.

Suppose that f is an analytic F-valued function defined on ah open neighborhood of zero in l~n and the Taylor series of f is given by (2.3). Then the unique monogenic extension f of f i s given by (2.4).

Let V h'''tu be monogenic polynomials defined by (2.5). For any ordered set ( l l , . . . , lk) of k integers belonging to {1 . . . . , n}, set

V~t,...tu (A) = ~ , ..... u . ( vh ' ' ' tu( t i l ," tin)). 1 , . . . , $ Z n " "

(3.2)

The polynomial V tl"''lu ( t i l , . . . , ti, n) is understood to be formed in the disentangling algebra II)(A1,..., Ah) by replacing the monogenic functions zj in (2.5) by t i j for each j = 1 , . . . , n . Then we have

1 v " tu(ti) = ~. ~ A~i...~i,u, O" 1 ~ . . . ,O*k

where the sum is over all distinguishable permutations of ( /1 , . . - , lk). Suppose that for each j = 1 , . . . , n, the index j appears exactly kj = 0 , . . . , n times in the

k! k-tuple ( l l , - - - , lk)- Then k = kl +. . "+kn and there are ~ distinguishable

Advances in Appl ied Clifford Algebras 11, No. 2 (2001)

permutations of ( l l , . . .

v . . . ~ ~ ( ~ ) _ kx

Vu h'''tk ( A ) - 1 ,..-,Un kl

The operators pkl...k. Ul ,..-,/~n The equality

fu, ..... u~ (A) = k=0

, Ik). It follows that

l- kn! 1 - - . A h ,

1 pk~...k. (A~ ! . . . k~! m ..... u~, ,"

(A) are given by formula (1.10).

253

(3.3)

(3.4)

atl...t~ v m ..... u~(A) (3.5) (/1,--.,/k)

holds if (2.3) converges in a suitable neighborhood of 7~(A). In the case in which the monogenic expansion of a function about a point

does not converge over all of 7t~(A), the Cauchy integral formula is useful, as for the Riesz-Dunford functional calculus for a single operator. Moreover, when the Feynman functional calculus for A is not defined, the Cauchy integral formula can be used to define functions of the n-tuple A, see [9] for the case of the Weyl calculus.

For any w E l~,~+l not belonging to 7~(A), there exists an open neighborhood Uo~ of 7~(A) in R,~+I such that the F(,0-valued function

1 W - - X z ~+ G ( ~ , z ) -

~ n I~ - x i n + l '

for each x # w belongs to C~176 Then G~(w, A) := ~~,A(G(w, �9 )) may

be viewed as an element of L;(n)(X(n)) for each w E i~~+l \ 7~(A).

E x a m p l e 3.1 Let n = 1. Then as in Example 2.1, ~(1) "" C and ]or each x, w E ~2, x # w, the equality G(w, x) = 1/2Ÿ 1/(w - x) holds.

Ir A is an operator of type (s, r) (we drop the index p) , then

G(w ,A) := JZA(G(w, .))

for aU w ~ 7(A). Both G(w, A) and (2Ÿ -1 ( w I - A ) -1 have the same Neumann series expansion for Iw I > IIAIIs Ir turns out that 7(A) = a(A) and

G(w ,A) = (2~')-1 ( w I - A) -1 , for all w E l~2 \ a(A).

This result is proved in [3, Theorer¡ 3.1.6] under quite general circumstances and in [1, Lemma 3.3] for hermitian operators on a Banach space.


Ir follows that the formula

f ( A ) = J [ c G ( r A)n(r162 dlr

= 21Ÿ - A ) - l f ( r

f o r a closed curve C surroundin9 the spectrum a(A) of A and a function f analytic in a neighborhood of C and its interior is associated with the Riesz functional calculus.

Actually, we ate supposed to be identifyin9 N with iN in C, so we should write G(w,A) = (27r)-X(wI- lA) -1 and end up with the functional calculus for iA.

Example 3.2 Let n = 3 and consider the simplest noncommuting example of the Pauli matrices,

( 0 1 ) ( ~ - i ) ( 1 0 l ) {TI = 1 0 ' 0"2 = 0 ' 0"3 = 0 - - '

viewed as linear trans… acting on H = C 2. Set 0" = (o"1,0"2,0"3). I f I~1 = #2 = 1~3, then by [6, Remark 5.6], we may suppose that P1,P2,P3 all equal Lebesgue measure and Yz•,t, is the Weyl functional calculus. We drop the subscript p.

