Multidimensional Second Order Stochastic Processes

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MULTIDIMENSIONAL SECOND ORDER STOCHASTIC PROCESSES

SERIES ON MULTIVARIATE ANALYSIS Editor: M M Rao

Published Vol. 1: Martingales and Stochastic Analysis J. Yeh

Multivcmate Analysis •—Vol. 2

•

MULTIDIMENSIONAL SECOND ORDER STOCHASTIC PROCESSES

Yuichiro Kakihara Department of Mathematics University oj California, Riverside

V f e World Scientific 3» i

Singapore'New Singapore • New Jersey • London* London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Kakihara, Yuichiro. Multidimensional second order stochastic processes / Yuichiro Kakihara. p. cm. — (Series on multivariate analysis ; vol. 2) Includes bibliographical references and indexes. ISBN 9810230001 1. Stochastic processes. I. Title. II. Series. QA274.K328 1996 519.2-dc21 96-37165 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library

Copyright © 1997 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface

Univariate stationary stochastic processes in the weak sense were initiated by Khintchine in 1934, and since then the theory has been developed by many authors. In many applications, however, nonstationary processes play an important role. As an extension of stationarity Loeve, in middle 1940s, introduced harmonizability and, in 1959, Rozanov gave another harmonizability, where Rao distinguished these two concepts by calling them strong and weak harmonizabilities in 1982. Another ex tension is to consider multidimensional stochastic processes. In late 1950s Wiener and Masani developed such a theory for stationary stochastic processes. This book is concerned with multidimensional (mainly infinite dimensional) nonstationary (especially harmonizable) second order stochastic processes on (mainly) a locally compact abelian group. Since harmonizable processes are the Fourier trans forms of Hilbert space (typically L2{^)) valued measures, our treatment uses func tional analysis as well as harmonic analysis instead of basing it on probability theory alone. Here is a brief description of each chapter. In Chapter I, a historical survey of stationary and nonstationary processes as well as multidimensional and other extensions are given. In Chapter II, normal Hilbert B(H)-modules are considered in detail, where H is a (separable) complex Hilbert space and B(H) is the algebra of all bounded linear operators on H. This kind of spaces is useful to describe infinite dimensional second order stochastic processes and is essentially the same as the spaces of Hilbert-Schmidt class operators between Hilbert spaces. In Chapter III, we investigate stochastic measures and operator valued bimeasures. Let L<1(Q.;H) be the Hilbert space of all //-valued random variables on a probability space (f2, 5, A4) which are ^-square integrable. This space is a typical example of a normal Hilbert Z?(/Z)-module and an LQ(Q ; //)-valued measure can be regarded as an infinite dimensional stochastic measure. Such a measure induces a bimeasure with values in a subspace of B{H) and, in a certain sense, the converse is also true. In Chapter IV, we give the fundamental theory of multidimensional stochastic processes. After defining stationarity, harmonizability and other nonstationarities, we discuss their integral representations and interrelations. As basic topics we consider series representations, moving average representations, approxi mation and convergence, and subordination. In Chapter V, some special topics are studied such as Wold and Cramer decom positions, and KF-processes. Uniformly bounded linearly stationary processes and periodically correlated processes are of interest and are also treated in some detail. vii

viii

MULTIDIMENSIONAL SECOND ORDER STOCHASTIC PROCESSES

Some basic results on isotropic processes, processes on hypergroups, and processes on locally compact groups are stated without proof since a greater preparation is needed to give a full treatment of these topics. Finally in Chapter VI, applications are made from probabilistic and statistical points of view to prediction problems, Kalman filter, sampling theorems and the strong law of large numbers. Each chapter has the "Bibliographical notes" section assigning proper, and hope fully accurate, credits to authors whose contributions supported this book. A numbering system of this book should be mentioned. An item such as III.4.5 denotes the fifth in Section 4 of Chapter III. In a given chapter, only the section and item numbers are used, and in a given section only the item number is used. This book was suggested by Professor M. M. Rao of Unversity of California at Riverside (UCR). He kindly read the manuscript and gave criticism, advice and encouragement. The author would like to express his sincere gratitude to Professor Rao. He also thanks the hospitality of the Department of Mathematics of UCR where he could finish writing and typing this book during the stay.

Yuichiro Kakihara Riverside, California September, 1996

Table of Contents

Preface

iii

C h a p t e r I. I n t r o d u c t i o n and preliminaries

1

1.1. Stationary processes 1.2. Harmonizable processes 1.3. Multidimensional and other extensions Bibliographical notes

1 4 9 15

C h a p t e r II. Hilbert m o d u l e s and covariance kernels 2.1. Normal Hilbert B(i/)-modules 2.2. Submodules, operators and functionals 2.3. Characterization and structure 2.4. Positive definite kernels and reproducing kernel spaces 2.5. Harmonic analysis for normal Hilbert 5(i?)-modules Bibliographical notes C h a p t e r III. Stochastic measures and operator valued bimeasures 3.1. Semivariations and variations 3.2. Orthogonally scattered dilations 3.3. Gramian orthogonally scattered dilations 3.4. The spaces L J ( F ) and L2{F) 3.5. The spaces £ x ( 0 and £ 2 ( M ) 3.6. Riesz type theorems 3.7. Convergence Bibliographical notes C h a p t e r I V . M u l t i d i m e n s i o n a l stochastic processes 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7.

General concepts Staionary processes Harmonizable processes K-bounded processes Cramer and Karhunen classes Series representations Moving average representations

17 17 21 26 32 46 52 53 53 74 86 107 119 129 136 145 148 148 151 155 166 173 178 183

ix

x

MULTIDIMENSIONAL SECOND ORDER STOCHASTIC PROCESSES

4.8. Approximation and convergence 4.9. Subordination Bibliographical notes

184 192 200

C h a p t e r V . Special topics

203

5.1. Wold decompositions 5.2. Cramer decompositions 5.3. The KF-class 5.4. Uniformly bounded linearly stationary processes 5.5. Periodically correlated processes 5.6. Final remarks (1) Isotropic processes (2) Processes on hypergroups (3) Processes on locally compact groups Bibliographical notes C h a p t e r V I . Applications 6.1. Prediction problems 6.2. Kalman 6.3. Sampling theorems 6.4. Strong laws of large numbers Bibliographical notes

203 208 212 218 223 234 234 237 241 245 248

filter

248 259 275 285 293

References

296

Indices

316

Notation index Author Index Subject index

316 322 326

CHAPTER I

INTRODUCTION

AND

PRELIMINARIES

In this chapter, a brief survey of second order stationary and non-stationary stochastic processes is given and a motivation to study infinite dimensional nonstationary, especially harmonizable, stochastic processes is also included. We begin with a one-dimensional second order stationary stochastic process. We give an out line of its integral representation as well as that of its covariance function. This serves as a motivation for the following work. As a generalization we present har monizable and other related processes. Finite and infinite dimensional extensions of stochastic processes are discussed in some detail to show how functional analysis, operator algebra, vector measure theory and harmonic analysis play basic roles in developing the theory of multidimentional stochastic processes.

1.1. Stationary processes We are concerned, in this book, with second order stochastic processes. Let (f2,5,/i/) be a probability space and consider the Hilbert space L2{Q) consisting of all (equivalence classes of) C-valued random variables on fi with finite second moment, where C is the complex number field. We denote the inner product and the norm in L 2 (fi) by

(f,gh = I /HsRM<M,

II/II2 = (/, f)l

Jn ioif,geL2(n). Let R be the real line and consider a one-dimensional or univanate second oder stochastic process {x(t)} = {x(t),t € R} on R, i.e., a mapping x(-) : R ->• L 2 (fi). We sometimes denote it by x or {x(t,u)}. We usually consider a centered process {x(t)}, that is, a process such that the expectation E{x(t)} = Jux(t)dn = 0 for every l e t , since the second order (or covariance) properties are of primary interest. Henceforth, putting L ^ ) = { / £ L 2 ( f i ) : £ { / } = 0}, 1

I. INTRODUCTION AND PRELIMINARIES

2

we treat Lg(f2)-valued processes. As we will see later, the index set R can be replaced by Z, the set of all integers, or any locally compact abelian group G. Weak stationarity was introduced to Lg(f2)-valued processes by Khintchine [1] in 1934. This notion is important in various applications and also it gives a beautiful theory of weakly stationary processes. Here we do not treat strong stationarity, so that we will use the terminology of "stationarity" to stand for "weak stationarity." Thus, let {a:(i)} be an Lo(fi)-valued process. Then it is said to be stationary if its covariance function y{s,t) = (x(s),x(t))„, s,t £ R depends only on the difference s — t and if, -y(s — t) = ~f(s, t), then 7 : R —> C is continuous. Now let us proceed to obtain an integral representation of a stationary process {x(t)} with the covariance function 7 (of one variable). Observe that 7 is posi tive definite in the sense that for any n 6 N (= the set of all positive integers), « i , . . . , an e C and t\, - . - , * « € R it holds that n

2

n

2^2^Qjafc7(iJ-.-tk) =

y^akx(tk)

j = i fc=i

fc=i

> 0. 2

By Bochner's theorem on the integral representation of such functions, there exists a finite positive measure v on (R, 03), 03 being the Borel cr-algebra of R, such that 7(t)

= f

itu e v{du),

tern.

(1.1)

Jn v is called the spectral measure of the process and is sometimes denoted by t/$. The time domain 1-L{x) of {x(i)} is defined by ■H(x) = G0{x(t)

:teR},

the closed subspace of Lg(fi) generated by the set {x(t) : t e R}. For each t e R define an operator U(t) on H(x) by U{t)x(s)

=x(s + t),

seR,

or more generally by /

n

= Y^akx(tk

akx(tk)\ k=l

'

+ t),

fc=l

where a t 6 C and tk e R for 1 < k < n. Then by the stationarity of {x(t)} we have for s, u e R {U(t)x(s),

U{t)x(u))2

= (x(s + t),x(u + t))2 = 7 ( s -u)=

(x(s),x(u))r

3

1.1. STATIONARY PROCESSES

Hence U(t) is well-defined and can be uniquely extended by linearity to a unitary operator on H{x), still denoted by U{t). It holds that U(s + t) = U{s)U{t), f/(0) = 1 (the identity operator) and t / ( i ) _ 1 = U(—t) for s,t 6 R. Since 7 is continuous, {U(t),t 6 R} forms a strongly continuous group of unitary operators on Ti{x). Thus by Stone's theorem there is a spectral measure E(-) on (R,25) such that

U(t)= I eituE{du),

teR.

JR

Consequently we have x(t) = U{t)x{Q) = [ eltu E(du)x{0),

t e R

JR

If we put £(A) = E(A)x(0) for A g 03, then £ is an Lg(f2)-valued bounded and countably additive (c.a.) measure, and is orthogonally scattered (o.s.), i.e, (£(A)1£,(B))2 = 0 for every disjoint pair A, B € 23. We denote it by £ e caos(
= feituZ{du),

teR.

(1.2)

Here and elsewhere integrals relative to vector measures are defined in the wellknown Dunford-Schwartz sense. Note that the process is expressed as the Fourier transform of a vector measure and the representing measure is unique. Moreover, for t E R it holds that 7(t) = (x(t),x(0))2

= ( f e"» £(<*«),£(«)) = I WR /2 JR

eUu{d(du),adu))r

By the uniqueness of the meausre in (1.1) we see that v{-) = (£(•), £(-)) 2 Let 6 5 = ©o{£(^4) : A 6 23}, the closed subspace of £ Q ( ^ ) generated by the set {i(A) : A e 53}. Consider the Hilbert space L2(v) = L2(R,v), which we call the spectral domain of the process {x(t)}. Then Kolmogorov isomorphism theorem states that for every stationary process {x(t)} the time and the spectral domains are isomorphic, denoted 'H(x) ~ L2{v), where the isomorphism U : L2{v) -> rL{x) is given by U(f)=

[ fd£,

f€L2(v).

(1.3)

JR

Here the integral is a suitably defined stochastic integral, which is equivalent to the Dunford-Schwartz (DS-) integral (cf. Dunford and Schwartz [1, IV.10]). Note that H{x) = ©£ and hence 64 ~ L2{v).

4

I. INTRODUCTION AND PRELIMINARIES

1.2. Harmonizable processes In many applications stationarity is too restrictive and somewhat weaker condi tions are needed. Harmonizability was first introduced by Loeve [2] in middle of 1940s. Later weaker harmonizability was introduced by Rozanov [1] in 1959. These two notions were distinguished by calling them weak and strong harmonizabilities in Rao [4] in 1982. Let {x(t)} be an Lg(fi)-valued process with the covariance functiuon 7. Then it is said to be strongly harmonizable if 7 can be expressed as

7(M)

ft

J

ei{™-tv)m(du,dv),

s,t£R,

(2.1)

JK2

for some positive definite bimeasure m : 93 x 93 —>• C of bounded variation, where 9 3 x 9 3 = { A x . B : A , . B £ 9 3 } and the variation (= Vitali variation) is given by l

n

|m|(A,B) = s u p ^ ^ | m ( A j , B f c ) | ,

A,Be
j=ik=i

the supremum being and {Bi,... , Bn} of i) m(A, -),m(-, A) every A £ 03; ii) 77i is a positive a>i,... ,an £ C

taken over all finite measurable partitions {Alt... , A^} of A B, respectively. That is, m satisfies the following conditions: e ca(93,C), the space of all C-valued c.a. measures on 03, for definite kernel on 03 x 03 in the sense that for every n £ N, and A 1 ; . . . , A„ 6 93 it holds that n

n

22 Yl aj®km(Aj,Ak) > 0; jf=ifc=i

iii) |m|(K,K) < 00. In this case m can be extended to a measure on 93®93, the Borel
Y^Y^aiPkm(A]>Bk"> }—l

A,fi£93,

k=l

the supremum being taken over all finite measurable partitions { A 1 ; . . . , A(} of A and { S i , . . . ,Bn} of B and for all aj,Pk 6 C with \a3\, \/3k\ < 1, 1 < j < £

5

1.2. HARMONIZABLE PROCESSES

1 < k < n. That is, m satisfies i) and ii) above and ||m||(K,K) < oo. [Originally the terminology "semivariation" was used for vector measures. However, the semivariation defined above for bimeasures plays exactly the same role as that defined for vector measures. Thus we adopt this terminology for bimeasures. Also for a bimea sure m one can show that |]m||(K,R) < oo is always true since 23 is a cr-algebra (cf. Chang and Rao [2, p. 21]).] In this case the integral in (2.1) is a (strict) MorseTransue (MT-) integral (cf. Morse and Transue [4] and for the restricted version see the preceding reference). In both cases the bimeasure m is called the spectral bimeasure of the process {x(t)}. Since the integrand el{sn-tv) = ersue-itv _ ^ 1 ( u ) ^ 2 ( v ) , say, in the integral (2.1) is the product of one variable bounded continuous functions, we can interpret the MT-integral as follows. For A, B 6 93,
ip2(v)m(A,dv),

Ae<3,

JR

ti2(B)=

I
BeX,

JR

we see that p.i,p.2 £ ca(23,C), and ifj is p,j integrable for j = 1, 2. Moreover, they have the same integrals, i.e., / ipidni ¥>i d/i1 = = // JR

f2dn2,

JR JR

which we denote by / / R 2 (pi{u)ip2{v) m(du,dv). To obtain an integral representation of a (weakly or strongly) harmonizable pro cess we have to work a little hard. Let {x(i)} be an L§(fi)-valued weakly harmo nizable process with the covariance function 7 and the spectral bimeasure m. Since m : 2 3 x 2 $ — > C i s a positive definite kernel, we can use Aronszajn's reproducing kernel Hilbert space (RKHS) theory to get a Hilbert space 9ym consisting of C-valued functions on 93, which belong to ca(Q3,C), such that (i) m(A, •) € 9)m for every A e *B; (ii) v(A) = (i>(-),m(A, ) ) m for every v 6 f>m and A e 93, where (-,-)m is the inner product in Sjm. (ii) is called the reproducing property (cf. Aronszajn [1]). The outline of obtaining the RKHS Sjm is as follows. Let

£ 0 = I J2akm{Ak,-) *-

:ak 6 C, Ak 6 03, 1 < k
neNi, '

6

I. INTRODUCTION AND PRELIMINARIES I

which is clearly a subset of ca(33,C). Define for »/(•) = X] ajm(A,-,-)

and

M")

=

n

£ (3km(Bk,-) e?)o fc=i £

n

(M)m = ^ ^ a ^ f c T n ^ j . B f c ) . j = l fc=l

Since m is a positive definite kernel, (-,-) m defines an inner product in Sj0. The RKHS fjm is obtained by the "functional completion" of SjQ, that is, by adding some elements of ca(23,C) to S)0 we make the resulting space f j m to be a Hilbert space. P u t 77(A) = m(A,-) for A <E 23. Then 77 is an fj m -valued bounded and c.a. measure on (K,25), denoted 77 g ca(53,fj m ), and m(A,B) = (r)(A),r)(B))m holds for A, B 6 58 by the reproducing property (ii). Define an f) m -valued process {y(t)} by y(t)=

[ etturi{du),

teM.

