White Noise Calculus and Fock Space (Lecture Notes in Mathematics, 1577) 3540579850, 9783540579854

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen

1577

Nobuaki Obata

White Noise Calculus and Fock Space

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Author Nobuaki Obata Department of Mathematics School of Science Nagoya University Nagoya, 464-01, Japan

Mathematics Subject Classification (199 I): 46F25, 46E50, 47A70, 47B38,47D30, 47D40, 60H99, 60J65

ISBN 3-540-57985-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57985-0 Springer-Verlag New York Berlin Heidelberg CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany

SPIN: 10130019

46/3140-5432] 0 - Printed on acid-free paper

Contents Introduction 1 Prerequisites

1.1 1.2 1.3 1.4 1.5 1.6

Locally convex spaces in general . Countably Hilbert spaces . . . . . Nuclear spaces and kernel theorem Standard CH-spaces of functions. Bochner-Minlos theorem .. Further notational remarks. Bibliographical notes

2 White Noise Space

2.1 Gaussian measure . 2.2 Wick-ordered polynomials . . . . . . . . . . . 2.3 Wiener-Ito-Segal isomorphism and Fock space Bibliographical notes . 3 White Noise Functionals

3.1 3.2 3.3 3.4 3.5 3.6

Standard Construction Continuous version theorem S-transform . Contraction of tensor products. Wiener product . . . . . . Characterization theorems Bibliographical notes

4 Operator Theory

4.1 4.2 4.3 4.4 4.5 4.6

Hida's differential operator. Translation operators . . Integral kernel operators Symbols of operators Fock expansion . . . Some examples ... Bibliographical notes

vii 1 1

3 7 11

16 17 18

19 19

23

28 32 33 33 38 48 53 58 65 69 71 71

76 79 88

98 .100 .107

CONTENTS

VI

5 Toward Harmonic Analysis 5.1 First order differential operators . 5.2 Regular one-parameter transformation group 5.3 Infinite dimensional Laplacians .. 5.4 Infinite dimensional rotation group 5.5 Rotation-invariant operators . . . . 5.6 Fourier transform . . . . . . . . . . 5.7 Intertwining property of Fourier transform Bibliographical notes . . . . . . . . . . . . 6 Addendum

6.1 Integral-sum kernel operators 6.2 Reduction to finite degree of freedom 6.3 Vector-valued white noise functionals Appendices

A B C

Polarization formula . . . . . . Hermite polynomials . . . . . . Norm estimates of contractions

109 · 109 · 118

· 121 · 126 · 132 · 140 · 145 .149

151 · 151 · 154

· 159 167 · 167 · 168 .169

References

171

Index

181

Introduction The white noise calculus (or analysis) was launched out by Hida [lJ in 1975 with his lecture notes on generalized Brownian functionals. This new approach toward an infinite dimensional analysis was deeply motivated by Levy [1] who considerably developed functional analysis on L2(0, 1) and actually analysis of Brownian functionalso The root of white noise calculus is to switch a functional of Brownian motion j(B(t); t E R) with one of white noise ¢(B(t); t E R), where B(t) is a time derivative of a Brownian motion B(t). Although each Brownian path B(t) is not smooth enough, B(t) is thought of as a generalized stochastic process and ¢ is realized as a generalized white noise functional in our language. We may thereby regard {B(t)} as a collection of infinitely many independent random variables and hence a coordinate system of an infinite dimensional space. The mathematical framework of the white noise calculus is based upon an infinite dimensional analogue of the Schwartz distribution theory, where the role of the Lebesgue measure on Rn is played by the Gaussian measure J.L on the dual of a certain nuclear space E. In the classical case where B(t) is formulated, we take E = S(R) and the Gaussian measure J.L on E" defined by the characteristic functional: exp (

=

k.

J.L(dx),

E E,

where is the usual L 2­norm Then the Hilbert space (L 2 ) = L 2(E",J.L) is canonically isomorphic to the (Boson) Fock space over L 2(R) through the WienerIto­Segal isomorphism and links the test and generalized functionals. Namely, in a specific way (called standard construction) we construct a nuclear Frechet space (E) densely and continuously imbedded in (L 2 ) , and by duality we obtain a Gelfand triple:

An element in (E) is a test white noise junctional and hence an element in (E)" is a generalized white noise junctional. The above picture is easily understood as a direct analogy of S(Rn) c L 2(Rn) c S'(Rn) which is a frame of the Schwartz distribution theory. Then, B(t) = x(t), x E E", gives us a realization of the time derivative of a Brownian motion and, in fact, x f-+ x(t) becomes a generalized white noise functional for each fixed t E R. In our actual discussion we do not restrict ourselves to the case of E = S(R) and H = L 2(R) but deal with a more general function space on a topological space T. Typically T is a time­parameter space and is often taken to be a more general

viii

INTRODUCTION

topological space where quantum field theory may be formulated. Again {x( t); t E T} is considered as a coordinate system of E* intuitively. In fact, within our framework we may discuss not only functionals in {xU); t E T} but also operators derived from this coordinate system. The coordinate differential operator at = %x(t) is well defined as a continuous derivation on (E). We have also multiplication operators by coordinate functions x(t), which are, in fact, operators from (E) into (E)*. Furthermore, a; is a continuous linear operator on (E)*. The operators at and a; correspond respectively to an annihilation operator and a creation operator at a point t E T and they satisfy the so-called canonical commutation relation in a generalized sense. The above mentioned formulation was consolidated in the basic works of Kubo and Takenaka [lJ-[4J and has been widely accepted. The main purpose of these lecture notes is to develop operator theory on white noise functionals as well as to offer a systematic introduction to white noise calculus. From that point of view it is most remarkable that we are free from smeared creation and annihilation operators. In other words, at and a; are not operator-valued distributions but usual operators for themselves. This leads us to an integral kernel operator: ='/

....... ,m

(K)

= JTl+nt r

K(S1'' .. 'si ti ... ,m t )0*81 .,. 0*St at 1 "

.,.at

m

ds 1 · · · dsidt, ... di.; ,

where K is a distribution in l-i-m variables. The use of distributions as integral kernels allows us to discuss a large class of operators on Fock space. In fact, every continuous operator := from (E) into (E)* admits a unique decomposition into a Sum of integral kernel operators:

:=¢ =

00

L

:=/,m(K/,m)¢,

¢ E (E),

l,m=O

where the series converges in (E)*. Moreover, if := is a continuous operator from (E) into itself, the series converges in (E). In the process we investigate precise norm estimates of such operators and obtain a method of reconstructing an operator from its symbol. The above expression is called Fock expansion and will playa key role in our discussion. Although applications of white noise calculus are widely spreading, the present lecture notes are strongly oriented toward infinite dimensional harmonic analysis. The clue to go on is found in the following three topics: (i) infinite dimensional rotation group; (ii) Laplacians; (iii) Fourier transform. Being almost as new as the white noise calculus, they have been so far discussed somehow separately. Since the very beginning of the development Hida has emphasized the importance of the infinite dimensional rotation group O(E; H), that is, the group of automorphisms of the Gelfand triple E C H C E*. In fact, it played an interesting role in the study of symmetry of Brownian motion and Gaussian random fields. There are various candidates for infinite dimensional Laplacians which possess some typical properties of a finite dimensional Laplacian. So far the Gross Laplacian Ll a , the number operator N and the Levy Laplacian LlL have been found to be important in white noise calculus, though the Levy Laplacian is not discussed in these lecture notes. As for Fourier transform, among some candidates that have been discussed Kuo's Fourier transform (simply called the Fourier transform hereafter) has been found well suited to white noise calculus.

INTRODUCTION

ix

In these lecture notes the above listed three subjects are treated systematically by means of Our operator calculus and are found closely related to each other. For example, the Gross Laplacian Lio and the number operator N are characterized by their rotation-invariance. The Fourier transform intertwines the coordinate differential operators and coordinate multiplication oprators just as in the case of finite dimension and, this property actually characterizes the Fourier transform. Moreover, the Fourier transform is imbedded in a one-parameter transformation group of the generalized white noise functionals (called the Fourier-Mehler transform) and its infinitesimal generator is expressed with Lio and N. These results would suggest a fruitful application of white noise calculus to infinite dimensional harmonic analysis. It is also expected that our operator calculus is useful in some problems in quantum field theory and quantum probability. As is well known, a lot of efforts to develop distribution theories on an infinite dimensional space equipped with Gaussian measure have been made by many authors. In fact, mathematical study of Brownian motion or equivalently of white noise is now one of the most important and vital fields of mathematics toward infinite dimensional analysis. Since the main purpose is to develop an operator theory on white noise functionals, the present lecture notes are mostly based on a functional analytic point of view rather than probability theory or stochastic analysis. In Chapter 1 we survey some fundamentals in functional analysis required during the main discussion and propose a notion of a standard countably Hilbert space which makes the discussion clearer. The purpose of Chapter 2 is to establish the well-known Wiener-Itd-Segal isomorphism between L 2 ( E*, J.l) and the Fock space. Chapter 3 is devoted to a study of generalized white noise functionals. In Chapter 4 we develop an operator theory on white noise functionals, or equivalently on Fock space, in terms of Hida's differential operators and their duals a;. By means of the operator theory we discuss in Chapter 5 a few topics toward harmonic analysis including first order differential operators, the number operator, the Gross Laplacians, infinite dimensional rotation group, Fourier transform and certain one-parameter transformation groups. Chapter 6 is added after finishing the first draft of these lecture notes. We discuss integral-sum kernel operators, the finite dimensional calculus derived from our framework and a generalization to cover vector-valued white noise functionals. These topics are expected to open a new area in infinite dimensional analysis.

at

ACKNOWLEDGEMENTS First of all I would like to express my sincere gratitude to Professor H. Heyer who invited me to Tiibingen to do research within his working group. The stay there was supported by the Alexander von Humboldt Foundation which I appreciate very much. In fact, the present work is based on a series of my lectures in the Arbeitsgemeinschaft "Stochastik und Analysis" during the Summer Semester 1991. The lectures aimed at providing fundamentals of white noise calculus and at introducing some new aspects of harmonic analysis and quantum probability theory. Professor H. Heyer encouraged me to write up the notes and to expand them for publication. My special thanks are extended to Professors B. Kiimmerer, E. Siebert and A. Wolfffor their kind hospitality in Tiibingen. lowe also special thanks to Professors Ju, G. Kondrat'ev, H.-H. Kuo, L. Streit and J.-A. Yan for interesting discussion. During my writing these lecture notes, I learnt many relevant works made under the name of quantum probability. Let me mention with special gratitude the names of Professors L. Accardi, V. P. Belavkin, R. Hudson, J. M. Lindsay, P. A. Meyer and M. Schiirmann. My basic references have been among others the works of Hida-Potthoff [1], Kuo [7], [9], Lee [3] and Yan [4] which I appreciated highly. The readers are recommended to consult the recently published monograph Hida-Kuo-Potthoff-Streit [1] which contains different topics and various applications. It will complement our discussion certainly. Finally I am extremely grateful to Professor T. Hida for his constant encouragement. He initiated the white noise calculus around 1975 and remains always a fount of knowledge. January 1994 Nagoya, Japan

Nobuaki Obata

Chapter 1 Prerequisites 1.1

Locally convex spaces in general

We first agree that all vector spaces under consideration are over the real numbers 1R or the complex numbers C. A topological vector space X is called locally convex if the topology of X is Hausdorff and given by a family of seminorms {II·IL,}"EA. Then the seminorms are called defining seminorms for X. Without changing the topology we may choose a directed family of defining seminorms for X, which means that for any a, /3 E A there exists "y E A such that < and :::; for all E X. In that case A becomes a directed set naturally. Unless otherwise stated, X 3'! !V means that two locally convex spaces X and !V are isomorphic as topological vector spaces. For a systematic study of locally convex spaces we introduce general notion of projective and inductive systems and their limits. Let {Xet}"EA be a family of locally convex spaces. The direct product

II X" =

"EA

E Xet}

is always equipped with the weakest locally convex topology such that the canonical projection P/1 : TI"EA X" - t X(3 is continuous for all/3 E A. The direct sum

EB Xet =

E

"EA

II Xet; = 0 except finitely many a E A}

"EA

is equipped with the strongest locally convex topology such that the canonical injection i/1 : X/1 - t $etEA X" is continuous for all /3 E A. Let be a family of locally convex spaces, with A being a directed set. Suppose that we are given a continuous linear map 1",(3 : X/1 - t X" for any pair 0.,/3 E A with a:::; 13. Then {X et ,I",(3 } is called a projective system of locally convex spaces if (i) I",,, = id.; and (ii) 1","1 = 1",/11(3,"1 whenever a :::; 13 :::; "y. Then proj lim X"

"EA

=

E

II X,,;

etEA

whenever a :::;

13}

with the relative topology induced from TIetEA Xet is called the projective limit of {X",I",/1}' SO far as the projective limit is under consideration, it suffices to consider

CHAPTER 1. PREREQ UISITES

2

a reduced projective system; namely, every canonical projection P(3 : proj lim"'EA X", -+ X(3 has a dense image. We now introduce a dual object. Let {X"'}"'EA be the same as above and suppose that we are given a continuous linear operator g",.(3 : X(3 -+ X", for all pair a, /3 E A with a /3. Then {X""g",,(3} is called an inductive system of locally convex spaces if (i) g""", = id.; and (ii) g"",,/ = g""(3g(3,,,/ whenever a /3 "t- Consider 2:",>(3 Ran(i(3 i",g",,(3) which is a subspace of ffi"'EA X", generated by the ranges of the Tinear maps i(3 ­ i",g",,(3, where a,/3 run over all pairs with a /3. If 2:",>(3 Ran(i(3 ­ iag",,(3) is closed, the quotient space ­

ind lim E; "'EA

=

ED X"'/ L

aEA

Ran(i(3 ­ i",ga.(3)

equipped with the quotient topology is called the inductive limit of {X a,g",,(3} . If 11·11 is a seminorm on a vector space X, then I)'l = E X; = O} becomes a subspace of X and the quotient space X/I)'l admits a natural norm which is denoted by the same symbol. The completion of X/I)'l with respect to this norm 11·11 is called the Banach space associated with the seminorm 11·11. Now consider two seminorms 11·lla and 11,11(3 satisfying :::; C E X, for some C O. Note that 1)'l(3 = E X; = O} c IJ1a = E X; = O}. Let X a and X(3 be the Banach spaces associated with 1I·ll a and 11,11(3' respectively. Then, the canonical map from X/I)'l(3 onto X/I)'la extends to a continuous linear map f",,(3 : X(3 -+ X a· Proposition 1.1.1 Let X be a locally convex space with a directed family of defining seminorms {11·1I",}aEA. Then, notations being as above, {X""fa,(3} becomes a reduced projective system of Banach spaces. If in addition X is complete, X pro j lim"'EA Xa. Let X be a locally convex space with defining seminorms {11·lla}"'EA. A subset SeX is called bounded if SUP{ES < 00 for all a E A. Let r be the dual space of X, i.e., the space of continuous linear functionals on X and we denote the canonical bilinear form on r x X by (', ­) or similar symbols. Unless otherwise stated, r always carries the strong dual topology or the topology of bounded convergence. This topology is defined by the seminorms: Ilxll s

= sup I (x, 0 I, {ES

x E X*,

where S runs over the bounded subsets of X. In that case r is called the strong dual space as well. For a continuous linear operator T from a locally convex space X into another !D its adjoint T* is defined by (T*y, 0 = (y, y E !D*, E X. Then T* becomes a continuous linear operator from !D* into X*. In accord with Proposition 1.1.1 we can discuss the dual space of a locally convex space. We keep the notations there. Since the canonical map Pa : X -+ X a has a dense image, its adjoint map : -+ r is injective and thereby is regarded consists of linear functionals on X which are as a subspace of X*. In that case continuous with respect to 11·lla' Therefore, as vector spaces.