A calculation [1, Theorem 4.1] shows that for a l l f E C~162 the matriz ir ( f ) is 9iven b v

Y a ( f ) = I f (f + n . Vf) d/~a + [ a . V f d/.�91 1. (3.6) J S 1 J S l

Here & = {x E N 3 : [xi = t} la the sphere of radius t > 0 centred st zero in R 3, Pz is the unir surface measure on SI and n(x) is the outward unir normal at x E SI. Thus, 7(0") = Sx.

For all w E N a such that w q~ SI C N a, Yr - )) E s is given by

f f a ( ~ , , , ) = i I_ (a(,o, .) + n . v a ( ~ , , . )) d~,, + ! VG(w, . )dUx.

,15 1 J ~ l

Let vi, v2 be the standard basis vectors of C 2. For each Xo E R, the funetion (x, t) ~ ~t~(G(x + xoeo, �9 ))vi is the solution of the Weyl equation


O t u t + a . Vu t - -O, t > O,

with initial datum Uo (x) = - v i | x+xoeo) = 1/a3vj | (xoeo- x) /Ix+xoeo 14 ]or all x E ]~3, x + xoeo ~ O. The ]unction w ~-~ :T:ta(G(w, �9 ))vi is le… and right monogenic on the set ~4 \ St.

The following statements are formulated in a context more general than that of Feynman's functional calculus. Proposition 3.3 and Theorem 5.4 below appear in [8, Proposit ion 5.2, Theorem 5.4] in the Hilbert space setting. The same proofs work for Banach spaces.

Suppose that X is a Banach space over the field F and T : C~176 (ll~ n) - 4 / : ( X ) is a distribution with compact support K. We use the same symbol T to denote the map which sends the element f -- ~]s f s e s of C~176 to the element ~;~~s T ( f s ) e s of/:(n) (X(n)), ra ther than the more descriptive notation T | I(n). In particular, T ( f ) E s is defined for a l l f E M(K,F(~)) .

P r o p o s i t i o n 3.3 Let U be ah open subset o] ~~+1 containing K = suppT. Suppose that w ~-~ F~ is a continuous map from U \ K into C~176 If ]or each open set V with V C U \ K , there exists a neighborhood Nv oŸ K, such that for each x E Ny , the F(n)-valued … w ~-+ F~(x) is le]t monogenic in V, then w ~-~ T(F~) is lc~t monogenic in U \ K.

Pro@ By Cauchy's theorem for monogenic functions [2, Theorem 9.6], for all intervals I contained in U \ K , rol n(w)F~(x) d#(w) = 0 for each x belonging to some neighborhood of K . The function w ~ F~,w E U \ K is continuous, and so Bochner integrable in C~ on all boundaries OI of intervals I contained in U \ K. Moreover, the function foz n(w)F~ d#(w) belongs to C~(K)(n) and vanishes in a neighborhood of the support K of T.

The distribution T : C~176 ~ s is a continuous linear map, so by (3.1), the equalities

J~oi n(w)T (F~) d#(w) = T ( f ~ n(w)F~d#(~z)) = 0

hold. By Morera's theorem for monogenic functions [2, Theorem 10.4], w ~-4 T(F~) is left monogenic in U \ K . []

The same result holds for right monogenic functions.


Coro l la ry 3.4 The E(n) (X(,))-valuedfunction w ~ G~(w, A) is left and right monogenic in ~n+, \ %(A).

T h e o r e m 3.5 Let T be an L(X)-valued distribution with compact support in E n. Let 12 be a bounded open neighborhood of suppT in ~n+l with smooth boundary 012 and exterior unir normal n(w) defined for all w E 012. Let p be the surface measure of 12.

Suppose that f is left mono9enic and 9 is right monogenic in a neighborhood of the closure 12 = 12 U 012 of 12. Then

T ( f ) = fo T(G(w, . ))n(w)f(w) du(w),

T(9) = fon9(w)n(w)T(G(w, " )) d#(w).

Proof. We consider only the case where f is left monogenic. The case where 9 is right monogenic is similar. The space C~(12)(n) of smooth F(,~)-valued functions defined on 12 is a separable Fr› space with the topology of uniform convergence of functions, and their derivatives, on compact subsets of 12. The continuous function w ~ G(w, �9 )n(w)f(w),w E 012 takes its values in C~176 and satisfies fo~P(G(w,')n(w))[f(w)[ d#(w) < oo for each continuous seminorm p on C~(12)(n), that is, it is Bochner integrable in C~(9t)(n).