(2.2)

and an operator V : H(jj) —t H(x) by Vy{t) = x(t),

t € R.

Then we see that (Vy(s),Vy(t))2

= (x(fl),i(t))2= / / = // J JR2 = ( f e^riidu), \ JR

c * ^ - ^ m(du, du)

e^-t^(r1(du)Mdv))m f eitvr,(dv)) JR

=

(y(s),y(t))m

/ Tji

for every s , t e R Thus V can be extended (by linearity) to a unitary operator from H(y) onto U{x). Put £ = V77. Then £ e ca(23,Lg(ft)) and for t 6 R x(t) = Vy(t) = V f eUu r,(du) = [ eltu VV(du) = [ eUu £(du), JR

JR

(2.3)

JR

since V and the integral commute (cf. Dunford and Schwartz [1, IV. 10]). Therefore we have an integral representation of the L^(fi)-valued weakly harmonizable process {x(t)} with the unique representing measure £ g ea(23,L§(f2)). Let us add some remarks here. Note that the spectral bimeasure m of the process is expressed as m{A, B) = (£{A),£{B))2 = m^{A,B) for A, B g 23. Also note that

7

1.2. HARMONIZABLE PROCESSES

the 5} m -valued process {y{t)} defined in (2.2) can be regarded as a weakly harmonizable process with the representing measure r\ 6 ca(iB,ft m ) and its image by any bounded linear operator is also weakly harmonizable. That is, if 9) and 8. are tv/o Hilbert spaces and {x(t)} is an $j-valued weakly harmonizable process, then, for any bounded linear operator T from 5} into &, a ^-valued process {Tx(t)} is also weakly harmonizable. But this is not necessarily true for stationary or strongly harmoniz able processes. Finally note that a weakly harmonizable process is stationary iff (= if and only if) its representing measure is o.s. Now we state a characterization of weakly harmonizable processes. Let {y{t)} be an Lg(f2)-valued stationary process and P : LQ(Q) —> R be the orthogonal projection of LQ(Q) onto some closed subspace R. Then, as was noted before, the process {Py(t)} is weakly harmonizable. The converse is also true and is known as stationary dilation. That is, for any Lg(f2)-valued weakly harmonizable process {x(t)} there exists a Hilbert space Lg(fi) on some probability measure space (f2, 5, A) containing Lj)(f2) as a closed subspace, and an Lg(f2)-valued stationary process {y(t)} such that x(t) = Py(t) for t € R, where P : Ll(&) -f LQ(U) is the orthogonal projection. This is based on the orthogonally scattered dilation (o.s.d.) of Hilbert space valued. measures and was first proved by Niemi [2] in 1975. The latter result, in an abstract setting, was actually proved by Sz.-Nagy [2] in 1955, extending an earlier result by Naimark [1, 2]. In the rest of this section we give definitions of classes of stochastic processes most of which have integral representations with respect to stochastic measures, i.e., elements of ca(Q3, L Q ( Q ) ) , and state their interrelations. Another extention of stationarity appeared in Karhunen [1] in 1947, which is of interest in applications. An Lg(f2)-valued process {x(t)} is said to be of Karhunen class if its covariance function 7 can be expressed as 7(M)=

g(s,u)g(t,u)v(du),

s.teR,

(2.4)

where v € ca(S3,R+) (R+ = [0,oo)) and {g{t,),t e K} C L2{R,v). Thus, if g(t,u) = eltu, then (2.4) reduces to the stationary case. A further generalization of the Karhunen class was introduced by Cramer [2] in 1951. A process {x(t)} is said to belong to strong Cramer class if its covariance function 7 can be written as j(s,t)=

//

g{s,u)g(t,v)m(du,dv),

s,(eR

(2.5)

for some positive definite bimeasure m of bounded variation and some family {g(t,-),t 6 R} of Borel functions. When the bimeasure m in (2.5) is simply of bounded semivariation, the process is of weak Cramer class, relative to the family {g(t,-),teR}.

8

I. INTRODUCTION AND PRELIMINARIES

Integral representations for these classes are available. If {x(t)} is of Karhunen class with a family {g{t,-),t 6 K} C L 2 ( R , J / ) , then there is a £ 6 caos(
( i ) = [ g(t,u)£(du),

teR

(2.6)

and i/(-) = (£(•),£(0) 2 - Similarly, if (a?(t)} is of (weak or strong) Cramer class with an individually bounded family of Borel functions {g(t, -),t 6 K}, then there is a f € ca( ( B,Lg(fi)) such that (2.6) holds. Thus, such a process has a Karhunen dilation by an o.s.d. of the representing measure. In 1953 Bochner [1] introduced V-boundedness which makes full use of Fourier analysis. An Lo(fi)-valued process {x(t)} is said to be V-bounded if, as a mapping x(-) : E -¥ L Q ( O ) , X(-) is bounded and norm continuous, and if the set

| f v(t)x(t) dt : Me* < 1 , p e l 1 ) is a bounded subset of Lg(fl), where the measure of L X (R) is the Lebesgue measure dt,

he

exists and is positive definite and continuous in h, where 7 is the covariance function of the process. In this case, by Bochner's theorem there is an v 6 ca(Q3,R + ), called the associated spectrum of {x(t)}, such that

m = [ elhu v(du),

hE

Such a process is said to be of KF-class or an asymptotically stationary process. Unfortunately, not every process of KF-class allows an integral representation. The following inclusion relations hold among the classes introduced so far, which

1.3. MULTIDIMENSIONAL AND OTHER EXTENSIONS

(t))\

teR,

are generally proper: weak Cramer class 3 strong Cramer class D Karhunen class

u F-bounded class = weakly harmonizable class U

KF class D strongly harmonizable class U

stationary class

1.3. Multidimensional and other extensions Let (fi,5, M) a n d ^ o ( ^ ) be- as before. Let q > 2 be an integer. Then we can consider q-dimensional second order stochastic process {x(t)} by setting x(t) = (x\t),...

,xq

where {xk(t)} is an LQ(f2)-valued process for 1 < k < q and "*" stands for the transpose. That is, {x{t)} is an [Xg(fi)] '-valued process. For x = (x1,... ,xqY and y = (y1,... , yqY € [LQ(Q)] the inner product and the norm are denned respectively by ^

(x,y)2,q = J2(xk,ykh,

||x||2,,= (X>*|liy.

fc=i

^k=i

'

Moreover, [ L ^ f i ) ] ' ' has a matrix valued or S(C)-valued inner product [•, ■],, called the gramian, defined by f{x\y1)* [x,y]q = xy% =

:

■■■ '.,

1

\(^,y )2

■••

(x\yq)7 : {xq,y«)2i

where B(Cq) is the algebra of all bounded linear operators on C* with the Euclidean norm. Thus we have to consider two kinds of covariance functions for {x(t)}. The scalar covariance function 7 is defined by 7 ( s , i ) = (x{s),x(t))2

s,te

10

I. INTRODUCTION AND PRELIMINARIES

as before and the matricial or operator covariance function T by

r(s,t)=[x(s),x(t)}q,

s,tem.

Clearly, operator covariance functions are more general than scalar ones in the theory of multidimensional processes because 7(s,t) = t r T ( s , t ) , where tr(-) is the trace of the matrix. At first let us consider [Lo(fi)]'-valued stationary processes. An [L§(fi)]'-valued process {x(t)} is said to be operator stationary if its operator covariance function T(s,t) depends only on the difference s - t and if, putting f ( s - t) = F(s,£), each of the components of f is continuous on R. As in the one-dimensional case we can show that there exists a hermitian positive 5 ( C ) - v a l u e d c.a. measure F, denoted F E ca((B,B+(Cq)), and an [Lo(f2)]'-valued c.a. gramian orthogonally scattered (g.o.s.) measure £, denoted £ e cagos(*B, [L§(fl)]'), such that f(t)=

[ eltuF{du),

(£l,

(3.1)

JR

x{t)=

[eitu(,{du),

teR.

JR

If we write £ = ( £ \ . . . , £ " ) \ then it is easily shown that £fc 6 caos(
teK.

Since every Lo(fi)-valued weakly harmonizable process has a stationary dilation, for each k = 1 , . . . , g there exists an L 2 -space Lg(flfc) o n some probability measure space (^fc,5fc,Mfc) containing L Q ( ^ ) ^ a closed subspace, and an Lo(fi fc )-valued station ary process {yk{t)} with the representing measure r\k e caos( ( B,Lg(r2 fc )) such that xk(t) = Pkyk{t), t £ R, where Pfc : Lg(f2fc) -> L§(0) is the orthogonal projection. Choose a probability measure space (fi, J, /2) such that Lo(fi) may contain L§(f2fc) as a closed subspace for fc = 1 , . . . ,q and consider an [Lg(fl)]'-valued process {y{t)}

11

1.3. MULTIDIMENSIONAL AND OTHER EXTENSIONS

defined by y(t) = ( j / 1 ^ ) , . . . , y * ( t ) ) \ t e E. Then we see that {y{t)} is operator stationary with the representing measure 77 = (n1,...

,JJ*)

e cagos(Q3, [Ljj(ft)] )

q

and satisfies that x(t) = Py{t), t g R, where P = 0 P

t

: [Xg(fj)]fl -4 [£§( f i )]

is

the gramian orthogonal projection, Pk being the extension of Pk to L Q ( Q ) . In a similar way, we can define q-dimensional processes of Karhunen, weak Cramer, strong Cramer and KF-classes in a componentwise manner. We now consider the case that the vector valued process has infinitely many components. To obtain an infinite dimensional extension, we have to work a little harder. For q e N observe the identification [ i g ( f i ) ] ' = LQ(Q ; V), the latter being the Hilbert space of all C-valued random variables x on fi with zero expectation such that JR \\X{W)\\Q, n(dui) < 00, where ||-||c« is the usual Euclidean norm in C. If q = 00, we want to consider the Hilbert space Ll(il; P) of all ^ 2 -valued (strong) random variables x on fi with zero expectation such that J R ||i(o;)|| 2 2 fJ-(du)) < 00, where || • \\p is the norm in the sequence Hilbert space i2. Hence, letting H be an arbitrary (infinite dimensional) Hilbert space, we can consider the Hilbert space Lg(fi ; H) and an L^Q ; H)-valued process {x(t)} as an infinite dimensional second order stochastic process. Put X = Z/g(f2;//) a n d study its structure. To do this we need properties of Hilbert-Schmidt class and trace class operators on a Hilbert space, which are seen in Schatten [1, 2]. The inner product (-,-).Y and the norm | | | | . Y are respectively defined by (x,y)x=

/ {x{u),y{w))Hii(dw), Jn

\\x\\x =

{x,x)x

for x,y € X, where (•, -)H is the inner product in H. The gramian \-,-}x in X is defined by [x,y]x=

f x{uj)®yJu)ii(duJ)

(3.2)

JVt

for x,y 6 X , where ® is the tensor product in the sense of Schatten [2], that is, (4>®ip )' = (', IP)H4> for , ip, ' 6 H. The operator defined by (3.2) is as follows: {[x,y}x4>,iP)H

= / ((i(w)®y(u))^,^) Ja

-L

^(dw)

(x(u),ip)H(<j>,y{uj))Hn{duj)

for =(<j>,x)H, 4>eH. (3.3)

12

I. INTRODUCTION AND PRELIMINARIES

Since / n | ( 0 , * M ) H | M < M < / n | | ^ | | 2 H | | x H | | 2 H M ( ^ ) = U\\2HMX < °°> T , is a bounded linear operator and, moreover, a Hilbert-Schmidt class operator, denoted Tx 6 S(H,Ll(Q)). In fact, letting {
,1TX=£||T^«|11 = £ | | ( ^ ) H | Q6I

.2

Q S I

5 3 / | ( ^ c „ a ; H ) H | 2 / x ( d w ) = / Y^\{^>c,x{tj))H\2 aex n ^ a e l

/

n{du)

z(w)|&M<M=NLx<°°.

(3-4)

where we used the Bounded Convergence Theorem and Parseval's equality. Further more, [X,J/]JC = T*Ty holds since for <j>,ip e H we have that ( T X * T „ * , V ) H = (Ty4>,T^)2

I

=

((4>,y)H,(i>,x)H)2

t>,y{u))H{i>,3;(u))HfJ,(dv)

= / ((zH®yM)0,V) H M<M = ([a:,2/]x
(3.5)

Therefore, [x,y]x is a product of Hilbert-Schmidt class operators and hence a trace class operator. It can also be checked that tr[x,y]x

= 5 3 ([x>y]xa,4>a)H = (x,y)x,

\\x\\x =

\\[x,x\x\\l

for x , y € X, where tr(-) is the trace and || • ||T is the trace norm. If x = f(f> and y = gtp, where / , g 6 £ Q ( ^ ) an< ^ 4>,'

,gip]x = (1,9)2 ®i>. Thus we obtained a T(H)-valued gramian [■, -]x on X. Clearly, it holds that (i) [x, x)x > 0, and [x, x]x = 0 iff x = 0; (ii) [x + y, z]x = [x, z]x + [y, z\x\ (iii) [ax,y]x iv

=

a[x,y]x;

( ) [x,y]*x = [y,x}x, for x,y,z € X and a € B(H), the algebra of all bounded linear operators on H. Since X is a left _B(#)-module under the module action (a,x) i-> a - x = ax for x € X and o e B(H), and is a Hilbert space with a T" (if)-valued gramian, we

1.3. MULTIDIMENSIONAL AND OTHER EXTENSIONS

13

call it a normal Hilbert B(H)-module. As we shall see later (Sections 2.1 and 2.3) Ll(il; H) and S(L\{Q), H) are isomorphic as normal Hilbert B(H)-modules by the isomorphism U : x H-> T* where Tx is defined by (3.3) (cf. (3.4) and (3.5)). We shall make full use of this space later in stochastic analysis. X = LQ(U ; H)-valued operator stationary processes are defined similarly as in the case of [Lg(fi)]"-valued processes. Now we want to define X-valued harmonizable processes. Let {x(t)} be an X-valued process with the operator covariance function T and suppose that F has an integral representation T(s,t)=

f[

2

el("u-tv)

M{du,dv),

(3.6)

s,t 6

JJR

for some T(H)-valued positive definite bimeasure M on 03 x 03. If M is of bounded semivariation, which is equivalent to sup {||M(>1, 5 ) | | r : A,B e 23} < oo, then the integral in (3.6) is a well-defined vector MT-integral developed by Ylinen [2]. In this case we can prove by an analog of RKHS theory for vector valued positive definite kernels that there exists a £ 6 ca(23,X) such that x{t)

s

itu

Z{du),

te

Hence {x(t)} may be termed "weakly harmonizable," but it does not have an oper ator stationary dilation in general. [See Chapter IV for more details.] Thus we need a stronger notion of harmonizability. In order to integrate oper ator valued functions with respect to a T(H)-valued bimeasure M or an X-valued measure f we introduce the following. The operator semivariation ||M|| 0 (-,-) of a T(.ff)-valued positive definite bimeasure M is defined by

||M|U(A,B)=sup

X ^ o j - M ^ - , £**)&;

A,BG23,

j=lfc=l

where the supremum is taken over all finite measurable partitions {Ax,. .. , Am} of A and {Bu... , Bn) of B, and a,j,bk € B{H) with ||aj||, \\bk\\ < 1 for 1 < j < m and 1 < k < n. If M is of bounded operator semivariation, i.e., ||M|| 0 (R,R) < oo, then the representing measure f of {x(t)} is also of bounded operator semivariation, denoted £ e 6ca(Q3,X). Here the operator semivariation ||£|| o (0 of £ is defined by \\„(A) = sup

Yla^(Ak)

A 6 23,

fc=l

the supremum being taken over all finite measurable partitions {Ai,... ,An} of A and ak € B(H) with ||afc|| < 1 for 1 < k < n. If this is the case, {x(t)} is said

14

I.