3

1.2. COUNTABLY HILBERT SPACES

Note also that C Xii for 0: :s: (3. Namely, in a purely algebraic sense X* is the inductive limit of In general, if {XC" f",,(3} is a projective system oflocally convex spaces, becomes an inductive system of locally convex spaces in an obvious ind lim",EA way. Unfortunately, with respect to the strong dual topology X* does not hold in general. While, it is true whenever X* and are equipped with the Mackey topologies r(X*, X) and X",), respectively. Instead of going into a detailed topological argument we note a class of locally convex spaces X for which the strong dual topology coincides with the Mackey topology r(X*, X). A locally convex space is called Frechet if it is metrizable and complete. Recall that a locally convex space is metrizable if and only if it admits a countable set of defining seminorms. A locally convex space X is called reflexive if the canonical injection X -+ X** is a topological isomorphism, where X** is the strong bidual of X. It is known that for a reflexive Frechet space X the strong dual topology on X* coincides with the Mackey topology r(X*, X). Since the projective limit of a sequence of reflexive Frechet spaces is again a reflexive Frechet space, we have the following Proposition 1.1.2 Let

be a reduced projective sequence of reflexive Frechet

spaces. Then, ( proj limXn)*

n-+oo

where the strong dual topologies are taken into consideration. We note another important property of a Frechet space (in fact, a characteristic property of a barreled topological vector space). Proposition 1.1.3 Let X be a Frechei space. Then for a subset S C X* the following

four properties are equivalent: (i) Sis equicontinuous, i.e., if {II·II",}"'EA is a directed family of defining seminorms for X, one may find C 0 and 0: E A such that I (x, e) I s C lIell", for all e E X and XES; (ii) S is (strongly) bounded; (iii) S is weakly bounded; (iv) S is relatively weakly compact.

1.2

Countably Hilbert spaces

A seminorm 11·11 on a vector space X over lR (resp. C) is called Hilbertian if it is induced by some non-negative, symmetric bilinear (resp. hermitian sesquilinear) form 2 (-,.) on X x X, namely if lIell = (e,e) for all e E X. Here it is not assumed that (e, = 0 implies = O. We further agree that a hermitian sesquilinear form is linear on the right and antilinear on the left. The Banach space associated with a Hilbertian seminorm becomes a Hilbert space in an obvious way. A complete locally convex space X is called a countably Hilbert space or a CH-space for brevity if it admits a countable set of defining Hilbertian seminorms. We first note the following

e)

e

CHAPTERl.

4

Proposition 1.2.1 Any CH-space is a projective limit of a reduced projective se-

quence of Hilbert spaces, and therefore, is a reflexive Frechei space. Then, in view of Proposition 1.1.2 we have Proposition 1.2.2 Let:r be a CH-space and let {Hn,fm,n} be a reduced projective

becomes sequence of Hilbert spaces such that:r projlimn_=Hn. Then, an inductive sequence of Hilbert spaces and r indliffin_= Moreover, is regarded as an increasing family of subspaces of rand r = U::'=o as vector spaces. We shall be mostly concerned with a particular class (or construction) of CH-spaces. The following general result will be useful. Lemma 1.2.3 Let A be a positive linear operator in a complex Hilbert space fl. Then

A is selfadjoint if and only if (1 + A) Dom(A)

= fl.

We first consider the complex case. Let fl be a complex Hilbert space with norm fl with (dense) domain Dom(A) C fl. Suppose inf Spec(A) > 0 and put 11.11 0 and let A be a selfadjoint operator in p

= (infSpec(A))-l.

(1.1)

According to the spectral theory we may define a (positive) selfadjoint operator AP for all p E IR with (maximal) domain Dom(AP) C fl. For the moment suppose p O. Since 0 f/. Spec(AP), by definition AP admits a dense range and bounded inverse. In fact, we see from Lemma 1.2.3 that the range of AP coincides with the whole fl because we have inf Spec(AP) > 0 by assumption. Therefore (APt l is everywhere defined bounded operator on fl. In that case (AP)-l = A-P, P 0, in particular, Dom(A-P) = fl, and (1.2) v > o. Note also that

».« o. It is known that the closure of APAq coincides with Ap+q, p, q We now introduce a family of Hilbertian norms: E Dom(AP),

Note that Dom(Aq) C Dom(AP) whenever q

p

O.

p E R,

(1.3)

O. In fact, by (1.2) we have

(1.4) Equipped with the norm 11·ll p the vector space Dom(AP) becomes a Hilbert space which we shall denote by Then, the inclusion Dom(Aq) C Dom(AP), q p 0, -t and fp,q} becomes a projective gives rise to a continuous injection fp,q : system of Hilbert spaces. In this case we have also a chain of Hilbert spaces: ... C

C ... C

C ... C

= fl,

q

p

o.

(1.5)

1.2. COUNTABLY HILBERT SPACES

5

Lemma 1.2.4 For any q p 0 the vectorsubspece fp,q} is reduced. the projective system

e, is dense in

In particular,

PROOF. Obviously, = Dom( N-P) is a dense subspace of S). Note also from onto S). Hence the inverse definition (1.3) that Ap is an isometric isomorphism from image of is a dense subspace of On the other hand,

Consequently,

is dense in

qed

By virtue of (1.5) a subspace of fj defined by (1.6) becomes a CH-space equipped with the Hilbertian seminorms is isomorphic to the projective limit:

Obviously

proj lim

p--+oo

To sum up, given a pair (S), A) where A is a selfadjoint operator in a complex Hilbert space S) with infSpec(A) > 0, we have constructed a CH-space is called a standard CH-space constructedfrom (S), A).

Definition 1.2.5 The above

As may be proved easily from definition, we have Lemma 1.2.6 Let A be a positive selfadjoint operator in S) with inf Spec(A) > O. Then, the standard CH-spaces constructedfrom (S), A) and (S), A") are isomorphic for any s > o. As for the dual space of ind lim p-loOO

it follows from Proposition 1.2.2 that P

and

=

as vector spaces.

is identified with the space of linear functionals on which are conRecall that tinuous with respect to 1I·llp ' With this identification the canonical bilinear forms on p 0, are denoted by the same symbol (.,.). x and on x By virtue of our particular construction and can be described more explicitly. We have already defined in (1.3) a Hilbertian norm II·II- p on S) for p O. Let be the completion of S) with respect to II·II-p ' Then the identity map from S) onto -+ whenever q p 2 0, and itself extends to a continuous injection f- q .- p : f-q,-p} becomes an inductive system of Hilbert spaces. Furthermore, thereby there is a natural inclusion relation: S)

=

C ... C

C ... C

C ...

q

p

o.

(1.7)

CHAPTER 1. PREREQUmITES

6 Recall that A-P : .lj

-+

IEp is a bounded operator and by definition it satisfies

Therefore A-P extends to an isometric isomorphism A-p from lE_ p onto .lj. The inner product (".)p of IEp is by definition given as (1.8) It follows from Riesz' theorem that there exists an isometric anti-isomorphism Rp : IE; -+ IEp such that

On the other hand, in view of (1.8) we have

Thus, hp = (A-p)-l such that

0

AP

0

R p : IE;

-----+

lE_ p becomes an isometric anti-isomorphism (1.9)

Moreover, using A-(q-p)A-p = A-qf-q,-p, one may prove easily that f-q,-p hq 0 f;,q for any 0 p q. Consequently,

0

hp =

Lemma 1.2.7 Two inductive systems {1E;,f;,q} and {1E_p,J-q,-p} of Hilbert spaces are anti-isomorphic under the isometric anti-isomorphisms {hp}. Therefore, IE* is lE_ p . anti-isomorphic to ind

To be sure we shall give the inverse h;l more explicitly. Let x E lE_ p , p :::: O. Then A-px E .lj and we obtain a continuous linear function f-+ (A-px, NO o, E IE. In

fact,

I(A-px, APOol

II A - px lla

= II xll_p

Therefore there exists x* E IE; such that

Thus (1.9) is reproduced and, as is easily verified, x* = h;l(X). The correspondence x f-+ x* yields an anti-linear isomorphism from Up>o lE_ p onto IE*. In that case, identifying x with x*, we come to

(1.10) namely, the union of the increasing chain of Hilbert spaces (1.7). This is a counterpart of (1.5) and (1.6).

1.3. NUCLEAR SPACES AND KERNEL THEOREM

Lemma 1.2.8 Let {ej

7

be a complete orthonormal basis of S). Under the identi-

fication (1.10) we have

PROOF.

In fact, identifying x with x: we see that

where (1.9) is taken into consideration.

qed

From the universal property of an inductive limit we may deduce the following

Proposition 1.2.9 A linear operator T from continuous if and only if the restriction ofT to

into a locally convex space X is is continuous for all p O.

We are now in a position to discuss the real case. Let S) be a real Hilbert space and its complexification is denoted by S)C. A densely defined operator A in f) admits a unique extension to a densely defined operator A c in f)c. If A c is selfadjoint with inf Spec(Ac) > 1, we say simply that A is a selfadjoint operator in f) with inf Spec(A) > 1. Taking the real part of the complex CH-space constructed from (f)c, Ac), we obtain a real CH-space imbedded in 5). This is called a CH-space constructed from (f), A). The above discussion for complex spaces are also valid for real spaces with obvious modification.

1.3

Nuclear spaces and kernel theorem

We begin with the following

Definition 1.3.1 A locally convex space X equipped with defining Hilbertian seminorms {II·IL,}.,EA is called nuclear if for any a E A there is (J E A with a :::; (J such that the canonical map !.,,{3 : X{3 - t Xc> is of Hilbert-Schmidt type. By definition a nuclear Frechet space is a CH-space. As for structural characterization of a nuclear Frechet space we mention the following

Proposition 1.3.2 A nuclear Frechet space admits a sequence of defining Hilbertian seminorms {1·ln} :::"=0 such that (i) leln :::; leln+l' e E e, with some 0; (ii) !n,n+l : Hn+l - t H; is of Hilbert­Schmidt type, where Hn is the Hilbert space associated with 1·l n ; (iii) {Hn,fm,n} is a reduced projective sequence of Hilbert spaces;

c,

c.

CHAPTER 1. PREREQUISITES

8

(iv) proj limn--+ oo Hi; Conversely, if {H n, fm,n} is a (reduced) projective sequence of Hilbert spaces with fn,n+l being of Hilbert-Schmidt type, then proj liffin--+oo H; becomes a nuclear Frechet space. Proposition 1.3.3 A Frechet space X is nuclear if and only if so is r

.

Proposition 1.3.4 A standard CH-space

constructed from (.5), A) is nuclear if and only if A-r is of Hilbert-Schmidt type for some r > O.

PROOF. Let 11·ll p be the defining seminorms of given as lIeli p = IINello and denote the associated Hilbert space. by Suppose first that A-r is of Hilbert-Schmidt type with r > O. Then, there exists a complete orthonormal basis for .5) contained in Dom(A) such that Aej = \Ajej WIt . h Aj \ > 0 satis . fvi ,",00 \ -2r \ -(p+r) ej }oo . a comp Iet e ymg L-j=O Aj < 00. Note t h at { Aj j=O IS orthonormal basis for and

----> is of Hilbert-Schmidt type for all p O. Hence the canonical map fp,p+r : Therefore, is nuclear. is nuclear. Let {1·ln};:"=o be a sequence of Hilbertian Conversely, suppose that seminorms described as in Proposition 1.3.2. Since proj liIl'lp--+oo as well, we may find n 0 and r 0 such that

lIell o::; C lel n , with some C, C'

O. Then we have a chain of canonical maps: '"

"'0

= ')c;

f--

Hn

1n,n+1 f--

H n+1

f--

'"

"'r'

Since fn,n+l is of Hilbert-Schmidt type, so is the composion of the three which is nothing but the canonical map ----> = .5). Let {ej be a complete orthonormal basis of.5). The obvious relation

means that

is an orthonormal sequence in is of Hilbert-Schmidt type, IIA-rej Hilbert-Schmidt type. ---->

= .5)

Since the canonical map < 00, that is, A-r is of qed

In particular, Corollary 1.3.5 A standard CH-space constructed from (.5), A) is nuclear if A is a

positive selfadjoint operator with Hilbert-Schmidt inverse.

1.3. NUCLEAR SPACES AND KERNEL THEOREM

9

For two vector spaces X and !D we denote by X®aIg!D their algebraic tensor product. Various locally convex topologies can be introduced into X®aIg!D if X and !D are locally convex spaces. Among others we shall be mostly concerned with 7r-topology. The 7rtopology on X ®aIg !D is the strongest locally convex topology such that the canonical map X x!D -+ X®aIg!D is continuous. Let {II·IL,} and be defining semi norms of X and !D, respectively. Then the 7r-topology is defined by the seminorms wE

= inf

X ®aIg!D,

j

where the infimum is taken over all finite pairs 7}j) satisfying w = L,j ® 7}j, E X, 7}j E !D. The completion of X ®a1g !D with respect to the 7r-topology is called the s-iensor product and is denoted by X ®".!D. The following facts are useful.

Proposition 1.3.6 Let X and!D be Frechei spaces. Then every element w E X ®". !D is the sum of an absolutely convergent series w = ® 7}j, where ),i=1 and (7}j),i=1 are respectively sequences in X and!D converging to zero. Proposition 1.3.7 If X and!D are nuclear spaces, so is X ®". !D. Let Hand K be Hilbert spaces with inner products ("')H and (',')K, respectively. Then one may define an inner product in H ®a1g K by

,

= I:

® 7Jj)

® 7}i ,

,

',)

)

H (7}i , 7Jj) K .

The completion of H ®aIg K with respect to this inner product is denoted simply by H®K. Obviously, H®K becomes a Hilbert space again and is called the Hilbert space tensor product. It is noted that the Hilbert space tensor product does not coincide with the 7r-tensor product (as locally convex spaces) whenever both H and K are infinite dimensional.