By the Cauchy integral formula (2.2), the equality

f (x) = ron G(w, x)n(w)f(w) d#(w)

holds for all x belonging to the open set 12. Combining this equation with (3.1), and the fact that the distribution T defines a continuous linear map (denoted by the same symbol) from C ~ (12)(n) into the space s with the uniform operator norm, it follows that the function w ~+ T(G(w, �9 ) )n(w) f(w), w E 012 is Bochner integrable in the space/:(n) (X(n)), with the uniform norm, and by equation (3.1) the equality

T ( f o G(w, . )n (w) f (w)d /~ (w) )= lo T(G(w, . ))n(w)f(w)d#(w)

obtains. The stated equality T( f ) = ron T(G(w, . ))n(w)f(w)d#(w) therefore holds. []


Coro l la ry 3.6 Let li be a bounded open neighborhood of Tu(A) in ~n+l with smooth boundary Of~ and exterior unir normal n(w) defined for all w E Ofl. Let # be the surface measure of li.

Suppose that f is left monogenic and g is right monogenic in a neighborhood of the closure f~ = f~ U Of~ of f~. Then

f,~ ..... ~.(A) = / Gt,(w,A)n(w)f(w) d#(w), o~

g~~ ..... ~.(A) = f g(w)n(w)G~(w,A)d#(w). of~

(3.7)

We mention here that the extension of these results to X-valued functions is straightforward. First, if f = ~'~~j~=l f jhj for monogenic functions l i and vectors hi E X , then T(]) = ~--~~j T(•j)hj and the above equality holds. In the limit, both sides of the equation converge because C~(suppT) | X is dense in C~(supp T; X) [16, Proposition 44.2].

THE /~-MONOGENIC SPECTRUM

For each w 6 ~n+l such that w ~ 0, let

G(~,x)=~ Z W,I ,~(~)V" '~(x) (3.8) k = o (t~ . . . . . t~)

be the monogenic power series expansion of G(w, �9 ) in the region [x I < [w[ [2, 11.4 pp77-81]. Here Wtl...t~(w) is given for each w E IR'~+l,w ~ 0 by (-1)~0~,1 . . .O~,G(w, 0) and V h'''t~ is defined as in Section 2. Each function Wta...tk is monogenic because

D~(-1)kO~t~ .. .0~, G(w,O) = (--1)ka~,~ .. .0~, D,,G(w,O) = O, for all w ~ O.

Let A be an n-tuple of bounded operators of Paley-Wiener type (s,r,/~) acting on a Banach space X. If [w[ > r, then the sum (3.8) converges normally for all x E i~,~+1 in the closed ball Br(0) of radius r centred at zero and also in C~(Br(O)). Aceording to the Paley-Wiener theorem [7, Proposition 3.3], the support %,(A) of .Ÿ is contained in Br(0) f3 ({0} • i~n).

258 Feynman's monogenic ca lculus . . . B. Jefferies and G. W. Johnson

It follows from formulas (3.8) and (1.11) and the continuity of -~'#,A on C~176 tha t

G,(w, A) = .T~,,A(G(w, .)) = ~ ~ , Wh...,k ~~J'" 'V"'lk, l,...,~,, ~,'A'J (3.9) k=0 (11 ,..., l~)

n for all w E n~ n+l such that IwI > max{r, (1 + v~)ll ~~=~ Aje~ll}. Formula (3.9) is adopted as a definition of the Cauchy kernel in [12, Definition 3.11] for the case in which the measures P l , . . . , # n are all equal. The sum converges in s because of the following result.

L e m m a 3.7 The sum

E:E: h=0 (ll ,...,l~)

IW. . . . l , (w) l ,1...'~ IIV,~, ..... ,,. (A)II

converges uniŸ I~,1 _> R,,,., ~ ~"+~, .fo,- each R > ( l+v~ ) l l ~~~1 A~e.711.

Proof. According to equation (3.3), the norm of V h'''l~ (.4) is bounded by

1 i iA l l lh l . . . I IA. I I h". kl ! ' " k,!