INTRODUCTION AND PRELIMINARIES

to be weakly operator harmomzable and we can show that {x(t)} has an operator stationary dilation, i.e., there exists a normal Hilbert B(H)-module Y = LQ(Q,;H) containing X as a closed submodule and a K-valued operator stationary process {y{t)} such that x(t) = Py{t), t e K, where P : Y -+ X is the (gramian) orthogonal projection. For a further investigation we need to develop integration theory of operator valued functions w.r.t. (= with respect to) an X-valued measure £ G ca(25,X) and a T(H)-valued bimeasure M. Let us add a remark here. Note that each x e X can be expanded as

X=^2{x,cj>a)H<j)a,

(3.7)

which converges in the norm of X, {4>a}aei being the CONS of H. If H is separable, then it has a CONS {k)kLi- For each X-valued process {x(t)} put n fc=i

Then, since \\xn(t) - x(t)\\x -* 0 for every £ 6 R by (3.7), we can regard as a finite dimensional approximation of {x(t)}.

{xn(t)}^'=1

There are other infinite dimensional extensions of second order stochastic pro cesses. Let X be a Banach space and consider the Banach space LQ(Q;X) of all X-valued strong random variables x on Q, with zero expectation such that fn\\x(uj)\\x ti(dbj) < co, where \\-\\x is the norm in X. Take an x e Lg(fi;X) and observe that x*[x()) 6 LQ(Q) for every x* e X*, the dual space of X, since / \x*{x{u))\\{du>)

 L Q ( O ) . Thus, instead of thinking about LQ(U ; X)-valued processes we can consider B(3£*,Lo(fl))-valued processes, or more abstractly, B(2), K)-valued processes, where 2) is a Bancah space and K is a Hilbert space. So far the index set of processes has been restricted to the real line M. When we consider Z, the set of all integers, then {x(n),n e 1} C LQ{Q) may be called a stochastic sequence. If we take Rk, then { x ( t ) , t e l ' } may be called a stochastic field. More generally, we can take an LCA (= locally compact abelian) group G as an index set and call {x(t),t e G} a stochastic process as before, and we shall mainly be concerned with this case. If G = Rk, then we can consider isotropy on processes, which concerns the rotational invariance about the origin. If G is a nonabelian locally compact group, some difficulties arise. To define and obtain

BIBLIOGRAPHICAL NOTES

15

integral representation of stationary or harmonizable processes, we have to define suitable Fourier transform by making use of C"*-algebra theory. A hypergroup was introduced by Jewett [1] and others, and has important applications. When G is a commutative hypergroup, we can consider various types of processes on G as well.

Bibliographical notes There are several books which deal with (weakly) stationary processes, e.g., Cramer and Leadbetter [1](1967), Doob [1](1953), Loeve [3] (1955), Rosenblatt [1](1985), Rozanov [2](1967), [3](1977) and Yaglom [2](1987). Those which treat nonstationary processes are very few. We refer to Priestley [1](1988) for time se ries and Yaglom [2]. Two Proceedings edited by Mandrekar and Salehi [4] (1983) and Miamee [4](1992) are good references for both stationary and nonstationary, or one-dimensional and multidimensional processes. We shall give a brief historical notes here. For one-dimensional second order stochastic processes, (weak) stationarity was first introduced by Khintchine [1](1934) as was mentioned in Section 1.1. Kolmogorov [1](1941) studied interpolation and extrapolation of stationary sequences. Cramer [1](1940) treated multidimensional stationary processes and gave a so called Cramer decomposition of the spectral measure of a process. An extensive study of multidimensional stationary processes was done by Wiener and Masani [1](1957) and [2](1958), Masani [1](1960), Helson and Lowdenslager [1](1958) and [2](1961). Masani [2] (1966) gives a survey for multidimensional stationary processes. Infinite dimensional (or Hilbert space valued) stationary processes appeared in Kallianpur and Mandrekar [l](1965). Later on Payen [l](1967), Nadkarni [1](1970), Mandrekar and Salehi [2](1970), Kallianpur and Mandrekar [2](1971) and Truong-van [2](1985) give several results in infinite dimensional stationary processes. Some of these au thors considered Hilbert-Schmidt class operator valued processes instead of con sidering L 2 ( f i ; _ff)-valued processes. Gangolli [1](1963) already treated B{H,K)valued stationary processes where H and K are Hilbert spaces. This idea appears in treating Banach space valued second order stochastic processes, which was seen in Section 1.3 (see also Miamee [1](1976)). Chobanyan and Weron [1](1975) stud ied B(2),i^)-valued stationary processes, where 2) is a Banach space and if is a Hilbert space. Loynes [3] (1965) considered stationary processes in V7/-spaces (cf. also Loynes [1, 2](1965)) and Saworotnow [4](1973) studied stationary processes with values in a Hilbert module over an //'-algebra. In this direction, we also refer to Rosenberg [3](1978) and Suciu and Valuses,cu [1](1979). As to a survey of infinite dimensional stationary processes one may refer to Salehi [l](1981) and Makagon and Salehi [2] (1989). It is interesting that classes of nonstationary stochastic processes considered so

16

I. INTRODUCTION AND PRELIMINARIES

far contain a class of stationary ones. Many authors have considered the extension of stationarity. Loeve [2] (1948) (see also references therein) defined harmonizability in the middle of 1940s and Rozanov [1](1959) introduced weaker harmonizability. Bochner [l](1956) defined V-boundedness which turned out to be equivalent to weak harmonizability if we consider weakly continuous processes. Karhunen [1](1947) and Cramer [2](1951) introduced new classes of processes which we termed as the Karhunen class and the (strong) Cramer class, respectively. Asymptotic stationarity was defined and studied by Kampe de Feriet and Frenkiel [1](1959) and [2] (1962) (see also Kampe de Feriet [1](1962)), Rozanov [1](1959) and Parzen [2](1962). All these notions were clarified, compared and classified by Rao [4] (1982), where he notified the importance of bimeasure integration theory developed by Morse and Transue in [2](1955) and [4](1956). An extensive study of bimeasure and nonstationary (espe cially harmonizable) processes was given in Chang and Rao [2] (1986) (see also Rao [7](1985) and [9, 11](1989)). When the parameter space is Kfc, Yadrenko [1](1983) gave an extensive study for stationary isotropic processes. Yaglom [1, 2] also treated such processes. Harmoniz able isotropic processes were introduced and studied by Rao [12] (1991) and Swift [1, 2](1994, 1995). If processes on a commutative hypergroup are considered, Lasser and Leitner [1, 2](1989, 1990) and Leitner [1, 2](1991, 1995) studied hyper station ary and hyper weakly harmonizable processes, where the definition of hyper weak harmonizability is due to Rao [10](1989). Study of processes on a nonabelian locally compact group is initiated by Ylinen [1](1975), where he introduced some kinds of stationarity. He also introduced harmonizability for such processes in [7] (1989). Rao [10] considered the case where the group is unimodular. As to infinite dimensional nonstationary processes Truong-van [1] (1981) defined (weakly) harmonizable (rather than F-bounded) processes and obtained (operator) stationary dilations under certain conditions. Kakihara defined weak and strong harmonizabilities and V-boundedness in [7](1985) and [9](1986). He also denned the Karhunen class in [11] (1988). A classification of various harmonizabilities is found in Kakihara [15] (1992) from which most terminologies are taken. Harmonizability and V-boundedness are also introduced to B(H, if)-valued processes and B(X, K)-val\ied processes by Makagon and Salehi [1](1987) and Miamee [2](1989), respectively, where H and K are Hilbert spaces and X is a Banach space. Rao [10] included a survey of harmonizable processes which are either of univariate, multivariate or infinite variate, and whose index sets are in varieties.

CHAPTER II

HILBERT MODULES A N D COVARIANCE

KERNELS

Special classes of Hilbert modules and operator valued positive definite kernels are considered. Let H be a Hilbert space and B(H) be the algebra of all bounded linear operators on H. Then a normal Hilbert B(i/)-module is defined to be a left B(H)-module with the trace class operator valued inner product as an abstraction of the space LQ(CI ; H) which was considered in Section 1.3. Fundamental properties of it will be studied. For infinite dimensional processes, covariance functions are positive definite operator valued kernels, and hence they will be treated in the frame work of the reproducing kernel space theory. Finally, a Stone and a Bochner type theorem is formulated for operator valued continuous positive definite functions on a locally compact abelian group. Also a Plancherel type theorem is established.

2.1.

N o r m a l Hilbert

B(H)-modules

Let H be a complex Hilbert space with norm \\-\\H and inner product (-,-)HB{H) denotes the algebra of all bounded linear operators on H with uniform norm ||-1|, and T(H) the space of all trace class operators on H with trace tr(-) and trace norm ||-|| T - For a left B(H)-modvle X we denote the module action of B(H) on X by B{H) x X B (a, a;) M> ax £X. In this section we define normal Hilbert £?(f/)-modules and give their elementary properties as well as examples. Definition 1. A normal pre-Hilbert B(H)-module is a left B(H)-module X with a mapping [•,■]: X xX —> T(H) which satisfies the following conditions: for x,y,z £ X and a € B{H) (a) [x, x] > 0, and [x, x] = 0 iff x = 0; (b)

[x + y,z]

(c) [a-x,y]

= {x,z} + =

[y,z};

a[x,y};

(d) [x,y]* = \y,x], 17

18

II.

HILBERT MODULES AND COVARIANCE KERNELS

where "*" stands for the adjoint of operators. The mapping [•, •] is called a gramian in X and we sometimes denote it explicitly by [,-}x- W e s e e t n a t bY ( c ) a n ^ (d) [x, ay] — [x, y]a* holds. In the above definition, if the gramian is B(#)-valued, then X has been termed a "pre-Hilbert B(//)-module." In our case, since the gramian is T(if)-valued and T(H) = B{H)m, the predual, we adopted the terminology of "normal" pre-Hilbert -B(i?)-module. The duality T(H)* = B{H) is realized in a way that, given p G T(H)*, then there is a unique ap e B{H) such that p(a) = tr(aa„) for a G T(H) with \\p\\ = ||a p ||, and conversely. This fact is frequently used hereafter. In a normal pre-Hilbert B(H)-moduie X we define ax = (al)-x,

(x,y)x

= tv[x1y],

\\x\\x = (x,x)x

= \\[x,x}\\^

for a e C and x,y G X, where 1 is the identity operator on H. Then we see that X is a vector space and (•, -)x is an inner product in X, so that X is a pre-Hilbert space. We need the following basic result. L e m m a 2. Let X be a normal pre-Hilbert B(H)-module. Then: (1) (a ■ x, y)x = (x, a* ■ y)x for a € B(H) and x,y G X. (2) ||a ■ XJ|JC < HalllNIx foraeB(H)

and x e

X.

(3) [y,x][x,y}< \\x\\x[y,y} for x,y e X. (4) \\{x,y}\\T<\\x\\x\\y\\xforx,yeX. (5) \\x\\x = sup {||[x,y]\\T : \\y\\x < 1} for

xeX.

Proof. (1) and (2) follow from the computation: (a ■ x, y)x = tr [a ■ x, y] = ti(a[x, y\) = tr([x, y]a) = tr([x, a* • y}) = {x, a* ■ y)x, \\a ■ xfx = ||[a ■ x,a ■ x]\\r = ||o[i,i]a*|| T < ||a|| 2 ||x||^. (3) Let S be the state space of B(H), i.e., the set of all positive linear functionals p on B{H) such that \\p\\ = p(l) = 1. Let x,y € X and p e S. Since p([-, •]) defines a semi-inner product in X, we have by the CBS-inequality p([y,x}[x,y})

=

p{[[y,x\x,y\)

< p([[y,x]x,

{y,x\x\yP{{y,y\y 1

= P{[y,x}[x,x}[x,y})

2

1

p([y,y\y

<\\[xfx}\\lp{[y,x][x,y]yp{[y,y])K

2.1. NORMAL HILBERT £(tf)-MODULES

19

since a'ba < \\b\\a*a and hence p(a*ba) < \\b\\p(a*a) for a, b € B(i?) with b > 0. Hence we get p(h/»*]I*.l/])< II^II 2 YP([2/^])Since this is true for every p e S, (3) holds. (4) For x, y 6 X and p 6 5 we have again by the CBS-inequality |p([x, 2 / ])|®ip for (f>,ip £ H, where (®V )' = W,i>)H for ' € H

as before. Then we see that tr [,ip] = ((/>, IP)H, the original inner product in H, and hence H is a normal Hilbert f?(.ff)-module. We shall see later that H has essentially a unique gramian in it which is the one given above (cf. Proposition 5 below). (2) Let q G N or q = oo and A" be a Hilbert space with the inner product (•, -)K and the norm || • \\K. Let H = C if q S N and H = £2 if q = oo. Consider the (inflated) Hilbert space Kq = {(xn)qn=1 For x = {xn)qn=v

y = {yn)qn=1 [x,y] = (ajk),

:xneK,l
IKIItf < o o } .

€ K" we define aik={Xj,yk)K,

l<J,k
That is, [x, y] is the (finite or infinite) Gram matrix determined by x and y. The module action of B(H) is naturally defined, so that Kq becomes a normal Hilbert B(iT)-module. (3) We have seen in Section 1.3 that X = L%{Q.\H) is a normal Hilbert B(H)module.

20

II.