Proposition 1.3.8 Let X and !D be locally convex spaces with defining Hilbertian respectively. Let X a and be the Hilbert seminorms {IHlcr}aEA and spaces associated with 11·lla and respectively. Then, {Xa ® becomes a projective system of Hilbert spaces. If in addition X or!D is nuclear, we have X e, !D

3!

proj lim (X a ®

Let X and !D be locally convex spaces with defining semi norms {11·lla}aEA and respectively. We denote by £(X,!D) the space of continuous linear operators from X into !D equipped with the topology of bounded convergence, namely, the locally convex topology defined by the seminorms:

IITlIs" = sup {ES 'fJ

fJ

T E £(X, !D),

where S runs over the bounded subsets of X and (3 E B. By definition X* or = £(X, C) according as X is a real or complex space.

= £(X, R)

CHAPTER 1. PREREQUISITES

10

Let 3 be another locally convex space with defining seminorms We denote by B( X, 3) be the space of jointly continuous bilinear maps from X x into 3. It is equipped with the topology of hi-bounded convergence, namely, the locally convex topology defined by the seminorms:

3),

¢J E

r.

where S1 and S2 run over the bounded subsets of X and respectively, and, E When 3 = C or = JR, we put = 3) for simplicity. There is a canonical correspondence between T E £(X 3) and ¢J E 3) given by E X, 'fJ This gives rise to a universal property of the 7r-tensor product.

3 it holds that

Proposition 1.3.9 For any locally convex spaces

as vector spaces. We then recall the canonical correspondence among tensor products, spaces of conbe two tinuous linear operators and spaces of continuous bilinear forms. Let X, locally convex spaces. --+ X). With each ( = L: 0 'fJi E X 0alg 1. X 0alg continuous linear operator T( E £(!D*, X) defined by

we associate a

YE 2. X* 0alg --+ £(X, With each w = L: Xi 0 'fJi E X* 0alg continuous linear operator T; E £(X,!D) defined by

we associate a

--+ B(X, With each z = L: Xi 0 Yi E X* 0alg 3. X* 0alg continuous bilinear form ¢Jz E defined by

we associate a

4.

--+

£(X,

With each ¢J E

=

,'fJ),

we associate T¢J E £(X, EX,

5. (X 0". --+ B(X, With each w E (X 0". bilinear form ¢Jw E B(X,!D) defined by E X,

by

'fJ

we associate a continuous 'fJ

This is a particular case of Proposition 1.3.9 and, in fact, as vector spaces.

(X 0".

1.4. STANDARD CR-SPACES OF FUNCTIONS

11

The above listed maps are all injective. From that viewpoint a nuclear space enjoys a significant property stated in the following Theorem 1.3.10 (KERNEL THEOREM) Let X be a nuclear Frechet space and let!D

be a Freehet space. Then the above listed five linear injections yields (topological) isomorphisms:

X ®,,!D

((!Do, X), ((X, !D), ®,,!D (X ®,,!D)*

r e, !D

r

0

B(X,!D)

((X, !DO).

Here is a quick remark on separately continuous bilinear maps. Let B.ep(X,!Di 3) be the space of separately continuous bilinear maps from X x !D into 3. If 3 = C or = R, we write simply B.ep(X,!D) for B.ep(X,!Di 3). In general, there is a crucial difference between B.ep(X,!D; 3) and B(X,!Di 3). In this connection the following result is useful. Proposition 1.3.11 Let X,!D, 3 be locally convex spaces. Then, B.ep(X,!Di 3) = B(X,!D; 3) holds if both X and!D are Frechet spaces; or if X,!D and 3 are all strong dual spaces of reflexive Frechet spaces.

In general, for any locally convex spaces X and !D there is a canonical isomorphism: as vector spaces, where is the dual space of !D equipped with the weak topology, i.e., the topology of pointwise convergence. Therefore, Proposition 1.3.12 If both X and!D are Frecne! spaces, then

as vector spaces.

1.4

Standard CH-spaces of functions

In §1.2 we discussed construction of a standard CH-space from a pair (fj, A), where A is a selfadjoint operator in a Hilbert space fj with inf Spec(A) > o. We discuss in this section the case where fj is an L 2-space. Let fl be a topological space equipped with a a-finite Borel measure u, (All the measures under consideration are assumed to be a-finite in these lecture notes.) Suppose that we are given a selfadjoint operator A in the (real or complex) Hilbert space fj = L 2(fl, v) with infSpec(A) > o. Then the standard CH-space constructed from (fj, A) is written explicitly as SA(fl). As usual, we understand that SA(fl) and SA('o) are spaces of test functions and generalized functions (or distributions) on fl, respectively. That the delta function (evaluation map) is a member of SA(fl) is not clear at all. On the contrary, continuity of a test function does not follow automatically. By construction each element of SA(fl) is merely a function on fl which is determined up to v-null functions. We thus need the first hypothesis:

CHAPTER 1. PREREQUISITES

12

(HI) For each function { E SAUl) there exists a unique continuous function {' on fl such that {(w) = ((w) for II-a.e. wE fl. Once this condition is satisfied, we always regard SA(fl) as a space of continuous functions on fl and we do not use the exclusive symbol (. The uniqueness in (HI) is equivalent to that any continuous function on fl which is zero v-a.e. is identically zero. Under (HI) we put two more hypotheses to keep a delta function in SA(fl): (H2) For each wE fl the evaluation map Ow : { linear functional, i.e., Ow E SA(fl).

I-t

{(w), { E SA(fl), is a continuous

(H3) The map w I-t Ow E SA(fl), wE fl, is continuous with respect to the strong dual topology of SA(fl). As will be clear in Proposition 1.4.3, hypothesis (H3) relates to a certain property of tensor products. Proposition 1.4.1 Let SA(fl) be a standard CH-space constructed as above and let be a sequence converging to 0 in SA(fl). If (Hi) and (H2) are satisfied, then the sequence converges pointwisely, i.e., limn_co {n(w) = {(w) for

en E SA(fl), n = 1,2,""

any W E fl. Moreover, if (HS) is satisfied in addition, the pointwise convergence is uniform on every compact subset of fl. PROOF. By (H2) the delta function

v >0 depending on w E fl. I{n(w) -

Ow

E

SA(fl), and hence

118wll_p < 00

for some

Then,

{(w)1 = I(ow ,en - 01

S; 11 8wlLplien

- ell p'

Hence limn_co {n(w) = e(w). Suppose (H3) is satisfied and let flo c fl be a compact subset. Then, by assumption {ow; w E flo} is a compact subset of SA(fl), and therefore it is bounded. It follows from Proposition 1.1.3 that there exists p :::: 0 and C :::: 0 such that

Hence we have

and therefore limn.... co {n(w)

= {(w)

uniformly on flo.

qed

Lemma 1.4.2 Let A be a selfadjoint operator in L2(fl, v) with A-r being of HilbertSchmidt type for some r > O. Assume that SA(fl) satisfies (Hi) and (H2). Then,

where 0 < Al :::; A2 :::; ... are the eigenvalues of A.

1.4. STANDARD CH-SPACES OF FUNCTIONS

13

PROOF. By assumption there exists a complete orthonormal basis {e, such that Aej = )..jej. Then, by Lemma 1.2.8, 00

IlowlI:r

=L

j=o

00

l(ow,A-r ej)1 2 = L)..j 2r l (ow,ej)

2

1

j=O

C

L2(fl, V)

00

= L)..j2rlej(w)12,

wE fl.

j=O

We then integrate both sides to get the result.

qed

It is noteworthy that properties (Hl)-(H3) are preserved under forming a tensor product. Recall that for two selfadjoint operators Ai in S)i, i = 1,2, their tensor product Al @ A 2 becomes a selfadjoint operator in S)I @S)2 in a canonical way. Moreover, note that if inf Spec(A i) > 0, i = 1,2, then inf Spec(A I @ A 2 ) > 0 as well.

Proposition 1.4.3 For i = 1,2 let fl; be a topological space with a Borel measure Vi and let Ai be a positive selfadjoint operatorin L 2(fl i, Vi) with Hilbert-Schmidt inverse. Then (1.11) SAl0A2(fl l x fl 2) SAl (flt} @". SA2(fl2) 2(fl 2( under the identification: L fl l x fl 2, Vl x V2; R) L2(fl b Vl;lR)@L 2, V2; lR). Moreover, if both SAl (flt} and SA2(fl 2) satisfy hypotheses (Hl)-(H3), so does SAl0 A2(fll x fl 2 ) . PROOF. We see from Corollary 1.3.5 that SA.(fl i) is nuclear. It then follows from Proposition 1.3.8 that the standard CH-space constructed from (£2(fl l, Vl) 1)9 L2(fl 2, /12), A l @ A 2) is isomorphic to the 1l"-tensor product of the standard CH-spaces constructed from (L2(fl l, vt},At} and (L 2(fl 2,v2),A 2). This proves (1.11). We now suppose that both SAl (fl l) and SA2(fl 2) satisfy hypotheses (Hl)-(H3). For (E SAl0A2(fl l x fl 2) SAl(fld @".SA2(fl2) we put

('(WbW2) = (OWl @OW2'()'

WI E fll ,

W2 E fl 2·

We note that (' is a continuous function on fl i x fl 2 • Indeed, (' is a composition of continuous maps:

Wi

1-+

ow; E SA.(fli),

Wi E fli,

i

= 1,2,

is continuous by (H3);

x,y

1-+

x@y E SAl (fll ) @",SA2(fl2 ) ,

x E SAl(fld,

y E SA2(fl2),

is continuous by the definition of 1l"-tensor product;

SAl (flt) @". SA2(fl2)

SA10 A2(fll x fl 2)

by the kernel theorem (Theorem 1.3.10). We then prove that (' coincides with ( for Vl x V2-a. e. Take an approximating sequence (n E SAl(flt}@algSA2(fl2) converging to (. We see from Lemma 1.4.2 that lI(n -

=

{

Jflt X 0 2

{

JOl

X02

\(n(Wt,W2) - (Wl,W2W VI(dwt} V2(dw 2)

I (OWl

@

0"12 , (n - () 12VI(dwt} V2(dw2)

< II(n-(lIi J( IIOw,II:Ivl(dwl) J( 1I 0W 0 for any non-empty open subset U en; (ii) every ej is a continuous function on n; (iii) there exists an open covering n = UI' nl' and a(-y) 0 such that

ThenJH1)-(H3) are satisfied. Moreover, for ¢ E SA(n) the unique continuous function ¢ on n such that ¢(w) = ¢(w) for v-a.e. wEn is given by the absolutely convergent series: . is a continuous function on E_p with ¢>(xo) > O. Hence there exists an open neighborhood U C E_p of Xo such that ¢>(x) > 0 for all x E U. On the other hand, since ¢>(x) = 0 for u-e:e. x E E* by assumption, we see that 1t(U) = O. But this will yield contradiction. In fact, we shall prove that any non-empty open subset U C E_ p is of positive measure. Since H C E_ p is a dense subspace, we may choose a countable subset

39

3.2. CONTINUOUS VERSION THEOREM

put

C H which is dense in E_ p • Let t

> 0 be an arbitrary positive number and

Obviously, 00

e., = U Bk(t).

(3.12)

k=O

Since the Gaussian measure j.t is quasi-invariant under translations by H (Proposition 2.1.6), if j.t(B", (t)) = 0 for some kl then j.t(Bk(t)) = 0 for all k. But this is impossible because of (3.12) and the fact j.t(E_ p ) = 1, which follows from Theorem 3.1.1. Hence j.t(Bk(t)) > 0 for all k and t > O. Since U C E_ p is a non-empty open subset, it contains an open ball B,,(t) for some k and e. Hence j.t(U) > O. qed We next prove

El

Proposition 3.2.3 Assume that a sequence in E L:::'=o n! < 00 for all p O. Then the series

n

,

n

=

0,1,2"", satisfies

00

"£ (: x 0 n : , in)

n=O

converges absolutely at each x E E*. For the proof we need a simple lemma. Lemma 3.2.4 If x E E* satisfies

IxL p < 00

with p > 1/2, then

vnT

+ IXI_pf .

I :x 0 n : i., PROOF. First note that Proposition 2.2.1,

lrL,
1/2. In fact, since r

00

00

00

j=O

j=o

j=O

=

ej

(g)Cj

by

2 p+2 = '" ,\-:-4p < ,,-4 ' " ,\-:-2 = p4 p- 282 < 00 Irl-p = LJ [eJ ® e·!2 J -p LJ J 0 LJ J '

whenever p > 1/2. Since . 0n._

.x

.-

(1)'"

[n/2J

n. t:o (n _- 2k)!k!2

k

r

0"

®x

0(n-2")

,

(3.13)

which is shown in Corollary 2.2.4, we have

I :x

[n/2]

0n

'" : I-p < t:o

r

n. (n _ 2k)!k!2 k

IT I"-p Ix In-p- 2" .

Using an obvious inequality:

vnr

1

-< -k k!2

-

(2k)!'

(3.14)

CHAPTER 3. WHITE NOISE FUNCTIONALS

40 we have

n! v'iiT I Ik I In - 2k (n _ 2k)! (2k)! T -p x -p

I .·x®n.' -,p <
1/2 such that Lemma 3.2.4 that

f= I(:x®n: ,In) I < n=O

+ IxL p < 1.

It then follows from

00

LI:x®n:l_pllnlp n=O < L v'iiT + lxl_ p) I/nlp n=O n) < n! 1/2 + 'X'_pf 1/2 00

n

(E

(E

Consequently, (3.15) qed

as desired. Suppose we are given ¢> E (E) with Wiener-Ito expansion:

¢>(x) = f(:x®n;,ln).

n=O

2

Then it follows from Theorem 3.1.5 that In E and n! I/nlp < p O. On the other hand, it follows from Proposition 3.2.3 that 00

==

L

n=O

00

for all

(:x 0n: ,In)

converges at every x E E*. Therefore, ¢>( x) = for u-s:e. x E E*. Thus, for the proof of Theorem 3.2.1 it is sufficient to prove that is a continuous function on E* whenever In E and n! lin < 00 for all p O. The statement will be formulated in Proposition 3.2.11.

41

3.2. CONTINUOUS VERSION THEOREM

It is much simpler to show that the restriction of if> to E_ p is continuous with respect to the norm '·Lp ' However, this is not enough to assert the continuity of if> on E* (equipped with the strong dual topology), because the inductive system is not strict. We start with construction of defining Hilbertian seminorms of E*. Let C be the set of sequences C = such that Co C1 .•• > O. For C E C we put 00

=L

1'=0

though possibly Denc

= 00.

eE E,

C;

Then define

sup {I (x ,e) I; DeDc ::; 1,

Ixlc =

(3.16)

eE E},

x E E*,

(3.17)

which is always finite. Obviously, for any C E C XEE*,

though DeDc =

00

eEE,

(3.18)

can happen.

Lemma 3.2.5 {I .

Ie }cEC

is a set of defining Hilbertian seminorms of E*.