By [5, Proposition 2.4], T,1 ..... ~. is a contraction on ~)(A1,. . . , Ah), so we have

IIV~',:::t.,~.. (A)I I < 1

k l ! . . , k . ! IIA1 II h I - - - I IA. I I h" .

n It suffices to show that for each R > (1 + v~)l[ ~~=1 Ajej[[, the sum

OO

hi ,...,kn=0

i a h ' - . . a~ :G(w, 0)111Alll h ' . . . IIA,,II h- k1 ! . - k,! ~1

(3.10)

converges uniformly for all [w[ > R, w G ~n+l . However, this follows from the normal convergence of the multiple power series

1 ~ (-1)k" 1 q _ xi,.,_ ~ = - - ~ . , ~x, x;'~,) k lyl¡

k=O


for Ix] < ( v f 2 - 1)IY] [2, p83] and the equality G(w,x) = 1---D-~-~~, ~ ~ , l valid for all w ~ x. []

We know from Corollary 3.4 that the function defined by formula (3.9) for all Iw I > max{r, (1 + v~)l] ~-~d~=l Ajejl]} is actually the restriction of an

s function m0nogenic in ~,~+l \ "yu(A). The question remains as to whether there is a larger 0pen set on which this function has a monogenic extension.

The spectrum of a single operator T is the set of 'singularities' of the resolvent function A ~-~ ( A I - T) -1. Similarly, the tt-monogenic spectrum spu(A) of the n-tuple A of bounded operators of Paley-Wiener type (s, r, ti) is the com- plement of the largest open set U C 1R ~+l in which the function w ~ Gu(w, A) is the restriction of a monogenic function with domain U. As indicated in Example 3.1, sp~,(A) = a(A) in the case n = 1.

T h e o r e m 3.8 Let A be an n-tuple of bounded operators of Paley- Wiener type (s,r ,#) acting on a Banach space X . Then sp~(A) = ~/u(A).

Proof. We have established in Corollary 3.4 that sp~,(A) C_ ~,u(A). Let x E spt,(A) c, let U C sp~(A) c be an open neighborhood of x in I~'~ and suppose that r is a smooth function with compact support in U.

Let x E X and x* E X*. A comparison with [2, Definition 27.6] shows that the F(n)-valued monogenic function w ~-~ (G~(w,A)x,x*), w E ~,~+l \ supp5~~,A, is actually the monogenic representation of the distribution (.~D,AX, X*) : f ~ (J:t~,A(f)x,x*), for all smooth f defined in an open neighborhood of 7u(A). Then (J:u,AX, x*)(G(w, �9 )) = (Gu(w, A)x, x*) and by Theorem 2.3,

(~-. ,Ax, x*) (r =

= hm~o" ~0+ / v [ ( G u ( Y + y o e 0 , A ) x , x * / - (G~(Y-yoeo, A)x,x*l]r

Because w ~+ G~,(w,A) is monogenic (hence continuous) for all w in U, the limit is zero, that is, (~u,AX, X*)(r = 0 for all x E X and x* E X* and all smooth r supported by U. Hence x E ~/t,(A) c, as was to be proved. []

C o r o l l a r y 3.9 Let A be an n-tuple of bounded operators o] Paley- Wiener type n (s,r ' ,~) acting on a Banach space X . Set r = II ~ j= l AjejlI. Then r~(A) ~ r

and

260 F e y n m a n ' s monogen ic c a l c u l u s . . . B. Jefferies and G. W. J o h n s o n

sp~,(A) g [-IIAj]I, IIA~II] c B~(0).

Pro@ Apply [7, Proposition 3.7] and invoke the equality sp~(A) = 7l,(A). []

This estimate for the spectral radius r~ (A) is better by a factor of v ~ - 1 than the one obtained from Lemma 3.7.

R e m a r k 3.10 The significance of the Ca~chy kernel w ~-~ G~(w, A) is that it is the monogenic representation of the distribution J:~,,A off ~/g(A)--the distribution J:g,A represents the 'boundary values' on {0} • II~ n of the monogenic ]unction

w ~ G,(w,A) , w E R n+l \ ({0} • 7,(A)).

In [10], the support of the distribution ira ]ora pair A of hermitian matrices is determined by examining discontinuities of the Cauchy kernel G(w, A).