HILBERT MODULES AND COVARIANCE KERNELS

(4) Let K be a Hilbert space and S(if, H) be the Hilbert space of all HilbertSchmidt class operators from K into H. For x,y e S(K,H) and a 6 B{H) define a ■ x = ax and [x,y] = xy* e T{H). Then we see that S{K,H) is a normal Hilbert £(ff)-module. (5) Let K be a Hilbert space and consider the tensor product Hilbert space X = H®K. This is obtained as follows. Let if © i f be the algebraic tensor product m

and define for x = J2 j ® ^ j

n

and

2/ = S 0* ® ^fc

j=i

(x,y)x

E

fc=i

-^ ® ^

the inner

Product

by 771

71

J=lfc=l

Then H ® A" is the completion of H O K with respect to the inner product (•, -)xA module action and a gramian are defined for elementary tensor products by a-(4>®ip) = a<j)®4>,

\®il),' ®ip'} =

{ip,i>')K(®¥)-

One can verify that H ® if is a normal Hilbert f?(ff )-module. By the way H®K and S(if, if) are isomorphic as Hilbert spaces, denoted H®K ~ S{K,H). To see this, fix a CONS j> Q } Q €Z of if. Then, every element in if ® if can be written in the form Yl 4>a®i>a with ^ ||0Q|IH < °°- D e m i e an operator U : i f ® i f -4 S(K,H)

by 1

V

V'c

a6Z

= 5Z$cr®^a,

^

Then we see that U is linear, onto and norm preserving, so that it is a unitary operator since ^a®1pa aex

S(K,H)

/3gz

" e»ex

ogl

ui2„.

Moreover, we can see that

for J^ 4>Q ® I/JQ and "}2 $'0® i>0 ^ aEZ

to ^

H

® K since both sides of the above are equal

/3€Z

$a ® 'Q- Hence, [/ is gramian preserving.

In this case we can say that

2.2. SUBMODULES, OPERATORS AND FUNCTIONALS

21

H ® K and S(K,H) are "isomorphic" as normal Hilbert fl(i/)-modules through the isomorphism U. The precise definition of an isomorphism will be given later in Section 3. We now prove the uniqueness of the gramian in H. P r o p o s i t i o n 5. Let [•,•] be a gramian in H such that tr [0o, 0o] = 1 for some 00 € H of norm 1. Then \<j),%p] = <j>®ip for every <j>,ip e H. Proof. Let 0 e H be such that \\4>\\H = 1 and observe that

[0,0] = [(0® 0)0,(0® 0)0] = (0®0)[0,0](0®0), which is a nonzero positive operator. Thus for each ip 6 H we have that [0,0]V>= ( 0 ® 0 ) [ 0 , 0 ] ( 0 ® 0 ) ^ = ( 0 ® 0 ) W > , 0 ) H [ 0 , 0 ] 0 = ([0,0]0,0)H(0,0)H0= ( [ 0 , 0 ] 0 , 0 ) H ( 0 ® 0 ) 0 .

Hence [0,0] = Q0 ® 0 for a = ([0,0]0, 0 ) H > 0. Then a = 1 since ||0O||H = 1 and 1 = t r [0o, 0o] = a. Thus for every 0 i , 02 € H it holds that [01,02] = [(01® 0 ) 0 , (02® 0 ) 0 ] = ( 0 1 ® 0 ) [ < M W > ® 0 2 ) = ^ 1 ® ^ 2 -

2.2. S u b m o d u l e s , operators a n d functionals In this section we consider submodules and gramian orthogonal complements in a normal Hilbert 5(i?)-module X. Among the bounded linear operators on X those which have gramian adjoints are of interest and they are characterized. Fi nally, T(i?)-valued functionals on X are considered and a Riesz type representation theorem is proved for bounded functionals. Definition 1. Let X be a normal Hilbert B(H)-modu\e. A subset of X is called a submodule if it is a left B{H)-mo&xAe. Every closed submodule is itself a normal Hilbert jB(#)-module. For a subset Y of X denote by &{Y) the closed submodule generated by Y, i.e., the closure of the set | ^TakVk *■

:ak € B{H),yk fc=i

eY,l
22

II. HILBERT MODULES AND COVARIANCE KERNELS

and by &o{Y) the closed subspace generated by Y. Note that every closed submodule is a closed subspace and &o(Y) C 6 ( F ) , the set inclusion being proper in general. For a subset Y of X the gramian orthogonal complement F # of Y is defined by Y* = {x £ X : [x,y] = 0, y £ Y}. The usual orthogonal complement of Y is denoted by F x and it holds that F # C F x . An orthogonal projection onto a closed submodule is called a gramian orthogonal projection. We have gramian orthogonal decompositions for elements in a normal Hilbert B(H)-module, which may be refered to as the "gramian orthogonal projection lemma." L e m m a 2. Let X be a normal Hilbert B(H)-module. Then for any subset Y of X the gramian orthogonal complement F # is a closed submodule. Hence, if Y is a closed submodule of X, then Y = F * and every x £ X can be decomposed uniquely as x = xi + x2, x1 £ F, x2 £ F # . That is, letting P : X —>■ F be the gramian orthogonal projection, it holds that Xi = Px and x 2 = (/ — P)x, where I is the identity operator on X. Moreover, the following equality holds: \\X-X-LWX

= min{||x-y||.Y : y £ F } .

Proof. Let F C X. If x%, x2 € Y# and a £ B(H), then [xi + X2,y\ = [xi,y]+

[X2,y] = 0,

[a-xuy]

= a[xj,y] = 0

for every y 6 F , so that Xi + x 2 , a ■ Zi € F * and hence F * is a left B(H)-module. To see that F * is closed, let {x n }^L 1 C Y* be a Cauchy sequence. Then there is an x E X such that ||x„ - x\\x —> 0 (n -+ oo). Now we see that, for y g F , [i„,y] = 0 (n > 1) and ||[x,2/]|| T < ||[x - x n , y ] | | T + ||[X„,J,]|| T < ||x - x n ||x||2/||jc ^ 0 as n -> oo by Lemma 1.2 (4), i.e., [x,y] = 0. Thus x e Y*. The rest of the lemma is now obvious. Definition 3. Let X be a normal Hilbert B(H)-modu\e and B(X) be the algebra of all bounded linear operators on X. An operator T e B(X) is said to have a gramian adjoint if there is an S £ B(X) such that

[Tx,y] = [x,Sy],

x,y£X.

23

2.2. SUBMODULES, OPERATORS AND FUNCTIONALS

S is unique if it exists, and is denoted by T*, which is equal to the usual adjoint T" of T. A(X) denotes the set of all operators in B(X) with gramian adjoints. Note that T** = T, {ST)' = T*S*, \\T*\\ = \\T\\, \\T'T\\ = \\T\\2 for S,T £ A(X). T £ B(X) is said to be gramian self-adjoint if T € A(X) and T* = T and to be gramian positive if [Tx,x] > 0 for x £ X. Thus, every gramian positive operator T has a gramian adjoint, i.e., T e A(X). U E B(X) is said to be gramian unitary if U E A{X), is onto, and satisfies that [Ux,Uy} = [x,y],

x, y € X.

1

Hence, U*U = I and U' = U*. T 6 B(X) is said to be B(H)-linear if T commutes with the module action, i.e., T{a ■ x) = a ■ (Tx) for a g B(H) and i e X . A S(iJ")-linear mapping is also refered to as a module homomorphism or a module map. Note that any gramian projection P satisfies that P2 = P* = P, that is, [P2x,y)

= [Par,y] = [x,Py] = [Px,Py],

x,y e X.

If Y is another normal Hilbert S(//)-module, then we denote by B(X,Y) the Banach space of all bounded linear operators from X into Y under the uniform norm and by A(X, Y) the set of all T E B(X, Y) with gramian adjoints, i.e., T € A(X, Y) iff there exists an 5 € B(Y, X) such that [Tx,y]y = [x,Sy]x for x E X and y E Y. B(H)-linearity of operators in B(X, Y) is defined in the same fashion as above. For each a E B(H) define an operator n(a) by TT(O)X = a ■ x for x E X. Lemma 1.2 (2) shows that n(a) E B(X) with ||7r(a)|| < ||a|| for a e B(H). Moreover, n(-) is a *-representation of B{H) on B(X), i.e., n(a + b) = ir(a) + Jr(fe), n(ab) = 7r(a)7r(6) and 7r(a") = it (a)' for a, b E B(H). \\n(a)\\ = \\a\\ holds for a G B(H) since 7r(-) is faithful, i.e., TT(-) is one-to-one. T e B ( X ) is S(i/)-linear iff T commutes with it(-). P r o p o s i t i o n 4 . Let X be a normal Hilbert B(H)-module. Then: (1) If u E B(H) is unitary, then it(u) is an isometry on X. (2) An operator T E B(X) is B(H)-linear iff T E A(X). (3) An operator U E B(X), which is onto, is gramian unitary iff U is unitary and B(H)-linear, which is so iff U satisfies [Ux, Uy] = [x,y] for x,y E X. (4) The norm of T E A(X) is expressed as \\T\\ = inf {a > 0 : \Tx,Tx\

x E X).

Proof. (1) Let u E B(H) be unitary, then for x, y E X we have (it{u)x,n(u)y)x

= (u-x,u-y)x

=

tt[u-x,u-y]

= ti(u[x,y]u*)

=ti(u*u[x,y})

=tr[x,y]

=

{x,y)x.

24

II. HILBERT MODULES AND COVAR1ANCE KERNELS

Thus 7r(w) is an isometry. (2) Assume that T £ B(X) is B(H )-linear. Then, for any a £ B(H) and x,y we have that tv(a[Tx, y}) = tr([a ■ (Tx),y\) = (a • x, T'y)x

= tr [T(a -x),y}

= {T(a ■

eX

x),y)x

= tr [a • x, T'y] = tr(a[x, T'y}),

where T" is the usual adjoint of T. Thus it follows that [Tx,y] = [x,T'y],

x,y£X,

from which we conclude that T £ A(X) with T* = T'. Conversely, suppose that T £ A(X). Then [T(a-x),y]

= [a-x,T*y]

= a[x,T*y] = a[Tx,y] = [a-

(Tz),y]

for a e S ( i f ) and x,y e X. Hence T(a ■ x) = a ■ (Tx). (3) Suppose that U is gramian unitary. Then, by definition and (2), U is B(H)linear and unitary. Suppose that U is unitary and S(H)-linear. Let x,y 6 X be fixed and observe the following two-sided implications: [Ux, Uy] = [x, y] <^=> ti(a[Ux, Uy]) = tr(a[x, y}) for every a 6 <=> (a ■ {Ux),Uy)x

B{H)

= (a • x,y)x

for every a £ -B(-ff)

<*=> (£/(a • x), £/«/) x = (a • x, y)x

for every a e 5 ( i f ) .

Since the last condition holds we have [Ux, Uy] = [x,y], as desired. Finally, suppose that U satisfies that [Ux, Uy] = [x, y] for x, y e X. We only have to show that U is S(/Z")-linear. We see that for a £ B(H) and x e X [U(a ■ x) - a ■ {Ux),U{a

■ x) - a ■ (Ux)}

= [U{a-x),U{a-xj] - [U(a-x),a-(Ux)] = [a-x,a-x]

- [a ■ (Ux), U(a ■ x)] + [a ■ (Ux), a ■ (Ux)}

- a [Ux, U(a ■ x)} - [U(a ■ x), Ux}a* + a[Ux, Ux]a'

= [a ■ x, a ■ x] - a[x, a ■ x] - [a ■ x, x]a* + a[x, x]a* = 0. Thus, U is B(#)-linear. (4) Let a > 0 be such that [Tx,Tx] < a2[x,x] for x £ X. Then ||[Tx,Tx]|| a 2 | | [ x , x ] | | r , i.e., \\Tx\\x < a\\x\\x for x € X. This implies that ||T|| < a.

<

2.2. SUBMODULES, OPERATORS AND FUNCTIONALS

25

We have to prove that [Tx,Tx] < ||T||[x,x] for x e X. Let x 6 X be fixed. Then for a state p e S of B(H) (see the proof of Lemma 1.2 (3)) we have by the CBS-inequality p([Tx,Tx})=p({T*Tx,x}) <

p{lT*Tx,T*Tx]yP(ix,x})1

< || [(T'Tfx,

{T*Tfx\

\\fp([x,x\f-^

<(w(T'Trnx\\\)^p{[x,x])1-^ =

\\T\\2\\x\\lrTp([x,x])1~^

for any n > 1. Letting n ->• oo, we get p([Tx,Tx\) < \\T\\2p{[x,x]). for every p £ <S, we deduce that [Tx,Tx] < \\T\\[x,x].

Since this holds

If X and Y are normal Hilbert B(.ff)-modules, then an operator T g B ( X , F ) is S ( i f )-linear iff T 6 A(X, F ) . This is verified similarly as in the proof of Proposition 4 (2) above. D e f i n i t i o n 5. Let X be a normal Hilbert S(H)-module. By a functional on X we mean a B(H)-linear mapping I : X —i T{H), that is, i satisfies that £(a -x + b-y) = a£(x) + b£(y) for a, b 6 B(H) and x,y E X. A functional ^ d.n X is said to be bounded if there is a constant a > 0 such that ||f(x)|| T < a||x||x for x € X. In this case the norm \\£\\ is defined to be the infimum of such Q'S. A Riesz type representation theorem holds for a bounded functional on X. P r o p o s i t i o n 6. For a bounded functional £ on a normal Hilbert B(H)-module X there exists a unique xe € X such that £(x) = [x, Xf\ for x e X with \\£\\ = \\xt\\xProof. Let £ be a bounded functional on X with norm ||£||. Then tr(£(-)) is a usual bounded linear functional on X with the norm < ||l||. Hence, by the usual Riesz representation theorem, there exists a unique xi 6 X such that tx£{x) = (x,xi)x for x 6 X. Now observe that, for a fixed x 6 X, we have tr(a£(x)) = tr(£(a • x)) = {ax, X[)x = tr [a • x, X(] = tr(a[x, X(\) for a 6 B(H). Thus we conclude that £(x) = [x,xt] for x £ X. \\i\\ = \\xe\\x is seen from Lemma 1.2 (5).

The equality

26

II. HILBERT MODULES AND COVARIANCE KERNELS

C o r o l l a r y 7. Let X be a normal Hilbert B(H)-module a bounded conjugate bimodule map, i.e., sup {\\L(x,y)\\r

and L : X x X -> T(H) be

: \\x\\x < 1, \\y\\x < l } = \\L\\ < oo,

L(ai - i i + 02 ■ i 2 , t i ■ yi + &2 -Jft) = aiL(xi,yi)bl

+

aiL(xi,y2)b*2

+ a2L(x2, y\)b\ + a2L{x2, y2)b*2 for every x1,x2,y1,y2 G X and alra2,bi,b2 T 6 A{X) such that \\T\\ = \\L\\ and L(x,y)

= [x,Ty],

6 B(H).

Then there exists an operator

x,y G X.

(2.1)

If L is hermitian, i.e., L(x,y)* = L(y,x) for every x,y G X, then T is gramian selfadjoint, and if L is positive, i.e., L{x,x) > 0 for every x G X, then T is gramian positive. Proof. For a fixed y € X consider a functional £(■) = L(-,y) on X. Then it is a bounded module map, so that by Proposition 6 there exists a unique y' € X such that £(x) = [x,y'] = L(x, y) for x E X. Define an operator T by Ty = y' for y e X. Clearly T is B(iJ)-lmear. That T is bounded follows from (2.1) with ||T|| = ||L||, so that T g A(X). The rest of the assertions are easy to show.