PROOF. For C = (C p E C let E( C) be the subspace of all e E E with De Dc < 00. Then, E(C) becomes a real Hilbert space with the norm n· Dc. Let f : E(C) -+ E be the natural injection which is apparently continuous (but does not have dense image in general). Then, j* ; E* -+ E(C)* becomes a continuous linear operator. We denote by (. ,.) and HE(c) the inner product and the norm of E( C), respectively. Note that E( C) E( C)* through the inner product (. , .) E(C)' Then, by definition, for x E E*,

Ixlc

=

e

sup {I (x ,!(e)) I; DeDc ::; 1, E E(C)} sup {I (j*(x) ,e)E(C) I; DeDc ::; 1, E E(C)}

e

1f*(x)IE(C) . Therefore I . Ie is a Hilbertian seminorm on E*. The strong dual topology of E* is defined by the seminorms X

1-+

sup {I (x ,e) I;

eE S},

x E E*,

where S runs over all bounded subset of E. Note first that {e E E; Denc ::; 1} is a bounded subset of E for any C E C. Hence, in order to prove that H . lc}cEc is a set of defining seminorms of E* it is sufficient to show that for any bounded subset SeE with S", {O} there is C E C such that S c {e E E; De Dc ::; 1}. In fact, given a bounded subset SeE, we put p

= 0,1,2"",

CHAPTER 3. WHITE NOISE FUNCTIONALS

42

and define inductively a sequence C

Co

=

1 In

y20'0

=

by

c, =

'

2p + l

O'p

,

CP_I},

P = 1,2,···.

As is easily verified, this satisfies the desired property. Using the Fourier expansion, we may give Lemma 3.2.6 For C

=

E

qed

Ix Ie explicitly.

C it holds that x E E".

PROOF. Since {Cj}f=,o is an orthogonal set for every (Cj,Ck)E(C)

= 0 whenever

1·l p '

it follows from (3.16) that

ej,ek E E(C). It is then easy to see that

is an orthonormal basis for E(C) s:: E(C)", where we understand = 00. On the other hand, each x E E* admits a Fourier expansion:

=0

if

co

x=L(x,ej)ej, j=O

which converges in E". Thus the result follows immediately.

qed

The topology of (E®n)" is defined in a similar way. Namely, for C E C put oo

=

L C;,' .. C;n

'

Pl"",Pn=O

(3.19)

where

L 00

=

I (w, ejl

0 ... 0 ejJ

il,···,jn=O

12 1eiJ 1:, .. ·Iejn I:n.

(3.20)

Then for FE (E®n)* we put

IFlc = sup {I (F ,w) I;

::; l,w E E®n}.

Then, in a similar way to Lemma 3.2.5 we see that {I . lc}cEc is a set of defining Hilbertian seminorms of (E®n)*. We next prove the following Lemma 3.2.7 Let C

though

IFI_ p = 00

= E C. Then I Fie ::; c:;n jF!_p ,

may happen. Moreover,

I . Ie is a cross norm, i.e., Xl, •.. ,Xn

E E*.

43

3.2. CONTINUOUS VERSION THEOREM PROOF. A similar argument as in Lemma 3.2.6 yields 00

IFlb= L

...

i1"",;n=0

It is then obvious that

I. Ie is a cross norm. =

ne.n-2

u Jue

L..

p

p=o

Since

< C-2X:-2p P J

J

-

for any p 2': 0, we obtain

L

p (F ,ejl ® ... ® ej")2 A;2 ... Aj,,2P = C;2n IFI:' p

00

IFlb

C;2n

jl,···,in=O

as desired.

qed

Lemma 3.2.8 Let q.

Then, by construction, C' S; C and < 1 - p2. Moreover, since Irlc' S; for all p 0 by Lemma 3.2.7, we have Irlc ' S;

=

lrL,

1 p2 S; - - - .

e

This completes the proof. Combining Theorems 3.2.1, 3.2.12 and 3.2.13, we come to the following

qed

48

CHAPTER 3. WHITE NOISE FUNCTIONALS

Corollary 3.2.14 The space (E) of white noise test functionals satisfies hypotheses (Hl)-(H3) introduced in §1.4. As an immediate application of Proposition 1.4.1, we obtain

= (zO

(fn,(A2Po®n)zn.

(3.32)

n=O

From Theorem 3.3.7 we see that (3.32) is the Taylor expansion of an entire analytic function. But by assumption (3.31) this is identically zero on {z = in; t E R.}. Hence every coefficient in the Taylor series (3.32) vanishes. Namely,

(fn, ( A2Po®n)

= 0,

for any

E E,

and hence, by the polarization formula,

(fn, for any

6,'" ,

E E and n

=0

QI) ••• QI)

= 0,1,2,· .. From this

we conclude that

Un , eil QI) .•• QI) ei = 0, namely, i« = 0 for all n = 0,1,2,· . '. Therefore = o. n )

qed

CHAPTER 3. WHITE NOISE FUNCTIONALS

52

Corollary 3.3.9 Let a E C with a =f:. 0 and let tP E (E)*. If StP(aO E E, then tP = o. PROOF.

Since = O.

all

By assumption, E E.

for all

tP

= 0 for

E E} spans a dense subspace of (E) by Corollary 3.3.8, we conclude that qed

As an application of S-transform we prove the following

Proposition 3.3.10 For any

4>dx)

=

E He,

f

(:x®n:,

n=O

= e(x,e)-(e,e)!2.

n.

In particular,

A.( 'l'e x PROOF.

JL(dx)'

This is obvious if

= O.

= e(x,e}-({,{}/2,

Suppose

E H.

E E,

We first prove that for

4>e(x)

x E E*,

x E E*.

(3.33)

=f:. O. It follows from Lemma 2.2.7 that

The last series is the generating function of the Hermite polynomials (B.I) and we obtain

which proves (3.33). Now for E He we consider the S-transform of 4>e. It follows from Lemma 3.3.2 that (3.34) S4>e(T/) = «4>e, 4>'1)) = e(e.TJ), 71 E Ee· On the other hand, consider the S-transform of the function:

4>( x)

= e(x ,e}-(e.e}!2.

In view of (3.33), for 71 E E we have

S4>(T/)

«4>,4>'1))

=

k.4>(x)4>TJ(x)JL(dx) [ e(x .eHe,{)!2 e(x,TJHTJ .TJ}!2JL(dx).

lE·

3.4. CONTRACTION OF TENSOR PRODUCTS

53

Using Lemma 2.1.6, one may immediately obtain

StP(e)

= e-(Ce}/2-(TI ''1}/2 e(UTI.HTI}/2 = e(e 'TI},

1] E

E.

(3.35)

We see from (3.34) and (3.35) that 1] E

E.

= tp.

Consequently, by Corollary 3.3.9 we conclude that tPe Proposition 2.1.6.

The rest follows from qed

The second half of Proposition 3.3.10 implies Corollary 3.3.11 If tp E (L 2 ) , then

=

Stp(O

L. tp(X + OdJL(x),

eE E.

Contraction of tensor products

3.4

Recall that {ej is a complete orthonormal basis of He such that ej E Ee and Aej = >"jej. Employing a simplified notation

one has

= L: I (f , e(j)} j

where the sum is taken over all possible j

lAP

=

= (jl,' .. ,jm).

In fact,

= L:1((A@m)1'f,ejl 0···0 ejmW j

= L: IU, APeil 0 j

'L..J " 1(1 '11 e·

=

.. ·0 APejm} 1

fJI . . . fJI 'Cf 'Cf

2

e·3m }1 2 >..21' ... >..21' 11 3m

j

L: I (f , e(j)} j

We now generalize this situation. Using a similar notation: i=(il,· .. ,i ,),

for

I

we put

E

111"m;1',q

=(

tr I

(f ,e(i) 0 e(j)}

,

p,q E JR.

(3.36)

54

CHAPTER 3. WHITE NOISE FUNCTIONALS

This is possibly infinite, however, is always finite for I E seen from definition, pE R, I/l p = 1/11,m;p,p ,

and

1/11,m;p,q

s /r+m'l/ll,m;p+r,q+s ,

As is immediately

r,s2':O.

p,qER,

In fact, for p, q E Rand r, S 2': 0,

=

1/1;,m;p,q

ij

< L I (f ,e(i) ® e(j)) 12 p21r le(i)I;+r p2ms

=

ij

p2(lr+m.) 1/12I,m;p+r,q+s .

The above relations are used without special notice. We now consider two elements I E and 9 E

Since

L (f ,e(i) ® eO)) (g, e(i) ® e(k)) i

converges absolutely, one may consider a new element in

I ®l 9 =

,e(i) ® e(j)) (g, e(i) ® e(k))) eO) ® e(k), J,k

where

defined by (3.37)

1

e(k) = ekl

® ... ®

ek n

k = (kI,'" ,kn ) .

,

We must show that (3.37) is well defined, namely the series converges. purpose it is sufficient to show that

For that

2I

(f ,e(i) ® e(j)) (g, e(i) ® e(k))1 e(j ) ® e(k)l; < J,k

for all p

00

1

2': O. In the following we prove a more general result.

Lemma 3.4.1 For any p, q, r E R it holds that

II ®l g!m,n;p,q PROOF. Since

jy

I

IfII,m;r,p Igll,n;-r,q,

le(i)l_r le(i)lr

(f , ,(i) @ ,(j)) (g,

E

9E

(3.38)

= 1, we have

,m

@

'(k))

I'

(f ,e(i) ® e(j)) 12Ie(i)I;)

(g ,e(i) ® e(k))

.

55

3.4. CONTRACTION OF TENSOR PRODUCTS

Hence, by definition (3.36) and (3.37) we have

1 2

If

rg/

(f, e(i) @ e(j)) (g , e(i) @ e(k)) le(j) le(k) J.k

J

L I (f , e(i) @ e(j)) 12 le(i)I; le(j)


o. Therefore, the map (f,g)

(3.39)

f @l 9 yields a continuous bilinear map: . E0(l+m)

\6I·C

E0(I+n)

Xc

E0(m+n)

•

PROOF. In view of Lemma 3.4.1, we obtain

If @l glp = If @l glm,n;p,p < IfI1,m;r,p IgI1,n;-r,p for any r E R Hence, putting r

= p we come to

If @l glp :::; Ifll,m;p,p Igll,n;-p,p :::; Ifl p p2pllglp' This proves (3.39).

qed

We have defined f@lg by taking a particular basis however, the definition is independent of the choice of basis. In fact, @l is characterized as a unique continuous bilinear map from x into satisfying the condition that for

56

CHAPTER 3. WHITE NOISE FUNCTIONALS

f = 6 0···0 el+m and 9 holds that

= 1]1 0··· 01]I+n,

= (6,1]1) ... (el ,T/I) el+1 0

f 0 19

where 6,'" ,el+m,1]1,'" ,1]I+n E Ee, it

... 0 el+m 01]1+1 0 ... 0 'TJl+n'

Using Lemma 3.4.1 we may extend 0 to a bilinear map from into In fact, it follows from Lemma 3.4.1 that

x

1

1

1

If 0 gl-p = If 0 glm,n;-p,-p :::; Ifll,m;r,-p IgII,n;-r,-p , Taking r

r E JR..

= -p, we come to 1

If 0 gl-p :::; Ifll,m;_p,_p IgI1,n;p,-p :::;

l/l., p2 pn Igll,n;p,p = p2 pn l/l., Igl p '

Therefore F 0 1 9 is defined for F E Lemma 3.4,4 Let F E

and 9 E

IF 0 1 g[_p In particular, the map (F, g)

and 9 E

f-t

:::; p2 p n

Moreover,

Then,

lFl-pIgl p,

p

2:: O.

(3.40)

F 0 1 9 becomes a separately continuous bilinear map:

01:

-+

X

We show the second half of the assertion. Given a bounded subset S C we see from (3.39) that

PROOF.

sup I(F 0 g, h)1 :::; sup /pn 1

hES

hES

lFl-p Igl pIhlp = M lFl_pIglp'

(3.41)

= p2 pn SUPhES Ihlp'

Suppose F E is fixed and choose p 2:: 0 with Then (3.41) implies that 9 f-t F0 19 is continuous. On the other hand, for we see from Proposition 1.2.9 that (3.41) implies the continuity a fixed 9 E qed of F f-t F 0 1 g. where M

lFl_p < 00.

The next result is also useful. Lemma 3.4.5 For F E

and 9 E

it holds that

IF 0 1 gil' :::; pqn 1F11,m;-(p+q),p Igl p +q , In particular, for F E

IF 0 PROOF.

and 9 E 1

q 2:: O.

(3.42)

it holds that

gil' :::; pqn lFl-(p+q) Igl p+q ,

P E JR.,

q 2:: O.

(3.43)

In view of Lemma 3.4.1, we have

IF 0 1 gil' = IF0 1 glm,n;p,p :::; Taking r

p E JR.,

IFII,m;r,p IgII,n;-r,p'

r E lR..

= -(p + q), we obtain 1

IF 0 gil' :::; 1F11,m;-(p+q),p IgII,n;p+q,p :::; IFII,m;-(p+q),p pqn Iglp+q , which proves (3.42). For (3.43) one need only to take m The left and right contractions are related as follows.

= 0 in (3.42).

qed

3.4. CONTRACTION OF TENSOR PRODUCTS

Proposition 3.4.6 For F E

9E (F fi/ g, h)

57

and h E

= (F ,g 0 n

it holds that (3.44)

h).

PROOF. Suppose F is fixed. It follows from Lemma 3.4.5 that both sides of (3.44) are continuous bilinear forms in 9 and h. Therefore we need only to check the identity for 9 = e(i) 0 e(k) and h = e(j) 0 e(k). But the verification is straightforward from definition. qed

One may employ (3.44) as the definition of F 0/ g. Given F E 9E we consider a linear form: h

hE

(F ,g 0 n h),

f-+

and

This is continuos by Proposition 3.4.3. Therefore there exists a unique element in which in fact coincides with F 0/ g. We call F 0/ 9 left contraction and again. In a similar way, F 0/ g, 10/ G and 10/ G are defined for I E GE Finally, we consider the symmetric case. For symmetric elements F E and 9 E

obviously it holds that

The symmetrization of F 0/ 9 is denoted by F@/g. Using an obvious inequality:

I

1111' ::; IIII"

E

p E lR,

we may restate (3.39), (3.40) and (3.43) for the symmetric case. Proposition 3.4.7 For p,q 2: 0,

I/@/gll' ::; p21'/ IIII' lsl, , IF@/gl-I' ::; p21'n

IFI_I' Igll' '

I

E

E0(1+m) C

,g

E E0(/+n)

c

F E (E®(I+m»)* C 8ym'

9

,

(3.45)

E E0(l+n)

c

, (3.46)

and lor p E R and q 2: 0 we have (3.47)

The following assertion shows a rule of associativity. Proposition 3.4.8 For F E that

G E

and h E

it holds (3.48)

58

CHAPTER 3. WHITE NOISE FUNCTIONALS

PROOF. Note first that both sides of (3.48) are continuous in h for fixed F and G. In fact, from (3.47) we see that for any p, q, q' 2 0,

and

IF®k(G®,h)l p $ pq/+q'(k+m) IFI-(p+q) IGL(p+q+q/) Ihlp+q+ql' Hence, it is sufficient to show identity (3.48) for h case, the verification is straightforward.

qed

and G E

Proposition 3.4.9 Let F E and 9 E

eE Ec. But, in that

=

Then, lor any

I

E

it holds that

(F®tJ, G®mg) = (F PROOF.