In the final part of this paper, we examine what can be said if a Paley- Wiener estimate (1.11) fails for ah n-tuple A of bounded operators. Rather than a C~-funct ional calculus like ~'l,,A, we could hope for a functional calculus defined for functions of n real variables analytic in a neighborhood of a compact nonempty set ~,~,(A). At least this is the case for the Weyl functional calculus when all the measures pi, j = 1 , . . . , n, are equal [9].

Let A be any n-tuple of boundedopera tors acting on a Banach space X. Let V l~''l~ (A~ be defined as in equation (3.2). Suppose that we set

�9 ] Z l , . . . , ~ n ~ ~

Gl'(w'A) = ~ Z Wt~'"t~(w)V~~~:::l.~,~~ (A) (3.11) k=0 (li ..... lk)

n for all ~ E ~n+l such that [w[ > (1 + Vf2)]l ~,j=l Ajejll" The sum converges in

l:(n)(X(n)) because )-~~~=o)-~~(tl ..... lk) IWtl...t~(w)] V t~'''t*. ~,~,...,~,, (A)I I converges uni- 2 n formly for I~{ _> R,w E ~,~+1, for each R > (1 + v~)ll Zj--1 Ajej[I by Lemma

3.7. Each function Wtl ...tk is monogenic, so (3.11) defines a monogenic s

valued function for all w e R n+l such that Iw] > (1 + v~)]l ~jn_-I Ajejll. It follows that the representation (3.7) is valid provided that the set ~ in the statement of Corollary 3.6 contains the closed unit ball of radius (1 +


x/-2)[[ ~"~~~=1 Ajejl[ centered at zero in ~n+l . However, this case is of little in- terest, because fa, ..... ~, (A) can also be expressed by the sum (3.5).

Although (3.11) makes sense for any n-tuple of bounded operators, the problem remains of enlarging the domain of definition of the monogenic function defined by (3.11) to be as large as possible in a unique way, such as in the case when the natural domain is connected or at least has no bounded component.

The Paley-Wiener condition (1.11) guarantees the existence of a compact nonempty subset ~~,(A) of i~n such that the function ~ ~ Gj,(w,A) has a monogenic extension to all of IR '~+1 \ ({0} x 3'~(A)). If all the measures pi , j -- 1 , . . . ,n, are equal, then it suffices to assume that the spectrum of the operator (~, A) is real for every ~ e ]R n [9].

A general p¡ of operator theory is that resolvent estimates ate equiv- alent to exponential estimates via the Laplace transform and its inverse

(AI - T) -1 = e-~te tT dt, o

1 f ( A I - T ) - l e txdA e~T = ~ C

f o r a suitable contour C. Let ~ E i~n be a unit vector. In our setting, the Paley-Wiener estimate (10A) implies the resolvent esimate

[l"~~(z~ - - i ( A , ~ ) ) - I I [ s - -

<

_<

/ :e-X~Tzeit(2'~) dt L(x)

floo TPeit(A'() s e -~~~ dt .] o

as ~ A ~ O + .

A similar formula holds for ~A < 0. The function A ~ Tz(A - i(A, ~))-1 therefore has a unique analytic contin-

uation from the set { A E C \ (iIl~) : [A~ > (~F,j=I [[Aj[[2) 1/2 } '~ to all of C \ (ill~).

We adopt this conclusion a s a definition in case the exponential bound (1.11) fails. The closed disk of radius r > 0 in C centered at zero is written as Dr.


Def in i t i on 3.11 Let A l , . . . , A,~ be bounded linear operators acting on a Ba- ( E n 2) 1/2 nach space X . Set r = j=x []Aj][ . Let ti = (tq, . . . , # , ) be an n-tuple of

continuous probability measures on B[0, 1] and let T m ..... ~. : D ( A 1 , . . . , Ah) --+ s be the disentangling map defined in Definition 1.1. We say that the n- .tuple A = (A1 , . . . ,Aa ) has real t t-joint spectrum ir for each ~ E ~n with I~1 = 1, the function

~ T ~ , ..... u . ( ~ - ( A , ~ ) ) - I , ~ e C \ i R , I)~[>r, (3.12)

is the restviction to C \ (D, U ~) of ah analytic function on C \ I~.

In view of the preceding discussion, we immediately have the following examples.