2.3. Characterization and structure In this section we discuss the characterization and the structure of normal Hilbert 5(H)-modules. Given a Hilbert space X with the inner product (-, •)% which is also a left B(H)module, when does X become a normal Hilbert B(//)-module? That is, when can we define a T(i7)-valued gramian [-, ■] for which tr [x, y] = (a;, y)x holds for x, y G XI Recall that a functional p on B(H) is said to be normal if it is
with such linear [-, •]

2.3. CHARACTERIZATION AND STRUCTURE

27

Proof. Define A x (x G X) and a-(a) (a G fl(if)) by A-x(a) = a • x = 7r(a)x. Since, for any x 6 X, px is positive linear on B(H), we see that px{a*) = px{a) and, hence, (a ■ x,y)x = (x,a* • J/),Y for x , y G X and a G B(H). Thus, TT(-) is a '-representation of B(H) on X . But every '-representation of a Banach '-algebra on a Hilbert space is continuous (cf. Rickart [2, p. 205]). So there is an a > 0 such that \\n(a)x\\x = \\Ax(a)\\x < a\\a\\\\x\\x, a G B(H), x G X. Now A x : B(H) H> X, so A* : X -> 5 ( H ) * . Define k y ] = AJ(j/),

x, j e i

Then, one verifies that [•,•] is a B(H)'-valued inner product on X. Since px is normal, i.e., px is a{B(H), T(i?))-continuous, we see that px G T(if) for x G X. Now for a G B(H) and x G X it holds that Pi (a) = (a ■ x, x)x = ( i , a* • x)x = {x,Ax(a*))x

=
where (•, ■) is the duality pairing of B(H)* and B(H). Consequently, we conclude that [x,x] G T(H) and px{a) = tr([x,x]a*) for a G B(H) and x e X. It is clear that [x,y] G T{H), [x + y,z] = [x,z] + [y,z] and [x,«/]* = [y,x] for i , j , z £ l . The equality [a-x,y] = a[x,y] is derived as follows: for every b G B(H) ([a-x,y},b)

= (A*a,x{y),b) = (y,Aa,x(b))x

= (y, (6a) • x)x

= (y,Ax{ba))x

=

= (Ax{y),ba)

= tv{[x,y]ba) = tr (a[x,y]b) =

([x,y],ba) (a[x,y],b).

To discuss the structure of a normal Hilbert B(H)-modu\e we need the following two definitions which give isomorphisms and bases for normal Hilbert B(H)-modules. Definition 2. Let X and Y be two normal Hilbert B(/f)-modules with gramians [-,-}x and [•,-]y, respectively. Then, X and Y are said to be isomorphic, denoted X = Y, if there exists a gramian unitary operator U : X -4 F . That is, [/ satisfies the conditions that U is onto, U{a-Xi+b-X2) = a - ( [ / x 1 ) + 6 ( f / x 2 ) , and [Cxj, Ux2]y = [a ; ii^2]x for a, 6 G 5 ( / f ) and x l5 X2 G X . Such a [/ is called an isomorphism. Let { X Q , Q G X} be a family of normal Hilbert B(H)-modules with gramians [', •]«) a G I . The A'recf sum X = 0 Xa of {X Q , a G 1} is defined by

X = 0I

a

= S (lo)ael

:

^a G X Q ,

a £ I, ^

ll^alla < 00 L

28

II. HILBERT MODULES AND COVARIANCE KERNELS

where ||-|| a is the norm in Xa for a € X. In X we define a module action and a gramian by a- x = (a- xa)aeX,

[x,y] = } ^ aex

[xa,ya]a

for a e B(H) and x = ( x a ) a 6 i , 2/ = (ya)aex £ -X- The well-definedness of them is easily verified and it holds that {x,y)x = tr[i,j/] = ^ (x a ,2/a)a> (■,•)<* being the inner product in X a for a 6 X. Since, as was seen before, H is a normal Hilbert B(H)-module, we can consider a direct sum 0 Ha, Ha = H (a € I ) . As we shall see later (the structure theorem) every normal Hilbert B(H)-modvde

can be expressed in this manner.

Definition 3. Let { x a } a e i C X be a set of elements of norm 1. Then, {x Q } Q e x is said to be gramian orthonormal if (i) [xa,x0] (ii) [xa,xa]

= 0 for a ^ fi; 2

= [xa,xa]

for

a e l

The condition (ii) says that, for each a € I , [xa, xa] is a usual orthogonal projection on H, and moreover it has finite dimensional range since it is in T(H). A maximal gramian orthonormal set is called a gramian basis, which exists by Zorn's lemma. The following theorem, analogous to a Hilbert space case, gives several equivalence conditions for a gramian orthonormal family to be a gramian basis. T h e o r e m 4. Let {xa}aex be a gramian orthonormal set in a normal Hilbert module X. Then the following statements are equivalent: (1) {xa}a£2

B(H)-

is a gramian basis.

(2) / / x 6 X and [x, xa] = 0 for a € I , then x = 0. (3) / / Xa = {a ■ xa : a € B(H)}

for a el,

then X =* 0 aex

Xa.

( 4 ) i = Yl [x, xa] ■ xa for x 6 X. (5) [x,y}=

J2 [x,xa][xa,y]

(6) [x,x] = ^2[x,xa}[xa,x]

forx,yeX. for x £ X.

a£X

Proof. (1) => (2). Suppose that there exists a nonzero x € X such that [x,xa] = 0 for a e I- Since [x,x] is a positive trace class operator, we can find an eigenvalue c > 0 with a corresponding eigenvector <j> € H of norm 1, so that [x, x\

2.3. CHARACTERIZATION AND STRUCTURE

a = c~^4>® 4> e B(H)

29

and put x0 = a ■ x. Then we see that for ip € H

[x0,xo\ip = a[x,x}a*rp = c~l(<j> ® (p)[x,x](ip,<j))H(j)

and hence [x0,a;o] = 4> ® = [xo,xo]2- Moreover, ||a:o||x = 1 and [x 0 ,x Q ] = 0 for a e I . Thus we have a larger gramian orthonormal family {xo,xa}aex, which contradicts the maximality of {xa}aEx. (2) =>- (3). Note that Xa = {a ■ xa : a 6 B(H)} is a closed submodule of X for a € I, and Xa and Xp are gramian orthogonal if a / /3, i.e., [x,2/] = 0 for x € -XQ and y € X/3. Hence we can construct a direct sum 0 Xa C X , which is also a closed submodule of X. If X ^ 0

X a , then there is a nonzero a; €

0

Xa

, the

gramian orthogonal complement. This implies that [x,x Q ] = 0 for a e I , so that x = 0 by (2), a contradiction. (3) => (4). Suppose that (3) holds and let x E X be given. Then it can be written as x = Yl aa ' xa for some aa 6 B(H) (a E I ) . Note that, for each a S I , [xQ, x Q ] • i Q = x Q as we can easily show: M^a 1 ***a:J ' Xa

x Q , [XQ , XQJ ' Xa

Now we have that [x,x a ] ■ xa = aa[xa,xa] holds. (4) =>■ (5) and (5) => (6) are obvious.

Xa\

U.

■ xa = aa ■ xa for a € I and hence (4)

(6) =>■ (1). If {a; Q } Q6 x is not a basis, we can find an element XQ E X of norm 1 such that [lo,!,,] = 0 for a E I . Then by (6) we have [XQ^O] — ]C [^OJ^QII^QI^O] = 0, aex a contradiction. Corollary 5. For a normal Hilbert B(H)-module X there exists a gramian basis {x Q } Q gx such that [xa,xa] is a one dimensional orthogonal projection on H for each a el. Proof. Let {x Q } Q gx be a gramian basis of X. Suppose that, for each a E I , [x Q ,x a ] is an na dimensional orthogonal projection, [xa, xa] — p Q ] i + h p Q | „ a say, where paj is a one dimensioanl projection (1 < j < na) and paj _L pak (j ^ k). Put Xot,j ~ P<x,j ' XCLI 1 < 3 < " a and observe that for x € X

X =

/ a£l

[X, XaJ

■ Xa

/

^ /

a£lj'=l

^ [^1 ^a j \ '

XQj.

30

II.

HILBERT MODULES AND COVARIANCE KERNELS

since {a; Q } Q e i is a gramian basis and Theorem 4(4) is applied. Now again by Theorem 4, we see that { x Q , i , . . . , i a , „ J Q £ i is a gramian basis, as asserted. We are now in a position to prove the structure theorem for a normal Hilbert B(H)-module using Theorem 4 and Corollary 5. T h e o r e m 6 (Structure T h e o r e m ) . For every normal Hilbert B{H)-module there exists an index set 1 such that X = 0 Ha, where Ha = H for a 6 2 .

X

Proof. By Corollary 5 we can choose a gramian basis {x Q }aex of X such that [xQ, xa) is a one dimensional orthogonal projection, 4>a ® Q for some <j>a e H of norm 1 say, for a e l Then, for each a £ I , Xa = { a i Q : a € B{H)} is a normal Hilbert B{H)module such that Xa = H since we have an isomorphism Ua : Xa -> H defined by Ua{a ■ xa) = aa for a e B{H). Now Theorem 4 (3) implies that X =* © Ha, Ha =

H{aeI).

Corollary 7. Every normal Hilbert B(H) -module X is isomorphic to the HilbertSchmidt class S{K,H) and hence to the tensor product Hilbert space H ® K for some Hilbert space K. Proof. By Theorem 6 we can find an index set 1 such that X = Y = 0

Ha,

Ha = H ( a e l ) . Let if be a Hilbert space which has a CONS {rf>a}aei- Define an operator V on Y by

Vy=^24>a<8>ipa, Since J2 II^QHH < ° ° ' ^V ^

a

y = (4>a)aax e Y.

well-defined Hilbert-Schmidt class operator from

Q£I

K into H for each y e Y, i.e., Vy e S(K,H). Clearly V gives an isomorphism Y = S{K, H). S{K, H) S H®K was shown implicitly in Example 1.4 (5). Therefore, we have X S 5(A", H)^H®K. F if Y of as

Let us make some remarks here. Let X be a normal Hilbert £ ( i i ) - m o d u l e and be a closed submodule. According to Corollary 7, we can find a Hilbert space such that X = H ® K. Hence there is a closed subspace Ki of K such that = H®K1. Moreover, B(X) = B(H ® K) = B(H) ® B(K), the tensor product C*-algebras. Then it follows from Proposition 2.4 (2) that we can identify A(X) the commutant of B{H) g) 1, which constitutes 1 ® B(K) = B{K).

Corollary 8. Let X be a normal Hilbert B(H)-module, and { X Q } Q € I and be two gramian bases of X. Then 1 and J have the same power.

{y0}

31

2.3. CHARACTERIZATION AND STRUCTURE

Proof. It follows from Corollary 7 that X = H ®K1 = H ®K2, where Kx and K2 are Hilbert spaces with dimensions | I | and \J\, respectively. Then it follows that K\ ~ K2, so that \1\ = \J\, i.e., I and J have the same power. D e f i n i t i o n 9. Let X be a normal Hilbert £?(#)-module. The power of any gramian basis of X is called the modular dimension of X, denoted Dim(X). X is said to be separable if Dim(X) < NoC o r o l l a r y 10. Let X and Y be two normal Hilbert B(H)-modules. Then they are isomorphic iff Dim(X) = Dim(Y). If X is a normal Hilbert 5(.ff)-module, then we have that dim(X) = Dim(X)dim(H), where dim(-) is the usual dimension of Hilbert spaces. Thus, if X = H ® K, then Dim(^C) = dim(i'C) and, hence, X is separable iff so is K. Let us give an example of a gramian basis of X = H ® K, K being a Hilbert space, as follows. P r o p o s i t i o n 1 1 . Let X = H ®K where K is a Hilbert psace, {ipa}a^z be a CONS in K and take £ H of norm 1. Then the family {
Proof. First we can see that [ ® ipp] = {4>a,4>g)K ® 4> = 0 if a / /3, [ ®ipa, = [® 4>a,® ipa]2 and | | 0 ® 4>a\\x = 1 for a g I . N let {<j>0}p£j be a CONS in H and observe that {0 ® ipa}aei,pej forms a CONS in H ® K. Take any x e H ® K and consider its Fourier expansion:

a,/3

Since the nonzero terms in the above sum are at most countable, we can choose an at most countable subset {0}i3£j (possibly N = oo) such that N

N

X = ^2^2{x,(j)0k k=laeX

®i>a)x4>0k

®4>a = Yl Yl (X>
Writing k = pk and putting ipk = J2 (z, <j>k ® 4'a)x4'a

£ K for 1 < k < N, w

TV

see that x = ^

4>k®i>k- Then for each a g I it holds that

fc=i r

[ x , $ ® ^ a ] ■ (® V"c

iV

^ ^ f c ® l/,fc,® V'a fc=l

®ipa-

i®1pa)

32

II.

HILBERT MODULES AND COVARIANCE

KERNELS

N fc=i N fc=l N

k=l

Consequently we have that N aelk=l

N

N

\

M=l

'

I

a£Xk=l

X

^s^<{'l>,i4>k)H{i>j^a)K$k®i)a

= a€X

k,j N

= 5 2 ILJ ^ f c ' ^Kk

® 4>a

a£Tk=\

= ^2{x,®il>a). Thus Theorem 4 (4) holds. Therefore {<j> Cg> V'ajaex is a gramian basis of X.

2.4. Positive definite kernels and reproducing kernel spaces C-valued positive definite kernels and their RKHS's were briefly discussed in Section 1.2. In this section we are concerned with T(//)-valued positive definite kernels and their reproducing kernel normal Hilbert B(i?)-modules consisting of T(H)-v&l\ied functions. Let 0 be a nonempty set. Definition 1. A function F : 9 x 0 —> T(H) (p.d.k.) if n

is called a positive definite

n

S X ^ r ^ . t k t e >o j=i

kernel

(4.i)

fc=i

for any n € N, O i , . . . , a„ € B(H) and tu ■ ■ • , tn € 0 , where the LHS (= left hand side) of (4.1) is in T+(H), the set of all nonnegative elements in T(H). [Note that

2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES

if T is B(if)-valued, above.]

33

then we can also define T to be a p.d.k. in the same manner as

P r o p o s i t i o n 2 . A function

F : 6 x © —» T(H) is a p.d.k. iff it satisfies that n

n

^^(r(i

f c

,^)^,^)

H

>0

(4.2)

J=lfc=l

for any n € N, 4>i,... , 4>n 6 H and * i , . . . ,tn 6 8 . Proo/. Suppose that (4.2) holds and let n € N, fli,... ,an € -B(#) and £ l 7 . . . ,£„ 6 9 be arbitrary. Then for any <j> 6 H it holds that

VV

;,fc

7

/H

j , k

Therefore F is a p.d.k. Conversely, suppose that T is a p.d.k. and let n € N and 4>\, ■.. , 4>n 6 H be arbitrary. We can choose a nonzero $ 6 H and Oi>... , a n g B(H) such that a*4> = 4>j, 1 < j < n. Then it holds that

]T (r(ttl tj)4>iAk)H = £ (r(tt,*i)o^, a^) H (j2akr(tk,tj)a;)'P,A

>0.

J.fc

Thus (4.2) is satisfied. Elementary properties of a p.d.k. are noted next. P r o p o s i t i o n 3 . Let F : © x © -> T(H) be a p.d.k.

Then:

(1) T(t,t) > 0 for every te ©. (2) T(i, *)* = T(s,t) for every t, s € ©. (3) r ( t , s)F(s, t) < | | r ( s , s) || T F(i, t) /or euen/ t, s G 0 . Proof. (1) is clear from the definition. (2) Let t, s € 0 be fixed. Then, for every , T/J e if it holds that ( r ( M ) ^ V ) H + {r{t,t),4>)H + {T{s,s)rl>,il>)H + (T(s,t)^,cP)H It follows from (4.1) that

(r(t,*)^)„ + (v>,r(*,t)'*)H

> 0.

34

II. HILBERT MODULES AND COVARIANCE KERNELS

is real. Putting ip = 4> and ip = i(f> (i = \f—l),

(r(t,s))H + (0,r(s,t)V) H ,

we have that

i{{r(t,8)4,4>)a

are real, so that (r(t,s), )H = (T(s,t)*^,)H. we conclude that T(t, s)* = F(s,t). (3) follows from Lemma 1.2 (3).