@

G, I

@n

(3.49)

g) .

In view of (3.46) and (3.47) we observe that for any p 2 0,

This implies that the left hand side of (3.49) is a continuous bilinear form in I and g. Obviously, the same is true for the right hand side. Therefore, for the proof we need only to show the identity for I = e®(l+n) and 9 = 17®(m+n), where 17 E Ec. But, in that case (3.49) is obviously true. qed

e,

Taking m

= 0 and

G

= 1 in (3.49), we come to the I

Corollary 3.4.10 For F E

E

following

and 9 E

it holds that

Some norm estimates discussed in this section are collected in Appendix C.

3.5

Wiener product

Lemma 3.5.1 For

PROOF.

e, 17 E Ec

and m, n 2 0, we have

We note first the obvious identity: s,t E C.

59

3.5. WIENER PRODUCT

Then by Proposition 3.3.10 we have

(3.50) The last expression is computed as follows:

Therefore, (3.50) becomes

L 00

m,n=O

(:x®m:,

=

s'" r (:x®n:, 1]®n)" m. n.

E

k! (

)

) (: X®(m+n-2kj:

The assertion follows by comparing the coefficients of smtn in both sides. Theorem 3.5.2 For any f E

and 9 E

qed

we have

PROOF. By the polarization formula the assertion is true if both f and 9 are algebraic tensor products by Lemma 3.5.1. Since 0k is a continuous bilinear map by Proposition 3.4.3, the general case follows. qed The formula stated in Theorem 3.5.2 is known as Wiener productformula. A simple application of Theorem 3.5.2 implies the following Proposition 3.5.3 Let ¢J,1jJ E (E) be given as

L (: x®m : ,Jm) , 00

¢J(x) =

m=O

=L 00

1jJ(x)

n=O

(:x®n: ,gn).

60

CHAPTER 3. WHITE NOISE FUNCTIONALS

Then the Wiener-Ito expansion of ¢J'Ij; is given by 00

(x)1jJ(x)

=L

(3.51)

(:X®l:, hi)'

1=0

where

Lemma 3.5.4 Let notations and assumptions be the same as in Proposition 3.5.3. Then for any 1 = 0,1,2,··· and 0.,13, P ? OJ

1! Ihd; s (l + 1)(p2a + p2(3)IIlII;+o II ¢ 11;+19

(1

t:

k

) p2k(+13+2 p).

(3.53)

PROOF. Note first that

Ifm+k®k9n+klp

s /pk Ifm+kl p 19n+kl p'

Then (3.52) is estimated as follows.

Vi! IhIl p S Vi!

=

{( m

mE=1

xJ(m


,1/J

E (E),

is a continuous bilinear form on (E). In fact, by Lemma 4.3.1 we have (4.19)

where

M

= p-p

(1Imm)1/2

p)

(-2pe log p p-

(l+m)/2

(4.20)

Therefore there is a continuous linear operator E1,m(K) E £((E), (E)*) such that

see e.g., Proposition 1.3.12. It then follows from (4.19) that

CHAPTER 4. OPERATOR THEORY

82 Then (4.18) follows from (4.20).

qed

The operator 5 I ,m(" ) is thus defined through two canonical bilinear forms:

1/))'

89

4.4. SYMBOLS OF OPERATORS

On the other hand, 0 we have I

, WTJ)

I
0, , > 0 and r 0 are chosen suitably. First of all we are given arbitrary p 0 and e > 0 with f < (2e382)- 1. Then take C 0 and q 0 so as the assumption (ii) is satisfied. We now choose r 0 with 4 logp and put

2 0:-(3---

-

Then,

-

logp'

p-fJ/ 2

p-Dt/2

1

-(3e log p = 2

-o:elogp and hence

Finally, we may take r number satisfying

0 to have A

6 4

4 --1

e 8 p2(Q+rol < 2

Then A < 1 for all r

< 1. In fact, let ro

- 2'

ogp

= ro(q) be the rO

o.

roo Consequently, (4.50) becomes

00

L II E I,m(lI:l,m)lI p_ l,m=O

1

CM(f,q,r) 1Illp+q+r+l'

E

(E),

minimum

97

4.4. SYMBOLS OF OPERATORS

where

qed

This completes the proof.

Simply combining Propositions 4.4.3, 4.4.4 and Theorem 4.4.6, we obtain an operator version of the characterization theorem for generalized white noise functionals (Theorem 3.6.1). Corollary 4.4.10 Let e be a function on Ee x Ee with values in C. Then, there

exists a continuous operator S E £((E), (E)*) with e = 5 if and only if (i) for any E Ee, the function z,w 1­4 +77d is an entire holomorphic function on C x C; (ii) there exist constants C 0, K 0 and p E JR. such that

In that case ¢ E (E),

where M(/{,p, q) is a (finite) constant for all q > qo(/{,p) > O. Similarly, by Propositions 4.4.3, 4.4.5 and Theorem 4.4.7 we obtain an operator version of the characterization theorem for test white noise functionals (Theorem 3.6.2). Corollary 4.4.11 Let e be a function on Ee x Ee with values in C. Then, there exists a continuous operator S E £((E)) with e = 5 if and only if (i) for any 6, 77, 771 E Ee, the function z, w + 6, W77 + 771) is an entire holomorphic function on C x C; (ii) for any p 0 and e > 0 there exist constants C 0 and q 0 such that

In that case

II S¢lI p _ 1

CM(f,q,r)

11¢ll p + 9+ r +l '

where M( e, q, r) is a (finite) constant for f < (2e382 ) - 1 , r

¢ E (E), ro(q)

O.

In some practical problems operators on Fock space are only defined on the exponential vectors {4>e; E Ee} due to the fact that they are linearly independent (Proposition 2.3.9). The above corollaries then give us criteria when such operators belong to £( (E), (E)*) or £( (E), (E)). A simple application will be illustrated in §4.6.

98

4.5

CHAPTER 4. OPERATOR THEORY

Fock expansion

Theorem 4.5.1 For any tions {x;I,mH;"m=o, X;l,m E

=

E £((E), (E)*) there exists a unique family of distribu-

such that 00

3¢> =

L

¢> E (E),

3 1,m(X;I,m)¢>,

l,m=O

(4.51)

where the right hand side converges in (E)*. If 3 E £((E),(E)), then each kernel &; = &; and the right distribution X;1.m belongs to hand side of ({51) converges in (E). PROOF.

For a given 3

E

1])

£((E), (E)*) we put

=

71)

= ((3¢>t., ¢>TJ)) ,

E Ee.

(4.52)

Then, by Propositions 4.4.3 and 4.4.4 we see that e satisfies the conditions (i) and (ii) in Theorem 4.4.6. Therefore, there exists a unique family of kernels {X;I,m}Fm=o, I8i h h t X;l,m E (E C (I+m»)* sym(l,m)' sue t a 00

1])

=L

l,m=O

e,7J

((3 1,m(X;I.m)¢>t. , ¢>TJ)) ,

E Ee·

Furthermore, as is stated in Theorem 4.4.6,

S'¢>

='

=

00

L

l,m=O

¢> E (E),

3 1.m(Kl,m)¢>,

converges in (E)*, E £((E),(E)*) and 2(e,1]) last identity and (4.52) yield

=

for all e,1/ E Ee· The

e,7J E Ee. Since the exponential vectors span a dense subspace of (E) and both 3 and 3' are continuous, we conclude that 3 = 3'. For the second half of the assertion we need only to employ Proposition 4.4.5 and qed Theorem 4.4.7.

Definition 4.5.2 The unique expression of 3 E £( (E), (E)*) given in (4.51) is called the Fock expansion of 3 and denoted simply by 00

3

=L

=1,m(X;I,m)'

l,m=O

Proposition 4.5.3 Let expansion. Then,

=E £(E), (E)*) and let 3 = = L 00

l,m=O

(X;I,m , 7J 18il &;

e

l8im

) ,

Erm=o

=l,m(lf­l,m) be its Fock

e,1/ E Ee·

(4.53)

99

4.5. FOCK EXPANSION

PROOF. It follows from Theorem 4.5.1 that

E¢e

=

00

L

E"m(KI,m)¢e,

l,m=O

converges in (E)*. Therefore, for

((E¢e, ¢J'I))

1/ E Ee,

=

00

L

((E"m(K"m)¢Je, ¢J'I))'

l,m=O

Then, in view of Proposition 4.4.2 we obtain (4.53).

qed

Thus, in order to obtain the Fock expansion of E E £((E), (E)*) one need only to compute the Taylor expansion of (4.53). Corollary 4.5.4 Every bounded operator E on (L 2 ) admits a Fock expansion.

PROOF. If E is a bounded operator on (L 2 ) , there exists some C 2 0 such that

IIE¢Jll o

C 1I¢lIo'

¢J E (E).

Hence E E £((E), (E)*) and it admits a Fock expansion.

qed

2

Even though E is a bounded operator on (L ) , the convergence of the Fock expansion can not be discussed within the framework of a Hilbert space. The following result also illustrates this remark. Proposition 4.5.5 Let K E If E"m(K) admits an extension to a bounded operator on (L2), then SI,m(K) = 0 or 1= m = O. Namely, except scalar operators no integral kernel operator admits an extension to a bounded operator on (L 2).

PROOF. We first observe the action of E"m(K) to an exponential vector ¢e. Since

by (4.24) in Proposition 4.3.3, we have

(4.54) Defining PI(t) by

(n + l)!tn = P(t) t " Ie, n.n. we can easily see that PI(t) is a polynomial of degree 1. (This PI appeared also in the proof of Lemma 3.2.9.) Hence (4.54) becomes LJ

n=O

(4.55)

CHAPTER 4. OPERATOR THEORY

100

We now suppose that E'/,m(l\;) admits an extension to a bounded operator on (L 2). Then, there exists some C 0 such that S;

C

=

(4.56)

E Ee·

Combining (4.55) and (4.56) we obtain (4.57) Suppose that 1#-0. In order that (4.57) is true, we have E Ee.

Then, for all

t,» E Ee,

0=

(I\; ®m

e: ,'f/®/) = (I\;, 'f/®/ ®

= (S/,m(l\;) ,'f/®/

®

This means that s/,m(l\;) = 0, and therefore E'/,m(l\;) = O. If m #- 0, applying a similar argument to E'/,m(I\;)- = E'm,/(tm,I(I\;)), we come to the same conclusion. Consequently, E'/,m(l\;) = 0 unless 1= m = O. qed Therefore, the Fock expansion of a non-scalar bounded operator on (L2) is always an infinite series of unbounded operators.

4_6

Some examples

We now assemble a few examples of Fock expansions and applications of the characterization theorems for operator symbols. For y E E- the translation operator is defined by Ty¢>(x)

= ¢>(x + y),

¢> E (E).

It was shown in Theorem 4.2.3 that t; E £((E), (E)). Theorem 4.6.1 For y E E- we have 1 '= (®n) T.y -_ L.. ,-O,n y . n=O

PROOF.

(4.58)

n.

Let f, 71 E Ee. First note that Ty¢>e(x) = ¢>dx + y) = exp ((x

Hence We therefore obtain

+y

-

= e(y,e)¢>e(x).

= ((Ty¢>e '¢>'1)) = e(y,e) ((¢>e '¢>'1)) = e(y,e)e(e,'1).

101

4.6. SOME EXAMPLES

On the other hand, if 00

t; = L

E/,m(II':/,m)

l,m=O

is the Fock expansion, it follows from Proposition 4.5.3 that

L 00

(4.60)

l,m=O

Comparing (4.59) and (4.60), we conclude that

=

if I 2: 1, 1 @m for m 2: O. m!Y

0

II':O,m

This completes the proof.

qed

The above result yields the Taylor expansion of white noise functional ¢J E (E). Corollary 4.6.2 Let Y E E*. Then, ¢J E (E), where the series converges in (E). In particular,

¢J(x + y)

00

1

n=O

n.

=L

x E E*.

PROOF. It follows from Propositions 4.3.10 that =O,n

(@n) Y

= o:y'

Y E E*.

Hence the Fock expansion of Ty (Theorem 4.6.1) yields ¢J E (E).

(4.61)

Since T y belongs to £((E), (E)), it follows from Theorem 4.5.1 that the series (4.61) qed converges in (E). We next give an example of a bounded operator on (L 2 ) . Proposition 4.6.3 Let 1l'n be the orthogonal projection from (L 2 ) onto 1i n (C) . Then its Fock expansion is given by

where

Tm =

L

(_l)m-n ),=m,m(Tm), n.

= m=n L n.'( m 00

1l'n

_

00

il,"',im=O

eit

® ... ® e.; ® eit ® ... ® ei m E (E@2m)*.

(4.62)

CHAPTER 4. OPERATOR THEORY

102 PROOF.

By definition,

Hence,

Then the assertion follows immediately from Proposition 4.5.3.

qed

Note that Tl = T. It will be proved in Proposition 5.4.6 that Sm,m(Tm) is a polynomial in a particular operator called the number operator. While, one may employ the following expression as well. Sm,m(Tm) = hm 8;1 ···8;m8tl · .. 8tmdtl .. ·dtm' Recall that any ifJ E (E)" gives rise to a continuous operator from (E) into (E)* by multiplication, see Theorem 3.5.8 and Corollary 3.5.9. Proposition 4.6.4 Let ifJ E (E)" be given with Wiener-Ito expansion:

=L 00

ifJ(x)

n=O

(:x 0n :, Fn) ,

where Fn E (E0 n):ym' Then as multiplication operatorifJ admits Fock expansion:

Moreover, ifJ E £((E), (E)) as multiplication operator if and only ififJ E (E). PROOF.

We first compute the symbol of ifJ:

((ifJ ,cP{+>7)) e({ ,>7)

n! \ Fn , (e

e(C>7) e(C>7)

f= (F

n ,

n=O

(e + q)0 n )

.

103

4.6. SOME EXAMPLES

Therefore, 00

L

n=O

(Fn,

7/)®n)

( r; , ( )

Ern( t k)

0 7/®(n-k) )

n

0 7/®n).

Since F n +k is symmetric, we have

Then, we need only to apply Proposition 4.5.3. Suppose (I> E £((E), (E)) as multiplication operator. Then it follows from Corollary 4.4.11 that for any p 0 and f> 0 there exist some C 0 and q 0 such that

Viewing 7/)

and putting

= (((I> , 0 be given arbitrarily. Then, by Lemma 4.6.6 there exists q 2 0 such that E Ec·

Therefore there exists an operator in £((E), (E)) whose operator symbol is particular, the action of this operator on exponential vectors is as in (4.63). Suppose next that A E JR. By definition, for E Ec and x E E*,

=

e.

In

= e(A2-1)(e,e)/2 e(x,AeH Ae,Ae)/2 =

For a general E (E) take an approximating sequence 1/Jn each of which is a linear comin (E), and therefore pointwisely, bination of exponential vectors. Then SA1/Jn -+ i.e., SA1/Jn(X) -+ for any x E E*. On the other hand, since SA1/Jn(X) = 1/Jn(AX), we conclude that = qed Definition 4.6.8 SA E £((E), (E)), A E C, is called a scaling operator. The symbol of SA is already obtained during the above proof. We record it as well as the Fock expansion which follows immediately from the Taylor expansion of the operator symbol.