E x a m p l e 3.12 (i) Suppose that A is o… Paley-Wiener type (s,r, tt). For ~ E R a and )t E C with ~)~ ~ O, set

f ~ - Rp(~, ( A , ( ) ) = - i eit)~T#e-it(A'~) dt, ~~ > O,

o

Rts(,X , (fix,~)) = i e-it)~T#eit(A'~) dt, ~)~ < O. o

Then )~ ~4 R~()~, (.4,~)), ~ E C \ ~, is ah analytic continuation of the function (3.12). In the case that .~~ > 0 and ])~[ > r, the equality

f f " (~ _ (.~, ~))-1 = - i eit)~e-it(A'~) dt

holds in the disentangling algebra ID(Al, . . . , Ah), so

(/o ~ ) Tg()~ - (A,~))-I = -i7-~ eit;~e -it(A'r dt = Ru(A , (A,~)).

Similar equalities hold for ..~A < 0 and [)~1 > r

(ii) In case #1 = . . . = #~, we drop the subscript tt and obtain the equality

T(,X - ( A , ~ ) ) -~ = ( ,xi - (A,~)) -1, I,Xl > r,

from [6, Remark 5. 6]. Then A has real joint spectrum ir and only i… the spectrum a((A,~)) of the bounded linear operator (A,~) is real … every ~ E ~n. The analytic continuation R ( . , (.4-,~)) of the function (3.12) is given by


R(A,(.4,~)) = ( M - (A,~)) -1, A E C \ a ( ( A , ~ ) ) .

In the case that A has real ~�91 spectrum, it remains to define the real Iz-joint spectrum ~h,(A) of A and define a functional calculus for functions real analytic in a neighborhood of ~h,(A) in II( n. This can be done along lines similar to the reasoning in [9] using the plane wave decomposition of the Cauchy kernel G(~, z).

R e f e r e n c e s

[1] Anderson R.. F. V., The Weyl functional calculus, J. Funct. Anal. 4 (1969), 240-267.

[2] Brackx F., R. Delanghe and F. Sommen, "Clifford Analysis", Research Notes in Mathematics 76, Pitman, Boston/London/Melbourne, 1982.

[3] Colojoax~ I. and C. Foia~, "Theory of Genera]ized Spectral Operators", Gordon and Breach, Mathematics and Its Applications Vol. 9, New York/London/Paris, 1968.

[4] Hille E. and 1~. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Colloq. Publ. XXXI, Providence, R.I., 1957.

[5] Jefferies B. and G. W. Johnson, Feynman's operational ca]culi for noncommuting operators: Definitions and elementary properties, Russ. J. Math. Phys. 8 (2001), 153-178.

[6] - - , Feynman's operationa] ca]culi for noncommuting systems of operators: tensors, ordered supports and disentangling ah exponentia] factor, Math. Notes 70, no. 6 (2001), 815-838.

[7] - - , Feynman's operationa] ca]culi for noncommuting systems of operators: spectra] theory, submitted.

[8] Jefferies B. and A. McIntosh, The Weyl ca]culus and Clifford ana]ysis, Bull. Austral. Math. Soc. 5'7 (1998), 329-341.

[9] Jefferies B., A. McIntosh and J. Picton-Warlow, The monogenic functional calculus, Studia Math. 136 (1999), 99-119.

[10] Jefferies B. and B. Straub, Lacuna in the support of the Weyl ca]culus for two hermitian matrices, submitted.

[11] Johnson G. W. and M. L. Lapidus, "The Feynman Integral and Feynman's Operational Calculus", Oxford U. Press, Oxford Mathematical Monograph, 2000.

[12] Kisil V. V., MSbius transformations and monogenic functional calculus, ERA Amer. Math. Soc. 2 (1996) 26-33.

[13] McIntosh A. and A. Pryde, A functional ca]culus for severa] commuting operators Indiana U. Math. J. 36 (1987), 421-439.


[14] Nelson E., Operants: A functional calculus for non-commuting operators, Func- tional analysis and related fields, Proceedings of a conference in honour of Professor Marshall Stone, Univ. of Chicago, May 1968, F.E. Browder (ed.), Springer-Verlag, Berlin/Heidelberg/New York 1970, 172-187.

[15] Taylor M. E., Functions of several self-adjoint operators, Proc. Amer. Math. Soc. 19 (1968), 91-98.

[16] Treves F., "Topological Vector Spaces, Distributions and Kernels", Academic Press, New York, 1967.

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Feynman’s operational calculi for noncommuting operators: The monogenic calculus