- (,r(s,t)*)H}

Since this holds for every <j> e H,

An analogy of the RKHS theory can be obtained by introducing reproducing kernel normal Hilbert B{H)-modules. Definition 4. Let r : 6 x 0 -> T(H) be a p.d.k. and X be a normal Hilbert B(if)-module with a gramian [•, •] consisting of T(H)-valued functions on 0 . Then r is said to be a reproducing kernel (r.k.) for X if (a) r ( t , ■) eX for each t e 9 ; (b) x(t) = [x(-),T(t, •)] for each t € 9 and x e X. The property (b) is called the reproducing property. In this case X is said to be a reproducing kernel (r.k.) normal Hilbert B(H)-module of T. In the case where X is a normal pre-Hilbert i?(//)-module, we can also say that T is a r.k. for X if (a) and (b) above hold. Note that, if H = C, the above definition reduces to the usual RKHS. Propositions 5-11 below give basic results on r.k. normal Hilbert B(.ff)-modules. P r o p o s i t i o n 5. / / a normal Hilbert B(H)-module is unique. Proof. If T, T' : 9 x 9 -> Tin) that, for t, s 6 0

X admits a r.k., then the kernel

are r.k.'s for X, then it holds by Proposition 3 (2)

r(s,t) = [r( s ,-),r'(t,-)] = [r'(*, •),!>, •)]* =r'(t,sy

= r'(s,t).

P r o p o s i t i o n 6. Let X be a normal Hilbert B(H)-module consisting of'T(H)-valued functions on 0 . Then, X admits a r.k. iff, for each t 6 0 , i(x) = x(t) (x 6 X) is a bounded functional on X. Proof. Suppose that X has a r.k. I \ For a fixed t € 0 we have that i(x) = x(t) = [&(•), r ( i , - ) ] for x e X. Hence | | i » | | T < | | i | U I | r ( t , - ) I U for x € X, so that i is bounded. Clearly, t is a functional on X. Conversely suppose that, for each t G 0 , i is a bounded functional on X. Then, by Proposition 2.6 there exists a unique xt g X such that i(x) = x(t) = [x,xt] for x 6 X. Putting T(t,s) — xt(s) for t,s 6 0 , we can check that T is a r.k. for X.

2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES

P r o p o s i t i o n 7. Suppose that F : 9 x 9 - > T(H) B(H)-module X.

is the r.k. for a normal HUbert

(1) If a sequence {i„}™ =1 C X converges strongly to x £ X,

||x n (t)-x(£)|| T ->0,

35

then

tee.

(2) / / a sequence {xn}^=i C X converges weakly to x € X, then {x„(t)}'^=1 T(H) converges weakly to x(t) E T(H) for every t 6 0 .

C

Proof. (1) For each t 6 0 observe that

\\Xn{t) - x(t)\\T = ||[xn(-) - x ( . ) , r ( t , -)]|| T < ||*n - x\\x\\r{t,-)||.v

-+ o.

(2) For each t € 0 and a G B ( i / ) we have that tr((xn(t)

~ x(t))a)

= tr [ » „ ( ■ ) - i ( - ) , a T ( * , - ) ] = ( i „ - i , a T ( ( , - ) ) x -> 0.

P r o p o s i t i o n 8. Let { x Q } a g z 6e a gramian orthonormal subset of X. Then it is a gramian basis of X iff T admits a representation r(s,t) = ^ x a ( s ) * x Q ( t ) ,

s,t€B.

Proof. Suppose that { X Q } Q 6 X is a gramian basis of X. 3.4 (4) and the reproducing property that for s e 6

(4.3)

It follows from Theorem

so that (4.3) hold. Conversely, suppose that (4.3) holds. If x € X and [x,x a ] = 0 for a 6 I , then we have that x(t) = [x(-),r(t,-)] =

s ( 0 . 5 3 *„(*)* ■««,(•) = ^ 3 Q £ I

[x(-),x Q (-)]x Q (i) = 0

QgZ

for i 6 0 , which implies that x = 0. Thus Theorem 3.4 concludes that {xa}aeJ a gramian basis of X.

is

P r o p o s i t i o n 9. Suppose that X(T) is a normal Hilbert B(H)-module admitting a p.d.k. r : 0 x 0 —► T(H) as a r.k. Let X be a larger normal Hilbert B(H)-module containing X as a closed submodule and P : X —> X be the gramian orthogonal

36

II. HILBERT MODULES AND COVARIANCE KERNELS

projection. x = Px.

For each x 6 X define x(t) = [x,F{t, ■)] for t € 0 . Then it holds that

Proof. For each x 6 X write x = xi + x2 where x\ £ X and X2 £ X* Then we have that

[*,r(t,-)] = [xi}r(t,-)] =xiW, Hence, xi(t) = (Px)(t)

= [x,T(t,-)],

= X Q X.

fee.

t e 0 , i.e., x = Px.

P r o p o s i t i o n 10. Let T : © x © -> T(if) fee tfte r.k. for a normal Hilbert B(H)module X. (1) Every closed submodule of X admits a r.k. (2) If X = Xi © X 2 and Tj : 0 x © —> T ( # ) is £/ie r.fc. /or a closed submodule

Xj {j = i, 2), then r = r 1 + r 2 . Proof. (1) Let Y be a closed submodule of X and P : X —> Y be the gramian orthogonal projection, and define r'(£, ■) = PT(t,) for t € 0 . Note that r" : © x © -> T(fl') is a p.d.k. Then, for each y e Y it holds that y(t)=

[y(-),r(tt-)]

= [Py,r(t,-)]=

[y,PT{t,-)}

= [y,V(t,-)},

tee.

Thus V is the r.k. for Y. (2) follows from (1). P r o p o s i t i o n 1 1 . Let T : 0 x 0 —> T ( / / ) 6e a p.d.fc. and X be a r.k. normal Hilbert B{H)-module of V. (1) .Eac/i bounded functional

I on X can be expressed as

£(x) = [x,X(\,

x £ X,

where xe(t) = e(T(t, ■))* for t 6 0 . (2) Each T 6 A(X) is expressible as

(TaO(t)=[*(-),rr(t,-)],

tee,

wftere T T ( i , ■) = T*T{t, ■) /or t e 0 . Proo/. (1) By Proposition 2.6 there is an xe 6 X such that £(x) = [ar,ar^] for x e X . Then we see that for t G 0 Xe(t)=

[xe,r(t,-)}

= [r(t,-),xe]'

=

e(r(t,-))'.

(2) If we define a kernel r T : 0 x © -> T ( H ) by rr(t,aJ=(TT(t,.))(a),

«,t€0,

2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES

37

then we see that for x 6 X and t 6 0

(Tx)(t)=[Tx,r(t,-)} = [i,rr(vl] = [*,rT(t,0]. Now given a p.d.k. r : 0 x 0 —>• T(H), we want to construct a normal Hilbert B(i7)-module Xp admitting T as the r.k. To do this we introduce the notion of "functional completion," which is different from the usual norm completion of preHilbert spaces. Definition 12. Let X 0 be a normal pre-Hilbert B(H)-module consisting of T(H)valued functions on 0 . Then the functional completion of Xo is to obtain a normal Hilbert B(H)-module X by adding T(H)-valued functions on 0 in such a way that the value of a function x e X at t e 0 depends continuously o n i e X and that X0 is a dense submodule of X. In this case the resulting space X itself is called the functional completion of XoT h e o r e m 13. Let T : 0 x 0 —> T(H) be a p.d.k. Hilbert B(H)-module X r admitting T as the r.k. Proof. Let us define a left B(H)-module 0 as follows:

Then there exists a unique normal

X0 consisting of T(ff)-valued functions on

Xo = I ^ a j r ( t i , - ) : oj e B(H), tj e 0 , 1 <j < n,n 6 N I ,

where the module action of B(H) is defined by n

n

a ■ ^2 ajV(tj,•)

= ]T)aa,-r(tj,0,

a e S(H).

If we define a mapping [■, -]o : Xo x Xo —¥ T{H) by

n

for x = XI a j r ( * j ' ' ) j=i

m an

d 2/ = S W£sfc,') 6 -X'o. t Q e n

we see

that [-, -] 0 satisfies all

fc=i

the properties of a gramian except that [x, x]o = 0 implies x = 0 (i.e., x(t) = 0 for f 6 0 ) . We shall show later that [-,-]o has this property and, hence, is actually a gramian on X 0 . P u t ||a:||o = |j[a;,x]o||*

for x

£ Xo-

38

II.

HILBERT MODULES AND COVARIANCE KERNELS

Now we can show that (a) and (b) of Definition 4 hold. In fact, (a) is obvious 71

and (b) is seen as follows: for x = J2 aj?{tj,

■) € %o a n d

t e

©^

nolds

tnat

,=1

[x(-),r(i,-

52 0,1% •), r(t, •)

= E ^'rfe-*) = XWj = i

i=i

Suppose that x 6 X 0 and [x,x] 0 = 0. [x(-),T(i, •)] implies that

Then the reproducing property x(i) =

l|xW|| T <||x|| 0 ||r(t,.)llo = o for every t £ 0 since Lemma 1.2 (4) holds for XQ. That is, x = 0 and hence [-, ]o defines a gramian in XQ- Thus XQ becomes a normal pre-Hilbert 5(H)-module. We are going to obtain a functional completion X? of X0. Let {i n }™ = 1 C XQ be a Cauchy sequence. It follows from the reproducing property that \\x„(t) - xm{t)\\T

= ||[xn-xm,r(t,-)]o||T < ||«„-im||o||r(t,-)||o->0

as n, 77i —> oo for t € 0 . Hence there exists a T(if)-valued function x on 0 such that ||x„(t) — x(t)|| T —> 0 for t 6 0 . Denote by X r the set of all such functions x obtained in this way. It is clear that Xp is a left /3(i/)-module and X0 C Xp. Define for x, y € ^ r [x,j/].Yr = lim [xn,yn]o

(in the trace norm),

n—too

where {x n }^L x and {j/nj^Li Q Xo are the determining sequences of x and y, respec tively. [■, -].Yr is well-defined. For, if {x^l}^°=1 and {y'n}^=1 C X 0 are other such deter mining sequences of x and y, respectively, then | | x J l ( t ) - x n ( i ) | | T , \\y'n(t)-yn(t)\\T —> 0 for f e 6 . Since {x'n - xn}^=1 and {y'n - y„}'^L1 are Cauchy in X0 and the limit functions are zero, we conclude that \\x'n - xn\\0, \\y'n - yn\\0 —i 0. Hence ||K.2/n]o - [a?„,j/n]o||T < | | K - xn,y'n]0\\T < \K

+ \\[xn,y'n -

yn]o\\T

~ S n | | o | | l & H o + | | x n | | o | | j / ; - 2/nHo - > 0.

Thus, [x, y]xr is independent of the choice of determining sequences of x and y, and is well-defined. Now it is easily seen that [■, -}Xr is a gramian on Xr. Moreover, A'0 is dense in Xr- For, let x € Xr be arbitrary. First we note that the equality \\y\\xr = \\y\\o holds for y 6 XQ- If {xn}^Li £ XQ is a determining sequence of x S Xr, then lim \\xn-x\\Xr= n—too

lim n—yoo

lim ||x n - xm\\0 = 0. m—too

2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES

39

Furthermore, to see that X-p is complete, let {x n }£° =1 C X r be a Cauchy sequence. Since X0 is dense in X r , there exists a Cauchy sequence {x'n}'^'=1 Q Xo such that ||x'n — x n | | x r —^ 0. This sequence determines an element x g Xr such that ||x'„(£) — x{t)\\T -> 0 for t € 6 , and \\x'n - x\\Xr -> 0. Therefore ||x„ - x\\Xr -> 0. Finally, we shall verify the uniqueness of X ^ Let X be a normal Hilbert B(H)module with the gramian [■,■] which consists of T(i?)-valued functions on 0 and admits T as a r.k. Since F(t, ■) E X for t g 0 by definition, we have X 0 C X . Moreover, it holds that [z>2/]o = [x,y],

x,y e X0-

(4.4)

Furthermore, XQ is dense in X since {T(t, ■) : t g 0 } is complete in X . Thus we must have Xp = X . For any x,y g X we can choose sequences {xnj^-^^, {yn}'^'-i C Xo which converge to a: and y in X , respectively. Then we have by (4.4) [x,y}=

\im[xn,y„]= n—>oo

lim [x„, yn]0 =

[x,y]xr-

n—foo

Therefore X and Xp are identical normal Hilbert B(i?)-modules. The above proof readily shows the following characterization of functional com pletion for a r.k. space. Corollary 14. Let X0 be a normal pre-Hilbert B(H)-module consisting ofT(H)valued functions on 0 . Then there exists a functional completion X of XQ iff the following conditions hold: (1) For each t g 0 the functional t on Xo defined by t(x) = x(t), x g Xo is bounded; (2) / / {xnj-^Lj C Xo is a Cauchy sequence such that \\xn(t)\\T —> 0 fort g 0 , then \\xn\\0 —> 0, where | | | | 0 is the norm in X0. If a r.k. is restricted to a subset 0 O x 0 O of 0 x 0 , then it is again a r.k. for some normal Hilbert B(H)-modu\e Xo, which is characterized by the following: T h e o r e m 15. Let ©o be a nonempty subset of 0 , Y : 0 x 0 —> T(H) be the r.k. for a normal Hilbert B(H)-module X, and T0 = r|e 0 xe 0 > the restriction of F to ©o x 0 O . Then, To is the r.k. for the normal Hilbert B(H)-module X0 = {x\&0 : x g X } and the equality [x0,x0} = miii{[x,x] : x g X, x | e 0 = x0} holds for x0 g X 0 , i.e., the operator inequality [X0,XQ] < \x,x] holds for x g X such that X\Q0 = XQ, the restriction of x to ©o, and for some x g X the equality holds. Proof. P u t X i = {x g X : x | e 0 = 0}. Then Xi is a closed submodule of X . To see this, let {x„}™=1 C X i be a Cauchy sequence. There exists an x g X such that

40

II.

HILBERT MODULES AND COVARIANCE KERNELS

\\xn ~ x\\x -» 0. By Proposition 7(1) it holds that \\xn(t) - x(t)\\T -> 0 for t E 9 . Since xn\e0 = 0 for n £ N, we have that x|e 0 = 0, i.e., x & X\. Let Ti and r f be r.k.'s for X i and X * (the gramian orthogonal complement of Xi), respectively, which exist by Proposition 10, and P : X -t Xf be the gramian orthogonal projection. Then F = Fi + r f again by Proposition 10. Note that V(t, ■) = rf{t, ■) for t E 9 0 . For x0 £ X0 we define [XQ] = {x e X : z | e 0 =

Note that PX\Q0

x0}.