105

4.6. SOME EXAMPLES

Proposition 4.6.9 For oX E C we have

The above method of defining an operator is useful. Here is another example. Proposition 4.6.10 For any T E £(Ee,Ee) there exists a unique operator r(T) E £((E), (E)) such that (4.64) E Ee. r(T)¢e = ¢re,

Moreover, for ¢ E (E) given with Wiener-Ito expansion: ¢( x)

it holds that

f: (: x®n : , fn)

n=O 00

r(T)¢(x) PROOF.

=

= 'L(:x®n:,T®nfn). n=O

We put E Ee·

For the first half of the assertion it is sufficient to verify the conditions (i) and (ii) in Corollary 4.4.11. In fact, (i) is immediate. For (ii) let p 0 and e > 0 be given. By the continuity of T there exist C 0 and q 0 such that E Ee·

It then follows that

for any r O. Taking r 0 large enough, we have Cp" /2 < f. Thus the proof of the first half is completed. The second half is proved in a similar manner as in the proof of Proposition 4.6.7. qed Recall that we used r(A) to construct the white noise functionals. This operator is, of course, a special case of the above. In general, for T E £(Ee, Ee) we define an operator dr(T) on (E). Suppose ¢ E (E) is given as ¢( x)

=

f: (: x®n : ,In) ,

n=O

x E E*,

CHAPTER 4. OPERATOR THEORY

106

as usual. Then we put 00

dF(T)¢J(x)

where

= L\:X0n;,in(T)fn),

(4.65)

n=O

in(T) = L:;;:6 [Ok 0 T 0 [0(n-l-k), { io(T) = O.

n 2: 1,

(4.66)

Proposition 4.6.11 dF(T) E £((E),(E)) [or any T E £(Ec,Ec). PROOF.

Let p 2: O. From definition we obtain 00

12

00

= L nln 2 !(I@(n- l ) 0 T)fn -:

= L n!

n=O

n=l

(4.67)

Supposing q 2: 1, we compute

j(I@(n-l) 0 T)fnl: = =L. 1((I0(n-l) o T)fn ,eiJ )1,"',Jn

=

L

L

s

02(n-l) L

n

h,'·-,in

·lejn

(4.68)

j

On the other hand, since T E £(Ec, Ec), there exist q 2: 1 and C 2: 0 such that

Then

I ir», ,e) I = I (Te, ej) I

ITelpH Iejl-(pH)

c lel p+q IeL(PH)'

and therefore Then (4.68) becomes ( [ 0(n- l ) 0 1

2

2 2 T)f,n 1 < - If,n /2p+q 02(n-l)'" L..J C21eJ'1 -(pH) IeJ'1 P

P

j

2 . = C 282n If,n1pH

BIBLIOGRAPHICAL NOTES

107

Consequently, (4.67) is estimated as follows. 00

L


E (E).

0

(5.3)

CHAPTER 5. TOWARD HARMONIC ANALYSIS

110 PROOF.

Given ,t/J is

PROOF. For simplicity we write W = wt/>,t/J. It then follows from Lemma 5.1.1 that 00

w(t,x) Given p

= V¢J(t,x). 'l/J(x) = Lej(t)Dj¢J(x)'l/J(x),

0 choose C

j=O

t E T,

x E E*.

0 and q > 0 such that

Then we obtain 00

L

IIDj¢J·

j=O

00

L


,t/J is defined as in {5.8}.

(5.10)

CHAPTER 5. TOWARD HARMONIC ANALYSIS

112 PROOF. Choose p

2:: 0 such as II¥II-p < 00. Then, by (5.9) we have

I((¥ ,wq",p)) I s II¥II-p II wq" ,p llp ::; MII¥II_p II¢IIp+ q+ l lI¢lIp+q, ¢, ¢

E (E),

for some q > 0 and M 2:: O. This means that (¢, ¢) 1-+ ((¥ ,wq",p)) is a continuous bilinear form on (E) x (E), and therefore there exists a unique operator E E .c((E), (E)*) satisfying (5.10). qed

Definition 5.1.5 The operator E defined as in (5.10) is called a first order differential operator with coefficient ¥ E (Ee 18) (E))* and is denoted (somehow formally) by

E

=

lr

1>t Otdt.

Here we put 4>t(x) = 4>(t,x). In fact, t 1­+ 1>t is an (E)*­valued distribution on T, namely, an element in E 18) (E)* (Ee 18) (E))*.

c

We are now interested in first order differential operators acting from (E) into itself.

Theorem 5.1.6 Let E be a first order differential operator (5.11)

with coefficient ¥ E (Ec if¥ E E 18) (E).

c

18)

= ¥(t,x).

(E))*, 1>t(x)

Then E E .c((E), (E)) if and only

PROOF. Let I< E .c(Ee, (E)*) be the operator corresponding to ical isomorphism (Ee 18) (E))* .c(Ee, (E)*). Then

((4>,( 18) ¢)) = ((I«, ¢)),

( E Ee,

¥ under

¢ E (E).

the canon(5.12)

c

It follows from the kernel theorem that ¥ E E 18) (E) if and only if I( E .c(Ee,(E)). Suppose that E is given as in (5.11). Then, by definition

¢,¢ E (E), where wq",p(t,x)

= Ot¢(x)¢(x).

On the other hand, we see from Lemma 5.1.3 that 00

Wq",p

= Lej 18) (D j ¢ · ¢) j=O

converges in Ee ® (E), Therefore

((E¢,¢))

=

00

L((¥,ej

18)

(D j ¢ . ¢ )))

j=O

=

00

L((I. Kejllp

ClIIDjif>ll p +q IIKejll,,+q'

Moreover, since K E £(Ee, (E)) there exist r 2': 0 and C2 2': 0 such that

Thus (5.13) becomes 00

1((Eif> , tP)) I < Cl C2 IItP"-" I: liD jif>lI p+qlejI,,+q+r j=o 00

Cl C2 I1 tP "-"

I: II D j if> lI p+qlejl,,+q+r+l .xjl j=o

11-,

< C, C, < C, C"

(E II (to lie, D

:+. kj 1:'.+,+,) 'I' (to >. j') '1'

@

D,

'I'

It then follows from Lemma 5.1.1 that

and hence

Consequently, E E £((E), (E)). Conversely, suppose that E E £((E), (E)). Then, for any p 2': 0 there exist q 2': 0 and C 2': 0 such that (5.14) if> E (E). II Eif>II" C 1Iif>lIp+q, Let E Ec be fixed and consider

s

if>(x) As is easily verified, w4>.'"

=

((Eif> , tP))

0

=

= (x ,0,

x E E·.

tP for any tP E (E). Hence by (5.12) we obtain ,w",.",)) =

0

tP)) =

tP))·

CHAPTER 5. TOWARD HARMONIC ANALYSIS

114 In view of (5.14) we obtain

and therefore

C 1Ilj1l p+q= C

,

E Ec·

This implies that K E £(Ec, (E)).

qed

Such an operator S described as in Theorem 5.1.6 is called a first order differential operator with smooth coefficients. This would be reasonable because in that case t t-t is an (E)-valued distributions on T. The symbol of a first order differential operator is easily obtained. Proposition 5.1.7 Let S be a first order differential operator given as S= where

=

E (Ec @ (E)*. Then it holds that

(5.15) PROOF.

We use the notation introduced in Lemma 5.1.3. By definition (5.8) we

have namely, Therefore,

17)

= ((SljIe ,1jIy/)) =

= e(CY/)

@

IjIHY/)) ,

which shows (5.15).

qed

c.

Corollary 5.1.8 Let JC E E Then the first order differential operator with coefficient JC @ 1 E E @ (E) coincides with SO,1 (x}. In particular, the first order differential operator with coefficient y @ 1, Y E E*, coincides with SO,1(y) = D y.

c

Such an operator described as in Corollary 5.1.8 is called a first order differential operator with constant coefficients. We are going into an algebraic counterpart. Since (E) is a topological algebra, we have a natural concept of derivations on (E). Moreover, since each E (E)* gives rise to a continuous operator in £((E), (E)*) by multiplication, the following definition is adequate. Definition 5.1.9 A linear operator S : (E)

S(IjIt/J)

-+

= SIjI· t/J + 1jI. St/J,

(E)* is called a derivation if

1jI, t/J E (E).

(5.16)

5.1. FIRST ORDER DIFFERENTIAL OPERATORS

115

It has been already observed in Corollary 4.2.7 that D y is a derivation for any y E E*. The rest of this section is devoted to answering a natural question: what are the derivations on white noise functionals? The main assertion is stated in the following

Theorem 5.1.10 Any continuous derivation in £( (E), (E)*) is a first order differen-

tial operator and vice versa. Furthermore, any continuous derivation in £((E), (E)) is a first order differential operator with smooth coefficients and vice versa. The essence of the proof lies in Fock expansion. We begin with a few lemmas.

Lemma 5.1.11 Let 5 E £((E), (E)*). Then, it is a derivation if and only if ( E Ee.

(5.17)

PROOF. Recall that the exponential vectors span a dense subspace of (E) and that multiplication (Wiener product) is a continuous bilinear map. Hence 5 is a derivation if and only if

((E( rperp,,) ,rp,))

=

+ (( rpe . 5 rp" ,rpd) = ((5rpe ,rp"rp,)) + ((5rp" ,rperp,)), ((5 rpe . rp" ,rp,))

Then with an obvious relation equivalent.

rperp" = e({,"}rpe+",

E Ee·

(5.18)

we see that (5.17) and (5.18) are qed

Lemma 5.1.12 Any first order differential operator is a derivation in £((E), (E)*). PROOF. Let E be a first order differential operator with coefficient Then, by Proposition 5.1.7 we have

fP

E (Ee®(E))*.

Then the verification of condition (5.17) in Lemma 5.1.11 is straightforward.

qed

Lemma 5.1.13 Let 5 E £((E), (E)*) be a derivation and let

5

be the Fock expansion. Then ("'I,m+l ,1/1&>1 ®

for all 1, m

o.

"'1,0

=

00

L

= 0 for all 1

=(1

(5.19)

5 1,m("'I,m)

l,m=O

0 and

m ) ("'I+m,l ,(111&>1 ®

,

E Ee,

CHAPTER 5. TOWARD HARMONIC ANALYSIS

116

PROOF. By assumption the symbol e-({,") S(e,

r/)

=

S

satisfies (5.17). On the other hand, we have

f \KI,m

0 e@m) ,

l,m=O

which is a general result, see Proposition 4.5.3. Then, it is easy to obtain

n ) \ K/,m+n , (@/ 0 e@m 0 T/@n )

( m

=(1

m ) \ KI+m,n , (@/ 0 e@m 0 T/@n )

+ ( 1+n

n) /\

K

i@/ 0 .,@n 0 t@m) , ",

for any e,T/,( E Ee· Then KI,O = 0, 1 2:: 0, follows by putting m We next put n = 1 and T/ = in (5.20) to obtain

e

l!(m

(5.20)

I+n,m,,,

+ I)! \ K/,m+1 ,(@I 0 e@(m+1l) = (1 + m)! \KI+m,1' ((@I 0 e@m) 0 e) +(1 + 1)!m! \KI+1,m, ((@/ 0 0 0

= n = 0 in

(5.20).

e@m).

Applying this argument to the second term successively, we come to l!(m + I)! \K/,m+1, (@/ 0 e@(m+1l)

= (m + 1)(1 + m)! \K/+m,1, ((@/ 0

e@m) 0 e).

This completes the proof.

qed

PROOF OF THEOREM 5.1.10.

Suppose that we are given a continuous derivation

E with Fock expansion 00

E

= L:

E/,m(KI,m)' l,m=O We first introduce a continuous bilinear form n on Ee x (E):

n(e, ¢»

eE Ee,

00

=

L: n! (K n,1' in 0 e),

n=O

¢> E (E),

(5.21)

where ¢>(x) = 2:::"=0 (:x@n: ,in). We shall prove the convergence of (5.21). In fact, for any p, q 2:: 0 we have 00

L: n! I(K in 0 e)1 n1 ,

n=O

00

s n=O L: n! IKn,11_(p+q+1l lin 0 el + +1 p q


llp+Q+1

n!

1/2Ielp+Q+1 1/2

(5.22)

117

5.1. FIRST ORDER DIFFERENTIAL OPERATORS Since 5 E £((E), (E)*), there exist C

0,

J(

0 and p

0 such that {,T/ E Ee·

It then follows from Theorem 4.4.6 that the kernel distributions "I,m of 5 satisfies

I

"

< C (1Imm)-1/2

1

I,m -(p+1) -

(2e302)(I+m)/2

2p L ( 2

+ J(

) (l+m)/2

In particular,

Therefore,

En! 00

C

for a sufficiently large q

E

2

00

'{

3 2p2 q 2e 0

2p

(

;-

+ J( ) }n+l < 00

(5.23)

O. In conclusion, we see from (5.22) and (5.23) that

00

L n! I("n,l , In @{)

n=O

1

C1 1{lp+q+1 1I¢>llp+q+1 '

for some C1 0, p 0 and q O. Therefore n in (5.21) is well defined on Ee x (E) and becomes a continuous bilinear form. Let E (Ee @ (E))* be the element corresponding to n, namely, { E

Ee,

¢> E (E).

Let 5' be the first order differential operator with coefficient Proposition 5.1.7 that

e-«

''1 )2 ({, r/) =

It then follows from

= n({,¢> 0 be fixed arbitrarily. Then, whenever 101 :::; 00 , !I"(O)I :::; (e20j (e ,0 I + eOI 7]) I + 1)2If(O)\ e200 2+ 1) I :::; (e200I (e ,0 I + eOol (e ,7]) I + 1)2 exp

C

I + eOol

I).

As is easily seen, the last quantity is bounded by :::; Cexp(Ie, r(g)¢>1/)) ((5 ¢>ge ,¢>g1/))

=

(5.62)

Moreover, from Proposition 4.5.3 we see that

g71)

=

e(g{ ,g1/)

=

e(e,1/)

L 00

l,m=O

(KI,m' (g71 )@I 0 (g{)@m)

L 00

(5.63)

l,m=O

From (5.62) and (5.63) one may derive the Fock expansion of r(g)"5 r(g):

=L

00

F(g)"Sr(g)

l,m=O

5 1,m «g@(I+m))"KI,m).

(5.64)

CHAPTER 5. TOWARD HARMONIC ANALYSIS

134 In particular,

(5.65) Therefore, by the uniqueness of the Fock expansion, E is rotation-invariant if and qed only if E"m(X;I,m) is rotation-invariant for alII, m = 0,1,2,··.. We say that F E is rotation-invariant if (g®n)*F = F for allg E O(E;H). During the proof of Theorem 5.5.2 we have established the following

Lemma 5.5.5 Let

Then E1,m(x;) is rotation-invariant if and

x; E only if x; is rotation-invariant.

Thus the proofs of Theorems 5.5.3 and 5.5.4 are essentially reduced to listing up the rotation-invariant distributions. The full list is, in fact, described satisfactorily as below, though the proof is long and deferred at the end of this section. For the definition of F" for F E and a E 6 n, see §1.6.

Proposition 5.5.6 Assume that F E

is rotation-invariant. If n is odd, then F = O. If n is even, say n = 2m, then F is a linear combination of (,®m)", a E 6 n. Moreover, the dimension of rotation-invariant distributions in (E®n)* is (n - I)!!.

Lemma 5.5.7 Let a, (3"

Then,

be non-negative integers and put 1= 2a + (3, m

- (®" 0 -=/,m'

'/3

0 T®,)

= 2, + (3.

A")* -=/3,/3 - ( T/3 ) DC' A, = ( DC

(5.66)

where T/3 E (E®2/3)* is defined in (4.56). PROOF. Since both sides of (5.66) are continuous operator from (E) into (E)*, it is sufficient to check that they coincide on the polynomials. But this is proved by a straightforward computation. qed

5.5.3. Suppose that E1,m(x;) is rotation-invariant. Without loss of generality we may assume that x; E Then, x; is rotationinvariant by Lemma 5.5.5. If 1+ m is odd, it follows from Proposition 5.5.6 that x; = 0 and hence E1,m(x;) = O. We next consider the case when I + m is even. It follows again from Proposition 5.5.6 that x; is a linear combination of (T®(l+m)/2)", a E 6 1+ m. For each a E 6'+m we may find at E 6/ x 6 m such that PROOF OF THEOREM

(T 0,

"C L...J {3"

=0

m

= 0,1,2, .. ·.

(5.68)

r is linearly independent

a> 0 and n + 2a = m.

whenever

•

=0

Ca ,f3., (m {3"

for any m 2a and e E Ec with e (5.67) becomes

=f.

O. Consequently, Cs».

=0

for a > 0, and

A,

::' -- "C L...J O,{3" N(J i..Je, {3"

qed

as desired. In the finite dimensional case where S(lRn ) C L2(lR invariant operators are generated by the Laplacian n

02

n

j=l

ax;

j=l

(

n

)

C S'(lR

a )* (- a )

aXj

aXj

n

)

the rotation-

136

CHAPTER 5. TOWARD HARMONIC ANALYSIS

and by multiplication with the Euclidean norm: n

R(x)

= LX]. j=l

On the other hand, we have a white noise analogue of the Euclidean norm:

R( x)

= (: X 0 2 : , T) ,

which is, as multiplication operator, expressed by Laplacians: R

= 2N + Ll a + Ll

a,

see Corollary 5.3.8. We have thus observed an interesting contrast between rotationinvariant operators on white noise functionals and those on a finite dimensional Euclidean space. The rest of the section is entirely devoted to the proof of Proposition 5.5.6. Let FE and consider the Fourier series expansion: 00

... KAe· ( F , e·11 KA \61 ... KAe· VY In )e·.1 KA I('y VY In·

F= il,···,in=O

Obviously, F is rotation­invariant if and only if (5.69) for every choice of i l , "

' ,

in and 9 E O(E; H).

5.5.6 (CASE OF n BEING ODD). Suppose that i l , ' " .i« are arbitrarily given. Since n is odd, there is some j appearing odd times in that sequence. Take k which is different from i l , " ' , in and let 9 be a (two dimensional) rotation defined by gej = -ej, gek = -ek and qe, = e, for all i =f. j, k, It then follows from (5.69) that PROOF OF PROPOSITION

(F ,eil 121··· 121 eiJ

Hence F

=­

(F ,eil 121··· 121 eiJ

= 0 as desired.

= O. qed

is rotationFrom now on we assume that n = 2m is even and that F E invariant. By the same argument as in the above proof, we see that (F ,ei l 121 ... 121 ein) can be non­zero only when i l , ' .. .i; consist of pairwise identical numbers, namely, only when there exists a permutation (1 E 6 n such that iU(I) = iU(2 )' ••• ,iu(n-l) = iu(n)' Let nn be the collection of such sequences (i l , ' .. , in)' Let 6 00 denote the group of all finite permutations of the non­negative integers. Then 6 00 acts on nn by means of the maps: (iI, ... ,in) t­+ (p(id, ... ,p( in)), where (i b· .. , in) E nn and p E 6 00 , Let 'In With these notations, we have

=6

00

\nn be the set of 6 00 ­orbit s in nn'

137

5.5. ROTATION-INVARIANT OPERATORS Lemma 5.5.8 If F E

n being even, is rotation-invariant, then

L:

F=

C(a)

E'tn

with some constants C(a) PROOF,

L:

eiI 0 , · · 0 ei n

(il t ••• ,i n )E

C.

E

We have already seen that such an F is expressed as

L:

F

(F ,eil 0 ... 0 eiJ eil 0· . , 0 ein

(il,···,in)Erln

L:

=

L:

(F,eh0···0ein)eiI0,··0ein·

E'tn (il,···,in)E

For any pair g E O(E; H) such that geil

= ei;,"

(F ,eil 0 ... 0

E a there exists a (finite dimensional) rotation . ,gein = ei;', Then, in view of (5.69) we see that

eiJ = (F , ei; 0

... 0 ei;') .

qed

This proves the assertion.

We need some notation. A sequence of non-negative integers 1r = (PJ,P2,"') is called a partition of a natural number m if (i) m PI P2 ... 0; (ii) PI + P2 + ... = tti. Let \Pm be the set of partitions of m. We introduce a lexicographic order into \Pm' For 1r = (PI, P2, ...) and 1r' = .. ) we write 1r < 1r' if PI

,

= PI'

for some 1 = 1,2" ". For simplicity we arrange all elements of \Pm as For 1r = (PI, P2, ... ) E \Pm we define W(1r)

L:

1r1

> 1r2 > . ",

00

=

0

0··· E (E®n)*,

(5.70)

il,h,···=O

For example,

1r1

= (1,1, ... , 1,0, ... ) and w(1l"J) = T®m.

For

1l"

E \Pm we define

With these notations we can state

Lemma 5.5.9 If F E

n being even, is rotation-invariant, then F

=

L:

L:

c(1l";O")w(1l")U

1rE13m uE6(1r)\6 n

with some constants c(7r;0") E C. Moreover, the constants c(1l";O") are unique.

(5.71)

CHAPTER 5. TOWARD HARMONIC ANALYSIS

138 PROOF.

= (Pl,P2,"')

First note that for -rr

((w( -rr) , eil ® ... ® eiJ)

={

1, 0,

E

=

if i 1 = ... = i 2P1 , i 2P1+l = ... = i 2P1+2P2' ••• , otherwise.

Therefore,

((w(-rr), eil ® ... ® eiJ)

= 0,

if

If (i 1 , •.. , in) E On, we may find (7 E 6 n and -rr'

where

it,h,'"

Iflm

(ill"', in)

= (p;,

rt. On'

(5.72)

(5.73)

... ) E Iflm such that

are mutually distinct. Then from (5.72) we see that if

-rr < -rr',

(5.74)

where -rr' E Iflm is the (unique) partition corresponding to (i 1 , ... , in) EOn' For -rr = (Pl,P2,' .. ) E Iflm and 8 E IR we put

17(-rr; 8) = ((sin 8)eo + (cos 8)el )02Pl ®

® ef2 P3 ® . .. E E0 n .

In view of (5.73) and (5.74) we obtain

(w(-rr), 17 (-rr'; 8)") (W(-rr),17(-rr;8)0")

= 0, (sin 8)2P1 + (cos 8)2p1, = { 0,

if -rr < x', (7 E 6 if (7 E 6(-rr) otherwise.

n,

(5.75)

We now go back to the proof of the original assertion. We first prove that {w( -rr)"; -rr E

Iflm,(7 E 6(-rr)\6 n } is linearly independent. In fact, suppose that

L

L

O"E6(1T)\6 n

c(-rr;(7)w(-rrt

=0

with some c(-rr; (7) E C. It then follows from (5.75) that, for any (7' E

0=

L

L

O"E6(1T)\6 n

c(-rr; (7) (w(-rr)O" , 17(-rrl; O)O"')

6 n,

= C(-rrl; (7').

Using the same argument successively according to the arrangement of partitions in = ... = O. Hence {w(-rr)O";-rr E Iflm,(7 E 6(-rr)\6 n } is linearly independent. Let W be the subspace of (El n )* spanned by w(-rr)0", where -rr E Iflm and (7 E 6(-rr)\6 n . We then note that dim W = 16n : 6(-rr)1. Let V be the subspace n of (El )* spanned by L(il .....in)EO' eil ® ... ® ein, where a runs over 'In. Let -rr = (Pl,P2,"') E Iflm. Since w(-rr) is decomposed into a finite sum:

Iflm, we conclude that C(-rr2;(7')

w(-rr) 11,12,'"

L + L + 3l,h=i3"" L +... + il=i2='" L ,

.11,12,'" di.tinct

1) =J2 ,)3 ,'"

distinct

diettnct

5.5. ROTATION-INVARIANT OPERATORS we see that w(7l") E V. Since VU other hand, since ' dim V:::;

=V

139

for any (7 E 6 n, we obtain W C V. On the

IXnl = L 16 n

:

6(7l")1

= dim W,

we conclude that W = V. We now suppose that F E is rotation-invariant. It then follows from Lemma 5.5.8 that F E V. As we have proved above, V = W is spanned by {w(7l")"; 7l" E E 6(7l")\6 n} which is a linear basis of W. Therefore F is uniquely expressed qed in the form (5.71).

Proposition 5.5.10 (r®m)u is rotation-invariant for any (7 E 6 n . PROOF.

for any

It is sufficient to show that

E Ee and for any 9 E O(E; H). We observe that

..

(r®m

Q9 ••• Q9

(r

... (r

Q9

Q9

...

qed

Similarly the left hand side is computed to get the same expression. PROOF OF PROPOSITION

5.5.6 (CASE OF n BEING EVEN). Suppose F is given as

in (5.71). First we prove

F'

F-

L

L

C(7l"l; (7)w(7l"t}u

uE6(1ftl\6n

L

c(7l"; (7)w(7l")"

= O.

"E6(1f)\6n

Take a (two dimensional) rotation go E O(E; H) defined by

go eo { goe}

= (cos 8)eo - (sin 8)el = (sin 8)eo + (cos 8)el,

with 8 E JR. It follows from Proposition 5.5.10 and the assumption that F' is rotationinvariant. Then,

8 E lR. Since 9r n71( 7l"2; 0)

c(7l"2; 0")

= 71(7l"2; 8), using

(5.72) and (5.75), we obtain

= (F' ,71( 7l"2; 0)") = (F' ,71(7l"2; 8)") =

((sin 8?Pl

+ (cos 8?Pl) c(7l"2; 0")

CHAPTER 5. TOWARD HARMONIC ANALYSIS

140

for all 0 E R., and therefore, c(11"2; a) = O. Inductively, we may prove that c(11"3; a) = ... = 0, namely, F' = 0 as desired. Since W(1I"1t" = (r l8im )U, a E 6(1I"1)\6 n , are independent, the dimension of the rotation-invariant distributions in is given by 16n : 6(1I"dl = n!/(2 m • m!) = (n - I)!!. qed During the above proof of Proposition 5.5.6, we have used only the subgroup of O(E; H) consisting of rotations which act identically on the subspace spanned by {en, en+! , ... } for a large n o. It would be interesting to investigate invariantdistributions under another subgroups of O(E; H), for instance, certain transformation groups of T which are naturally imbedded in O(E; H).

5.6

Fourier transform

We begin with the following Lemma 5.6.1 For each 0 E R. there exists an operator \!So E .c((E), (E)) such that

\!Soe PROOF.

= exp

Ge

iO sin 0

.o) eige,

E EtC.

(5.76)

It is sufficient to verify that

satisfies conditions (i) and (ii) in Theorem 4.4.7. In fact, (i) is obvious. To see (ii) we need only to note that E Ee,

and to apply Lemma 4.6.6. As a result, we see that e is the symbol of an operator qed in .c((E), (E)) of which action on exponential vectors is given as in (5.76). Lemma 5.6.2 For 0, OJ, O2 E R.,

\!So = I, This is immediate from (5.76). It is now obvious that jo = \!So becomes a continuous linear operator on (E)· and forms a one-parameter group of linear transformations acting on (E)· satisfying jo

= I,

Definition 5.6.3 The operators j = j-1r/2 and jo are called (Kuo's) Fourier transform and Fourier-Mehler transform (with parameter 0 E R.), respectively. These operators are characterized as follows.

5.6. FOURIER TRANSFORM

141

Proposition 5.6.4 For each 0 E R Fourier-Mehler transform ator satisfying

= StP(eiOe)exp GeiOsinO (e ,0),

e E E e,

is the unique opertP E (E)*.

(5.77)

In particular, the Fourier transform is characterized by

eE Ee, PROOF.

tP E (E)*.

Let tP E (E)* and e E Ee . Then by definition of

=

4>e))

(5.78)

we have

= ((e)) = ((tP, e)) .

Then, in view of (5.76), we obtain

,0) StP( eiOe) exp Ge iOsin 0 (e ,0) . ((tP ,4>.;ge)) exp Ge iOsin 0 (e

((tP ,e))

This completes the proof of (5.77). For (5.78) we need only to put 0

= -7r /2.

and are easily derived.

The explicit forms of

Proposition 5.6.5 Let

tP(x)

=

f

(:x®n: ,Fn), n=O be the Wiener-Ito expansion of tP E (E)*. Then the Wiener-Ito expansion of given as

=

00

n=O

where

PROOF.

First recall that StP(O

=

f (Fn ,e®n),

n=O

eE Ee.

Hence by (5.77) we obtain

=

qed

StP(ei°e)exp GeiOsinO(ce)) E(Fn,(eiOe)®n)

f f m,n=O m.

m,n=O

G

GeiOsinO)m (com

sinO) m (Fn ,e®n) (T ,e ® e)m

(i sinO)m (Fn0 2

T®m

,e®(n+2m»).

is

142

CHAPTER 5. TOWARD HARMONIC ANALYSIS

Changing parameters, we obtain

The assertion then follows immediately.

qed

By duality argument we have

Proposition 5.6.6 Let

¢lex)

=

f (:x®n: ,fn),

n=O

be the Fock expansion of rP E (E). Then the Wiener-Ito expansion of®orP is given as ®OrP(x)

=

f(:x®n:,fn(®orP)), n=O

where

The symbol of ®o appeared already during the proof of Lemma 5.6.1. The Fock expansion is then straightforward by Taylor expansion. Moreover by duality one may obtain easily the symbol and Fock expansion of the Fourier-Mehler transform.