= XQ, i.e., Px E [XQ] for x E [XQ] because xo = x\e0 = Px\e0 + {x - Px)\e0

=

Px\e0-

Moreover, Px = Py for x,y £ [x0] because Px = Py -^ Px\e0 = Py\e0can define x'0 = Px for x E [xo] independent of x. We also note that [x'0,x'0] < [x,x],

Hence we

x £ [x0]

because [x,x] = [x'0,x'0] + [x - x'0,x - x'Q] > [x'0,x'0]. Define [ZQ.^OIO = [^'o^ol for x0 E Xo- Then, the operator U : (X 0 ,[-,-]o) -> (Xf, [•,•]) defined by t/xo = x'0, xo E Xo gives an isomorphism between these two normal Hilbert B(.H")-modules. Note that (Uxo)(t) = xo{t) for XQ E Xo and t E OoWe now prove that T 0 = F | e 0 x e 0 *s the r.k. for (X 0 , [-, -]o)- Take any XQ E X 0 and t 6 0 o . Then it holds that x0(t)

= (Ux0){t)

= [Cteo, i f («,-)] = [ i o , r 0 ( t , - ) ] 0

since r f (t, •) £ X* and UT0{t, ■) = r f (*, ■) for t € 0 . Thus T 0 is the r.k. The sums and differences of r.k.'s are considered below. T h e o r e m 16. Let Yj : 9 x 9 —> T{H) be a r.k. for a normal Hilbert B{H)-module (Xj, [-, •];, | H | j ) , j = 1,2. Tftera F = r x + r 2 W a r.k. for the normal Hilbert B{H)module X = {x\ + x2 : x\ £ X i , 2:2 6 X2}, where the gramian and the norm are respectively given by [x,x] = min{[a;i,a;i]i + [x 2 ,x 2 ] 2 : a; = Xi + x2, 11 £ Xi, x2 € X 2 } ,

(4.5)

||x|| 2 = min{||x 1 ||2 + || a ;2||i : x = xx + x2, xx £ Xlt x2 £ X 2 } .

(4.6)

Proof. Observe that X = {(x%,x2) ■ Xi € X i , x2 £ X 2 } is a normal Hilbert module, where the module action, the gramian and the norm are defined by a- ( i i , i 2 ) = {a-xua-x2),

[{xi,x2),{yuy2)]x

= [x1,yi}1

+

[x2,y2}2,

B{H)-

2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES

41

ll(zi,x2)|i;r = iKiii + i M i , respectively. That is, X is the direct sum of Xi and X%. Define X0 = Xlf)X2)

X0 =

{{X>~x):xeX0}.

Then it is easily seen that X0 is a closed submodule of X. Define U : X —> X by (f/(xi,x 2 ))(-) = a;i(-) + x2(-)>

(xi,x2)

e X.

Then f/ is jB(#)-linear. Moreover, U\y# is one-to-one and onto, i.e., X = \x' + x" : (x',x") e X^}.

Define for x 6 A" with x = x' + x", (x',x") € X^ [x,x] = [{x',x"),(x',x")]x,

\\x\\ =

\\(x',x")\\x.

Now we prove (4.5). For x = xi+x2 there correspond (x\,x2) € X and (x',x") £ X*, so that a; = xi+x2 = x'+x". Hence, x2—x" = —(xi—x1) and (xi—x1, x2—x") E X0. Moreover, it holds that [xi,Xl]l +

[x2,X2]2

=

[(xi,x2),(x1,x2)]x

= [(x',x"), > =

(x',x")]x

+ [ ( m - x',x2 - x"), (Xl - x',x2 -

x"))x

{{x\x"),{x',x")}x \x,x).

Thus (4.5) is proved. (4.6) is clear from (4.5). Finally we show that V = r x + T 2 is the r.k. for X. Let x € X and t € 8 . Then we have that x(t) = x'{t) + x"(t),

= = =

where x <-> (x',x") €

Xf,

[x'(),r1(t,-)]1+[x"(-),r2(t,-)]2 [(x',x"),(rl(t,-),r2{t,-))]x [(x',x"),(F'(t,-)X'(t,))]x

+ [(x',x"), (Tiff, •) - r'(i, ),r 2 («, •) - r"(t, ■))]x, where T(t,-) =

^ {r'(t,-),T"(t,-))

e X*,

[(x',x"),(T'(t,-),T"(t,-))]x, since (Vi(t, ■) - r'(t, •), r 2 ( t , ■) - r " ( t , ■)) e * 0 and ( x ' , i " ) e * * ,

= [z(.),r(t,-)].

42

II. HILBERT MODULES AND COVARIANCE KERNELS

Definition 17. Let T ( 9 ) be the set of all p.d.k.'s r : 9 x 0 -> T{H). Tx, T 2 e T ( 9 ) define

For

Ti >r,-rier(e). Then ( T ( 0 ) , < ) is a partially ordered set. P r o p o s i t i o n 18. Let Fx, F2 € T(0) and Xx, X2 be the corresponding r.k. normal Hilbert B(H)-modules, respectively. If Fx < F2, then Xx C X 2 and [zi,Xi] 2 < [xi,%i]i for Xi £ Xx. Proof. By assumption T = T 2 - Ti e T ( 9 ) . Let X = Xr be the r.k. normal Hilbert B ( # ) - m o d u l e of F. Since T 2 = Ti + F, we have X 2 - {xx + x : xx € Xx, x £ X) by Theorem 16. Hence ^ C I 2 - Again by Theorem 16 it holds that for x2 6 X 2 [ x 2 , i 2 ] 2 = min{[xi,a:i]i + [ar.ar]: x2 = xx + x, xx € Xx, x € X } . Thus [xi,xx]2 5* [^li^ili f° r ^l £ X i . To consider the converse of Proposition 18 let F € T ( 9 ) be the r.k. for a normal Hilbert S(//)-module A" and X j C X be a normal Hilbert £?(#)-module such that [xi,a;i]i > [ x i , i i ] ,

i i 6 Xi,

(4.7)

where [•, ]i is the gramian in X\. L e m m a 19. With the above notation and assumption which is the r.k. for Xx.

there exists a Fx g T ( 9 )

Proof. By Proposition 6 for each t € 9 there is an at > 0 such that ||a;(t)|| T < o;(||xj| for x e X. Hence ||i 1 (t)|| T < a*II*Hi for xx e X i by (4.7). Thus, again by Proposition 6, X i admits a r.k. Pi e T ( 9 ) . To show that the above obtained Pi satisfies the relation Fx < F we shall construct a normal Hilbert B{H)-modu\e (X 2 , [•, ]2) admitting r 2 = F - Fx as the r.k. Define an operator L : X —> Xx by [xi,x] = [xuLx]i,

xi 6 X i , x € X.

To see that L is well-defined, observe that, for each x 6 X , l{xx) = [xi,x] (xx e Xx) is a bounded functional on X and hence on Xx by (4.7). Thus by Proposition 2.6 there exists a unique Lx € X^ such that l(xx)

= \xx,x\

= \xx,Lx\^

xx e Xx.

2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES

Now L € B(X) x<E X \\Lxf

n B{X,Xi),

Xx C X, with ||L|| < 1 (cf. Definition 2.3) since for

< \\Lx\\\ = WlLx^xhl

= \\[Lxtx]\\r

< \\Lx\\\\x\\ <

\\LxUx\\

and hence ||Lx|| < ||Lx||i < ||x||. We also have that L E A(X) n A(X,XX) Definition 2.3) and L is gramian positive because [La;,a;] = [Lx,Lx\i Moreover, I — L € A(X)

43

> 0,

(cf.

x 6 X.

is also gramian positive because for x € X

[(I — L)x,x]

= [x, x] — [Lx,x] = [x,x] — [Lx, Lx}\ > 0

by ||L|| < 1 and by Proposition 2.4 (4). Thus there exists a gramian positive operator V e A(X) such that I - L = L'2. Here we define X2 = {L'x:xeX},

X'=

{xeX

:L'x = 0},

X" = (X')*

=xex'.

Then we see that L' : X" —> X2 is a gramian unitary operator, where the gramian in X2 is defined by [x2,a;2]2 = [Px,Px]

for a;2 = L'a; with x 6 X".

Here P : X —> X" is the gramian orthogonal projection. Consequently (X", [•, ■]) = (^2,[-,-]2).

We shall show that r 2 = F - Ti is the r.k. for X2. For t e Q

r2(i,■) = r(t,■) - r,(t,■) = (/-£)r(t,•) = L'(L'F(i,■)) e x 2 . Moreover, for t g 0 and x 2 £ -X'2, letting x 6 X be such that L'a; = x2, the reproducing property is seen as follows: x2(t) = [ar a ,r(i,-)] = [ £ ' x , r ( t , - ) ] = [x,LT(t,-)] = [Px,PLT(t,-)] = [L'i,L'LT(t,-)]2

= [x 2 ,r(i,-)-r 1 (t,-)] 2 = h,r 2 (t,-)] 2 . We summarize the discussion into the follwoing. L e m m a 20. Under the assumptions

of Lemma 19 the kernel V\ satisfies Ti < F.

II.

44

HILBERT MODULES AND COVARIANCE KERNELS

From Lemmas 19 and 20 we deduce the following. T h e o r e m 21. Let F € T(O) be the r.k. for a normal Hilbert B(H)-module X. If F = Fi + T 2 , r i , r 2 e T ( 0 ) is a decomposition of F, there is a decomposition I = Li + L*2 of I into two gramian positive operators in A(X) given by (LlX)(t)

= [i(-), T1%-)],

(L2x)(t)

=

[x{-),F2(t,-)]

for ( 6 6 and x 6 X. Conversely, if I ~ Li + L% is a decomposition of 1 into two gramian operators in A(X), then there correspond p.d.k. 's Ti, F 2 G T ( 0 ) such that

Fj(t,-) = L3F{t,-){tee,j

= i,2)

and

positive

r = r i + r2.

As seen from Theorem 13, every F € T ( 6 ) admits a r.k. normal Hilbert / ? ( / / ) module Xp. According to Corollary 3.7 there exists a Hilbert space K such that Xr = H ® K. How can this Hilbert space K be realized? Can it be obtained as a RKHS? To answer this question we begin with the following definition. Definition 22. Let F 6 T(6) and S) be a Hilbert space consisting of //-valued functions on ©. Then Sj is said to be a reproducing kernel Hilbert space (RKHS) of F if the following conditions are satisfied: (a) L(-, t)(f> 6 f> for each t € 6 and e H, (b) (p{t),4>)H = (p(-),r(',«)^) i 5 for each t e 0 ,

h)J ■ tj 6 0 , 4>j 6 H, 1 < j < n, n € N | . Then f>0 becomes a pre-Hilbert space if we define an inner product (■, ) 0 by (P.9)o = X !

(r(sk,tj)4>j,4'k)H

j,k

n

for p(-) = J ] r (->*i)^j j=i

m and

"(■) = E r(-,sfc)Vfc € £o- That (p,p) 0 = 0 implies fc=i

p = 0 can be seen after checking that F)0 satisfies (a) and (b) of Definition 22. Thus we can obtain the RKHS 9)r by the functional completion of Sjo-

2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES

45

Note that Proposition 23 also holds for B(H)-valued p.d.k.'s. Now the RKHS S)r obtained above will play the role of K of Corollary 3.7. T h e o r e m 24. Let T £ T ( 6 ) , and Xr and Sjr be the normal Hilbert and the Hilbert space which admit T as a r.k., respectively. Xr^

B(H)-module

Then it holds that

H® Sir-

Proof. Let {pa}a£i

be a CONS in f j r . Then we claim that r(s,t)

= Y^Pa(s)®P~Jt),

s,teQ.

(4.8)

In fact, for <j> £ H and t 6 0 we have that T{-,t)4> 6 fjr by Definition 22 (a) and

r(-,i)tf = £(r(-,t)*.p*(0) ar P«(-) aex = Y^ (*,Pa(*)) H Pa(-).

by Definition 22(b),

= X](pa(-)®M*)) £ H define a T(.ff )-valued function 4> ® P on 6 by

(0®p)(-) = ^®pTTThen we claim that, for 4>,ip £ H and p, g G f>r, [0®P,V , ®g] = ( 9 1 p ) $ r ( 0 ® ^ ) . where [•, •] is the gramian in Xroo

(4-9)

In fact, since p and q can be written as p(-) =

oo

53 ^{-,tj)4>j and g(-) = 53 r(-,sfcfc)i/'fc Er(-,tj)0,-and «,(■)= Er(-,s )v for some sequences {tj}^lt

jJ == li

{skj^Li

£ @

fc=i fc=l

{ ^ } ~ x , {fc}~=i C i / , we see that

J2 {® ^)r(i„ ■)> £ > ® ^)r( ajb ,•

[0 ® p, V> ® q]

i ^

* ( ® ^ ) [r(tj, •), r(s f c , •)] (V'fc ® ? ) .

^(^®?j)r(t,-,*fc)(v»fc®?)

easily justified,

and

46

II. HILBERT MODULES AND COVARIANCE KERNELS

= ^

(<£ ® F(s fc , t,)^-)(V'fc ® V')

j,k

= ^(T/Jfc,r(Sfc,i,)^)H(<^®^) },k

= {q,p)f,r(®4>)This implies that (<j> ® p)(-) e Ay for $ 6 # and p € fjrNow take €// of norm 1. Then {0 ® p Q }aeJ forms a gramian basis of XrIn fact, by (4.9) we see that [

® p/j] = 0 for a / /? and [ ® pQ, ® Pa] = 4>® 4> = [4>®Pa,4>® Pa}2 for a € l Moreover, we have that by (4.8)

J2(®pa){sy(®pa)(t) = ]C p «( s )®p°W = r(s,i) for s , ( 6 0 . Thus Proposition 8 implies that {<j> ® pa}aex Therefore Xr £* H ® Sjr.

is

a

gramian basis of Xp-

2.5. Harmonic analysis for normal Hilbert B(.rY)-modules In this section we consider a T(i/)-valued continuous positive definite function denned on a topological group and its relation to a gramian unitary representation on some normal Hilbert B(H)-module. If the group is LCA (= locally compact abelian), we can prove a Stone type and a Bochner type theorem for such a positive definite function. Fourier transform is defined for a normal Hilbert B(i7)-module valued function on an LCA group, and a Plancherel type theorem and an inversion formula are established. Definition 1. Let G be a topological group and X be a normal Hilbert B(H)module. A T(ff)-valued function r on G is said to be positive definite (p-d.) if YJajr{t]tk1)al ^ ° f o r a n y n e N , tti,... ,o„ € B(H) and t z , . . . , tn e G. F : G ->• i,k T(H) is said to be weakly continuous if tr(aF(-)) is continuous for a € B(H). A gramian unitary representaion (g.u.r.) of G on X is a mapping [/(■) from G into A(X) for which U(s) is gramian unitary for every s e G and satisfies that U(e) = I and U(st) = U{s)U(t) for s,t 6 G, where e is the identity of G. A g.u.r. [/(•) of G on X is said to be weakly continuous if (f/(-)ar,y) is continuous for x,y E X. A vector i 0 6 I is said to be cyclic for a g.u.r. [/(■) if <5{U(s)xQ : s e G] = X, i.e., the closed submodule generated by the set {U(s)xQ : s € G) coincides with the whole space X.

2.5. HARMONIC ANALYSIS FOR NORMAL HILBERT B(H)-MODULES

If r : G -> T{H), we put F(s,t) = Fist'1) for s,t € G. r : G x G —> T(H) is a p.d.k. in the sense of Definition 4.1.

47

Then F is p.d. iff

P r o p o s i t i o n 2. Let G be a topological group and F : G —> T(H) be p.d. Then there exist a normal Hilbert B(H)-module Xp with the gramian [•,], a g.u.r. [/(■) of G on Xp and a cyclic vector xo G Xp such that F(s) = [U(s)xo,x0],

s E G.

It also holds that for s,t £ G

r( 5 - 1 ) = r( s )*,

||r( s )|| T < ||r(e)||T,

||r( s )-r(i)||?<2||r( e )^r( s r 1 )|| T ||r(e)|| T . Furthermore,

F is weakly continuous iff so is £/(•)•

Proof. P u t F(s,t) = r ( s i _ 1 ) for s,t 6 G and let X r be the r.k. normal Hilbert B(H)-mod\ile of F. Then we have that r ( a ) = f ( a , e ) = [f(s, •), f (e, •)], Let Xo be the set of all T{H)-valued

s G G.