Proposition 5.6.7 Let 0 E 1R. For (, 7] E Err. we have

Ge

=

exp

=

®O('f/,

iO sinO ((,0

°=

+ eiO (( ,7])),

exp GeiOsinO{'f/,7])

+ eiO

For the Fock expansions we have 1 (i I,Eo l!m! 2"e 00

= '"

te :

l'

• [1\ (Z"2 eiO sin l,m=O .m. 00

where

71

) m ( iO

0)

)I

e - 1

smO

I

(iO e

-

-=1,1+2m(n

1) =m

07

®m

),

(7 ®I 07m ),

E (E®21)* is defined in (4.62).

Moreover, by similar argument as in the proof of Theorem 5.3.11 one may obtain the following interesting result.

Theorem 5.6.8 {®O} OEl! is a regular one-parameter subgroup of GL( (E)) with in-

finitesimal generator iN + Lla.

5.6. FOURIER TRANSFORM

143

There is a close relation between the Fourier transform and the S-transform. For simplicity we put E Ee,

Then, E{

and, in particular,

E{

= e-({ ,{)/2¢!i{,

x E E*.

(5.79)

E Ec,

E (E).

Definition 5.6.9 The T-transform of

Ed),

=

Proposition 5.6.10 For any

E (E)* is a function on Ec defined by E Ec.

E (E)* it holds that

E Ee.

PROOF. In view of (5.79) we have

which proves the assertion. Corollary 5.6.11 Let

qed

= 0 for

E (E)*.

all

E E, then

PROOF. By Proposition 5.6.10 we have = 0 for all Corollary 3.3.9 we see that = O. (Hence it follws that

= o.

E E, and then from E Ed

= 0 for all

qed

It is also interesting to ask how is characterized as a function on Ec. But since the T- and S-transforms are connected directly as in Proposition 5.6.10, we obtain the following

Theorem 5.6.12 Let F be a C-valued function on Ec. Then F E (E)* if and only if (i) Z f-+ + 11) is an entire holomorphic function on C for any (ii) there exist C 2:: OJ K 2:: 0 and p E IR such that

IF(OI

C exp

(K

=

for some

E Ec;

E Ee·

In other words, the function spaces E (E)*} and E (E)*} are the same, see Theorem 3.6.1. Moreover, since T-transform is injective by Corollary 5.6.11, T-1S is a linear operator from (E)* into itself. (In fact, S-transform is also injective, see Corollary 2.3.9.) Theorem 5.6.13

= T-IS.

CHAPTER 5. TOWARD HARMONIC ANALYSIS

144

PROOF. By Proposition 5.6.4 we observe

E Ec·

Hence

= T. Since

=

we see that

=

1

qed

= T- S.

Finally we remark extension of T-transform. If f E Up >1 U(E*,/i;C), then I/>f E (E)* is defined by ¢J E (E),

(5.80)

see Theorem 3.5.12. Hence, Tl/>f(O

=

r f(x)ei(x, } /i(dx),

(5.81 )

E Ec·

lE'

Apparently, this is more like the Fourier transform on a finite dimensional Euclidean space than the Fourier transform x. While, (5.81) suggests us to define an integral transform:

7f(0

=

r f(x)ei(x,e} /i(dx),

lE'

Note that 7 f is defined only on E (not on Ed instead f can be taken from a larger class of functions, namely, f E Ll(E*, u; C). In this connection we only note the following fact. Proposition 5.6.14 Let f E L1(E*,/i;C)' If7f(0

= 0 for all

E E, then f

= o.

PROOF. Let be the sub a-field of generated by the functions: x f--+ (x, ej), j = 1,2" .. , and let denote the conditional expectation of f respect to see the proof of Proposition 2.3.2. Then, it is known that

= f(x),

lim

n_oo

Now suppose that 7 f(O

o

= 0 for all

u-e,e. x E E*.

(5.82)

E E. Then, in particular,

7 f(tl el + ... tnen) f( x )ei(tt{x .ei }+ ..+tn(x ,en}) /i(dx)

r = r

lE'

lE'

...+tn(x,en})/i(dx).

Then the uniqueness of the Fourier transform of L 1 (lRn) implies that for u-e.e. x E E*. Consequently, we see from (5.82) that f = O.

=0 qed

Corollary 5.6.15 For f E Up >1 LP(E*, u; C) let I/>f E (E)* be defined as in (5.80). Then the map 1/>: Up > 1 U(E*,/i;C) -+ (E)* is a linear injection. PROOF. Suppose that I/>f = 0, f E LP(E*,/i;C), p > 1. Then, TI/>f = 0 and therefore 7 f = O. (Here we note that LP eLI.) Then, applying Proposition 5.6.14, we see that f = O. qed

5.7. INTERTWINING PROPERTY OF FOURIER TRANSFORM

5.7

145

Intertwining property of Fourier transform

We next consider characterization of the Fourier transform in terms of its intertwining properties. For that purpose with each { E Ec we associate operators Pe and qe by

Pe = qe

=

h {EO,l«() + E1,o«()} = h {EO,l(O - E1,o(O} =

i

{(t)(Ot - on dt,

i

{(t)(Ot

+ ondt.

That Pe,qe E £«E),(E)) follows from Theorem 4.3.9. By a simple computation we obtain (E Ec, Pe = -Pe, qe = qe, and the canonical commutation relation: (, rJ E E.

Here we note that the symbol means the adjoint with respect to the complex bilinear form «. ,.)). For a real { E E we may write also 0

Pe

=

- De) = De

+

Lemma 5.7.1 For

0n) _ ( •X • , Uj -

2n1/ 2 H n (Xj) J2 -_

n

Xi

= OJ,

i=

0,

we obtain

+ ' ,..

Hence (E) contains every polynomial in Xj and therefore in Xl,"', XD since (E) is closed under pointwise multiplication. 2 As is easily verified, if . e- 0, the Fourier transform

converges absolutely at any E cP and becomes an entire holomorphic function on CD. Using this fact and the characterization theorem (Theorem 3.6.2 or Theorem 5.6.12), we may prove Theorem 6.2.1 A continuous function :]RD 2 (i) . e- 0; (ii) for any t > 0 there exists C ::::: 0 such that

-+

C belongs to (E) if and only if

Recall that (E) is obtained from L 2(]RD, /.l) where /.l is the Gaussian measure. On the other hand, there is a natural unitary isomorphism from L 2(]RD, /.l) onto L 2(]RD, dx) given by (6.5)

156

CHAPTER 6. ADDENDUM

Let V denote the image of (E) under the unitary map U. Then, the Gelfand triple (E) c L 2(lRD , /l) C (E)* is translated into a new Gelfand triple

All the results obtained within the general framework of white noise calculus can be restated in terms of the above finite dimensional calculus. We shall discuss a few interesting cases. We first mention a characterization of the space V, which is an immediate consequence of Theorem 6.2.1.

Theorem 6.2.2 A continuous function ¢ : lRD (i) ¢. eU­')lx I2 E LI(lRD,dx) for any E > 0; (ii) for any E > 0 there exists C 0 such that

­+

C belongs to V if and only if

Using the fact that S(lRD) is invariant under the Fourier transform, one may prove the following assertion easily.

Lemma 6.2.3 If E (E), then . e­'l xI E S(lRD) for any 2

E

> O.

As an immediate we see D C S(lRDJ. In. fact, the inclusion is proper. To see that It IS sufficient to consider ¢(x) = e- 1x/ /8 with Theorem 6.2.2. In the theory of operators on white noise functionals a principal role is played by annihilation (Hida's differential) and creation operators. In the present context Hida's differential operator is defined by

a"'( J'/'

In other words,

)­1'

x ­

(x+OOj)-(x) 0 '

a

aj=-a' Xj

j=1,2, .. ·,D.

a;

The adjoint with respect to the Gaussian measure /l is the creation operator by definition. Since OJ is not a distribution but belongs to E = lRD , the creation operator belongs to £((E), (E)) as well as j . The relation

a;

a

follows from gerenal theory though a direct proof is also possible and easy using the fact that ¢(x)(x)e-lxI2/2 E S(lRD) for any ,¢ E (E). Moreover,

This is the well known canonical commutation relation (CCR) in case of D degree of freedom.

6.2. REDUCTION TO FINITE DEGREE OF FREEDOM

157

Using the unitary operator U : L 2(lRD,J.L) ---+ L 2(lRD,dx) introduced in (6.5), we study a few interesting operators in .e«E), (E)*). Note that if S E .e«E), (E)*) then USU- 1 E .e(D, D*). A simple computation yields the following

ua.u- 1 = Xj J 2

tunr: = Xj _

+ ax'J

2

J

ax/

In particular,

Pj

1( -1 = -2' uap Z

*

a

1

uaju ) = -:--a' Z Xj -1

Qj

= uap-1 + ua;u- 1 = Xj

are the Schr6dinger representation of CCR on L 2(lR D , dx) with common domain D. An integral kernel operator §4.3 is merely a finite linear combination of compositions of creation and annihilation operators with normal ordering:

SI,m (K)

= '" L.J K(i1' ... " i/

)'1 , .. , ,m )'

I]

.. · e:I{ a31 .. , a1m'

(6.6)

where it, .. ·, i/,ill'" .i.: run over T = {I, 2"", D}. This is a differential operator with polynomial coefficients. It follows from Proposition 5.2 that US/,m(K)U- 1 is again a differential operator with polynomial coefficients: (6.7) It is also expressed in terms of Pj and Qj introduced in Proposition 5.2: USI,m(K)U- 1

I:

=

C(o.,(J)QapfJ.

(6.8)

lal,lfJI9+m

Applying the theory of Fock expansion (see §4.5), we see that every operator SE .e(D, D*) is expressed in an infinite linear combination of operators of the form (6.7) or equivalently (6.8). Namely, USU- 1

=

I:

I:

00

l,m=O

C/,m(o.,(J)QapfJ.

(6.9)

lal,lfJl$/+m

Formally we may rearrange the above infinite series according to the usual order of multi-index notation: USU- 1

= I:C(o.,(J)QapfJ, a,fJ

though the meaning of the convergence becomes unclear. Incidentally we note that (6.9) leads us to a statement of "irreducibility" of the Schr6dinger representation of CCR on L 2(lRD,dx ), where the common domain of Pj and Qj is taken to be D. The Gross Laplacian and the number operator are defined respectively by

= I: a;, D

.1a

j=1

D

N

= I:a;aj . j=1

CHAPTER 6. ADDENDUM

158 Since we have

D

j=1

1) = L\o + L\a + 2N.

-

The left hand side is "renormalized" Euclidean norm. The white noise analogue was discussed in §§5.3 and 5.5. By a straightforward computation we obtain

L (0x) j=1

oU-

UL\

L (0

1

0

2

D

x)

2

j=1

0

x)

xJ

LD

2 ( --2

D

j=1

2

l

x x

1)

+2 '

2

l-21) '

0 +2x __1) .

OXj

On the other hand, for the usual Laplacian L\

2

4

2

D

=L )=1

02 ox 2 on £2(JR D, dx) we have

1) ='"

2

D ( 02_ x U- 1 L\U = '" X)O+-.1.._LJ ) ) 4 2 )=1

D (

LJ

)=1

)

))

x

2

4

1)

2

.

This expression motivated Yamasaki [1] to introduce an infinite dimensional Laplacian (in our terminology -N) by omitting the divergent terms x;/4 - 1/2. In §5.4 we discussed an operator x(s)Ot - x(t)o. = O;Ot in connection with the infinite dimensional rotation group. The translation into our finite dimensional calculus is easy.

0;0.

U(XOk ) - XkO)U)

1

0 = x) OXk ·- -

0 Xk-, OXj

which coincides with an infinitesimal generator of rotations on JRD. Finally we discuss Kuo's Fourier transform and Fourier-Mehler transforms () E JR, which is a one-parameter group of transformations on (E)* involving Kuo's Fourier transform as = for details see §5.6. By a step-by-step computation one obtains an explicit form of as follows: for (}:f:. 0 (mod 1l'),

'?:f( x ). iO • - ( - 2 1l'ze sm

1>0

mD

f() Y exp (-i(IXI

2+IYI2)COS(}+2i(X'Y))d 2 . () Y,

and for () == 0 (mod 1l') we have

=

:

In particular,

=

1 )D = ( ..j2i

. kD f(y)e-,(x'Y)dy.

6.3. VECTOR- VALUED WHITE NOISE FUNCTIONALS

159

These operators are defined (in the sense that the integral is absolutely convergent) on L1(]RD,dx). Moreover, by the above explicit expression we see that J'o is nothing but the Fourier-Mehler transform on R" discussed in Hida [2:Chap. 7] and Wiener [I]. However, in fact, this is Kuo's original idea of finding the white noise version of FourierMehler transform. Incidentally we note that {J'o }OElI\ becomes a one-parameter group of automorphisms of S(]RD). We then easily see that

and in particular,

UJ'U-t'= e-lxI2/4 0 J' 0 elxI2/4.

It follows from the characterization of Kuo's Fourier transform (Theorem 5.7.4) that the operator = UJ'U- 1 is charactreized by the following intertwining properties:

6.3

Vector-valued white noise functionals

Our study of white noise functionals has been so far restricted to the case of scalarvalued functionals, however, the extension of the theory to vector-valued functionals is non-trivial and important from various aspects. Such generalization is indispensable to discuss, for example, quantum interacting systems such as "System + Reservoir" models (see Accardi-Lu [I] and references cited therein), infinite dimensional Dirac operators defined on Boson-Fermion Fock space toward supersymmetric quantum field theory (see Arai [1], Arai-Mitoma [1)), and so on. In the recent work Obata [10] has been proposed a theory of vector-valued white noise functionals where the values lie in a standard CH-space. The choice of a standard Cll-space is based on the following reasons. First of all, taking applications into account, we must not exclude Hilbert spaces. This means that our theory will cover the case of Hilbert space-valued distributions on Gaussian space. Second, a standard countably Hilbert space possesses nice properties from the viewpoint of topological vector spaces, in particular, the theory of topological tensor products can be applied effectively, see §1. Finally, notation and results established so far for scalar-valued functionals help the study of the vector-valued case very much. The present section is then devoted to a quick review of the results in Obata [10] with no proofs. Let fj be another complex Hilbert space whose norm is denoted by 1·10 again. This is often called an initial space. Let B be a positive selfadjoint operator on fj with inf Spec(B) > 0 and let be the standard CH-space constructed from (fj, B). Then (E) ® is the space of test white noise functionals and its dual space (( E) ® = (E)* ® consists of -valued generalized white noise functionals. We must keep it in mind that (E) is a nuclear Frechet space. Set (7

= (infSpec(B)t 1 = IIB-11I op > o.

CHAPTER 6. ADDENDUM

160

If we take the identity operator on .f) for B, identifying fi" with .f), we obtain .f)-valued test and generalized white noise functionals, By Proposition 1.3.8 the standard CH-space construed from ((L 2 ) 0.f), F(A) 0 B) is isomorphic to (E) 0 IE. This fact enables us to employ basic notations used so far x ((E) for scalar-valued functionals. The canonical bilinear form on ((E) 0 is denoted by ((" .)) again. Similarly, the norms of (E) 0 IE are denoted again by 11·ll p ' i.e., = II(F(A) 0 E (E) 0