(5.1)

functions on G of the form

n

Yl ajF(sj, ■),

a.j e B(H),

Sj e G, 1 < j < n, n e N.

Then Xo is a dense submodule of Xp- For each s e G define an operator U(s) on X0by n

U(s) ] T ajF{s},-)

n

= Y2 ajFisSj, •)■

3=1

i=i

It is easy to see that U{s) : X0 -> X 0 is onto and gramian preserving, indeed, [U(s)x,U(s)y] = [x,7/] for x, y € Xo- Hence f/(s) can be uniquely extended to a gramian unitary operator on X r , still denoted U(s). It is easily verified that U{st) = U{s)U{t) and (/(e) = / for s, t e G. Thus (/(■) is a g.u.r. of G on X r . If we put x 0 = f (e, ■) G X r , then we see from (5.1) that F(s) = [U{s)x0,x0],

seG.

The equality r ( s - x ) = F{s)* for s £ G follows from r ( s ~ 1 ) = f (e,s) = F{s,e}* = F{s)*. The inequality ||r(s)|| T < ||r(e)|| T for s G G results from | | r ( S ) | | T = ||f( S ,e)|| T = |lff(. S ,-),r(e,-)]|| T

48

II.

HILBERT MODULES AND COVARIANCE KERNELS

<||f( S ,-)IUI|f(e,-)IU = ||[f( s ,-),f(a,-)]||J||[f( e ,-),r( e ,-)]||J = ||f( S , S )||?||f(e,e)||?

= ||r(e)||?||r(e)||? = ||r(e)||T, where ||-||x is the norm in Xp- Now the last inequality follows from

||r(«)-r(t)||; = ||f(« 1 e)-f(t 1 e)l|; =

\\[f(s,-)-t(t,-),r(e,-)}\\l

<\\t(s,-)-f(t,-)\\2x\\r(e,-)\\2x = || [f (a, ■) - t(t, •), f (s, •) - f (t, ■)] ||T || [f (e, •), f (e, ■)] ||T = | | f ( S ) S ) - f ( S , t ) - f ( t , S ) + f(t,t)|| T ||f(e, e )|| T < (||r(e) - r(«t- x )l| r + ||r(e) -

T(ts-l)\\T)\\r(e)\\T

= 2||r( e )-r( S f- 1 )|| T ||r(e)|| T . The statement about the weak continuity is a consequence of the following twosided implications: U(-) is weakly continuous •£=>■ ({/(■)(a ■ x),y)x ti(a[U{-)x,x\) tr(a[U(-)xo,xo\)

is continuous for x,y £ Xr and a 6 B(H) is continuous for x € Xp and a G B(H) is continuous for a € B(H),

since x 0 is a cyclic vector, tr(ar(-)) is continuous for a € S ( H ) r is weakly continuous, where (-, ) x is the inner product in Xp. Definition 3 . Let X be a normal Hilbert S(H)-module and G be an LCA group with the dual group G. Denote by 2$g the Borel cr-algebra of G. A mapping P(-) : 23g -f A ( X ) is called a gramian spectral measure on G if P is gramian orthogonal projection valued such that, for x,y 6 X , (P(-)a:,y) is a C-valued measure on 93g. A gramian spectral measure P is said to be regular if, for x, y e X , the C-valued measure (P(-)x,y) is regular. Now a Stone type theorem, an extension of Stone's theorem for a g.u.r. of an LCA group, is formulated as follows

2.5. HARMONIC ANALYSIS FOR NORMAL HILBERT B(H)-MODULES

49

P r o p o s i t i o n 4. If U(-) is a weakly continuous g.u.r. of an LCA group G on a normal Hilbert B(H)-module X, then there exists a regular gramian spectral measure P(-) on G such that (U{s)x, y)

= / (s,x)

{P{dx)x, y)x,

seG,x,yeX,

JG

where (•, ■) is the duality pairing of G and G. Proof. Since U(-) is a weakly continuous ordinary unitary representation of G on the Hilbert space X, it follows from the classical Stone theorem that there exists a regular spectral measure P(-) on G such that (U(s)x,y)x=

l(s,x)(P(dx)x,y)x,

x,yeX.

J G

We have to show that P is gramian orthogonal projection valued. Let x, y € X be fixed. It holds that for a e B{H)

[(s,X){P(dx)(a-x),y)x={U(s)(a-x),y)x

=

(a-{U(s)X),y)x

JG

= {U(s)x,a*-y)x=

[

(s,x){P(dx)x,a*y)x

JG

= f

(S,x)(a(P(dX)x),y)x.

JG

It follows from the uniqueness of the Fourier transform of measures that (P(-)(a ■ x),y)x

= (a ■ (P(-)x),y)x,

ae

B(H).

Since this holds for x, y 6 X we conclude that P ( ) is B(H )-linear and hence gramian orthogonal projection valued. If G = R, then G = R and the above expression takes the following familiar form: (U(s)x,y)x

= f etsu{P(du)x,y)x,

s e R, x,y € X.

JR

A T(if)-valued c.a. measure F is said to be weakly regular if tr(aP(-)) is a Un valued regular measure for a e B(H). Then we prove a Bochner type theorem as follows.

50

II.

HILBERT MODULES AND COVAR1ANCE KERNELS

P r o p o s i t i o n 5. For a T(H)-valued weakly continuous p.d. function F on an LCA group G there exists a unique T+ (H)-valued weakly regular countably additive mea sure F on G such that F(s)=

f (s,x)F(dx),

seG.

JG

Proof. It follows from Proposition 2 that there exist a normal Hilbert B(H)-module Xr, a weakly continuous g.u.r. [/(■) of G on Xr, and a cyclic vector x0 € Xr such that F(s) = [U{s)xo,x0], s e G. By Proposition 4 there is a regular gramian spectral measure P on G such that U(s)=

f (s,X)P(dx),

seG.

JG

Thus, putting F(-) = [ P ( - ) X O , I Q ] , we see that F is a T + (if)-valued c.a. measure by the Orlicz-Pettis theorem since F is weakly c.a. (cf. Diestel and Uhl [1, p. 22]), i.e., tr(aF(-)) = [P(a ■ x0),x0)x is c.a. for a 6 B(H), and hence c.a. in the trace norm. Moreover, F is clearly regular. Now it holds that for s € G F(s) =

[U(s)x0,x0]

/

(s,x)P{dx)xo,x0

JG

= f (s,X)[P(dX)x0,x0} JG

= f

{s,X)F(dX).

JG

Again in the case that G = E , the above result has the following familiar form: F(s) = f e%suF{du),

s 6 R.

JR

Compare with (1.1.1) and (1.3.1). Now let G be an LCA group with the dual G. Q and g denote Haar measures of G and G, respectively, so that the Plancherel's theorem holds. That is, the Fourier transform T : L2(G, g) -> L2(G, Q) is a unitary operator, where T is defined by

(^/)(X)= [ (t,x)f(t)s(dt), JG

feL2(G,g)

and j | / | | 2 = H-77II2 for / G L2(G, g). We may write L 2 (G) = L2(G, g) and L2(G) L2(G, g) if g and g are fixed.

=

2.5. HARMONIC ANALYSIS FOR NORMAL HILBERT B(tf)-MODULES

51

Let X be a normal Hilbert 5(H)-module and L2(G;X) be the Hilbert space of all X-valued strongly measurable functions $ on G such that

I l * l l 2 = ( j f ||*(OIL*0(
[*(*),*(*)] g{dt),

$,*eL2(G;I).

./G

Similarly, we can consider the space L2(G; X), which is also a normal Hilbert B(H)module. Then, the Fourier transform T : L2(G;X) -4 L2(G;X) is defined by

(^>)(x) = [ (t,X)*(t)Q{dt),

$£L2(G;I).

./G

Finally, we have the following Plancherel type theorem. P r o p o s i t i o n 6. Let G be an LCA group and X a normal Hilbert B(H)-module. Then, the Fourier transform T : L2(G;X) —> L2(G;X) is a gramian unitary oper ator, and hence L2{G ;X) = L2{G \X). Proof. Observe that L2{G ;X) ^ L2(G)® X and L2(G;X) ^ L2{G)®X for elementary tensors / ® x,g ®y e L2(G) ® X we have that [.F(/ ® i ) , .F(s ® j/)] 2 = [(.F/) ® x, {Tg) ®y}2 = =

hold. Now

{Tf,Tg)2\x,y]

{f,g)2[x,y}={f<2>x,g®y}2

by Plancherel's theorem. Thus, T is gramian preserving on the algebraic tensor product L2(G)QX and can be extended to a gramian unitary operator on L2(G)®X 2 onto L (G) ®X. It follows from tha above proposition that the Fourier inverse transform T~x L (G ;X) —> L2(G ; X) is a gramian unitary operator given by 2

(^- 1 *)(t)= / (tTx)*(x)e(dx), JG

teG,*eL2(G;X).

Thus we have the following inversion formula:

*(*)= [Wx){F*)(x)e{dx), JG

teG,^eL2(G;X).

:

52

II. HILBERT MODULES AND COVARIANCE KERNELS

Bibliographical n o t e s For a general theory of inner product modules we refer to Istra^escu [1, Chapter 12](1987) and Lance [1](1995). Inner product modules over an operator algebra (C*-algebra and ylW-algebra) with values in the same algebra was first considered by Kaplansky [1](1953). Later on Saworotnow [1](1968) studied Hilbert modules over an # "-algebra, and Paschke [1](1973) and Rieffel [1](1974) treated modules over C*-algebras. For more infor mation relevant to this chapter we refer to Ambrose [1](1945), Giellis [l](1972), Kakihara [4](1983), Saworotnow [5](1976) and Smith [1](1974). 2.1. Normal Hilbert B(H)-modules. A (normal) Hilbert 5(if)-module was intro duced by Kakihara and Terasaki [l](1979) to treat Hilbert space valued stochastic processes. As was seen in Section 1.3, a normal Hilbert 5(.ff)-module is a natural abstraction of Lg(fi; H). Lemma 1.2 is esssentially due to Kaplansky [1] and Pashke [1]. Proposition 1.5 is due to Ozawa [1](1980). 2.2. Submodules, operators and functionals. Lemma 2.2 was noted by Kakihara and Terasaki [1]. Proposition 2.4 is the collection of results in Kakihara [l](1980), Ozawa [1] and Paschke [1]. Proposition 2.6 was proved in Kakihara and Terasaki [!]■ 2.3. Characterization and structure. The characterization theorem (Theorem 3.1) is in Kakihara [1]. Theorem 3.4, the structure theorem (Theorem 3.6) and Corollary 3.7 are due to Ozawa [1]. Kakihara [17](1994) gave an alternative proof to Theorem 3.6 using Corollary 3.5. Corollaries 3.8 and 3.10 are also given by Ozawa [1]. Proposition 3.11 is in Kakihara [9](1986). 2-4- Positive definite kernels and reproducing kernel spaces. The original idea of RKHS goes back to Moore [1](1916). The entire idea of Section 2.4 is taken from Aronszajn [1](1950); see also Burbea and Masani [1](1984), and Pedric [1](1957). Definition 4.1 through Theorem 4.21 are formulated in Kakihara [10](1987). Propo sition 4.23 is in Kakihara [ll](1988) and Theorem 4.24 is in Kakihara [17]. Some related topics can be seen in Miamee and Salehi [1](1977) and Umegaki [1](1955). 2.5. Harmonic analysis for normal Hilbert B(H)-modules. Propositions 5.2, 5.4 and 5.5 were formulated in a more general setting in Kakihara [4]. Proposition 5.6 is noted by Kallianpur and Mandrekar [2](1971) in the particular case G = Z and B and X = S{K,H). We also refer to Saworotnow [2](1970) and [3](1971).

CHAPTER III

STOCHASTIC MEASURES OPERATOR VALUED

AND

BIMEASURES

The contents of this chapter is fundamental and extensively used in the follow ing chapters since we consider those processes which are representable as integrals w.r.t. X = LQ(Q ; #)-valued measures and their properties are totally determined by representing measures. Thus, in this chapter, we first discuss some technical proper ties involving semivariations and variations of X-valued measures and T(H)-vaiued bimeasures. These variations and semivariations are essential for integral represen tations of various processes and their classifications since the latter depends on the behavior of these different variations and semivariations. Orthogonally scattered dilation and gramian orthogonally scattered dilation are then considered, which are then used to obtain stationary dilations of harmonizable processes. For T(//)-valued measures and bimeasures we can construct L 1 -spaces and L 2 -spaces. Some prop erties of these spaces are studied. When the measurable space is locally compact, we can consider Riesz type representation theorems for operators on (vector val ued) continuous functions. Finally, convergence and approximation properties are obtained for X-valued measures.

3.1.

Semivarations and variations

Let X = Ll(Q;H) be as in Chapter I. That is, the Hilbert space of all Hvalued strong random variables x(-) on the probability space (f2, J, fi) such that E{x) = Jn X(UJ) ii(dui) = 0 and Jn \\X(LU)\\2H fi(dio) < oo, where H is a complex Hilbert space with the inner product (•,■)// a n d the norm j|-||#. Throughout this chapter H is assumed to be separable. Recall that X is a normal Hilbert S(i?)-module with the T(H)-valued gramian [x,y]=

/ x{ui) ® y(u) n(dw), in 53

54

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

the inner product {x,y)x = tv[x,y] and the norm \\z\\x = [x,x)x = \\[x, z]\\* for x,y e X (cf. Section 2.1). In this section we consider various semivariations and variations of X-valued mea sures and T(H)-valued bimeasures. ^"-valued measures are sometimes referred to as stochastic measures. Interrelations among them are obtained mainly as inequalities. Several examples are presented to distinguish them. Let ( 0 , 21) be a measurable space where 0 is a set and 21 is a cr-algebra of subsets of 0 . We begin with general definitions of Banach space valued measure theory. The reader should refer to Diestel and Uhl [1], Dinculeanu [1] and Dunford and Schwartz [1, IV]. Definition 1. Let X be a Banach space with the norm | | - | | j . ca(2t,3t) denotes the set of all ^-valued c.a. (in the norm) measures on (0,21). For each A 6 21, 11(A) denotes the set of all finite measurable partitions of A. Let £ £ ca(2l, X). The semivariation \\£\\(A) of £ at A 6 2t is defined by

ueii(A)=Sup

Ea^A'

■■aAeC,

|QA| < l , A G w e n ( A n .

A6vr

The variation |f |(A) of £ at A 6 21 is denned by

|e|(A) = suP

Ell^ A )Hx^en(^)

Denote by -uca(2l,3£) the set of all £ 6 ca(2t,3E) of bounded variation, i.e., |£|(0) < oo. R e m a r k 2. Let X be a Banach space and X* be its dual. (1) Let £ e ca(%X). Then ||£||(-) is monotone, i.e., A C B implies ||?||(A) < CO

°°

HfH(B) and countably subadditive, i.e., ||£||( U i „ ) < E "-

1

UClPn) for {An}™=1 C

n=l

21, while |f|(-) is c.a. iff £ e nca(2t, 3£). For A e 21 the following relations hold: U{A)\\x

< U\\(A) < |£|(A),

sup ||£(A n B)ll* < UW(A) < 4 sup U(A n 5 ) | | 2 , sea Bea IKH(A) = sup {|z'£|(A) : * * € £ * , ||x*|U- < 1 } , where (i*£)(-) = a* (£(•)) e ca(2t,C) and |af£|(-) is its variation, | | - | | j . being the norm in X* (cf. Diestel and Uhl [1, pp. 2-5]). Since £ is denned on a cr-algebra, it is bounded, i.e., sup ||£(A)|| 2 < oo by the Nikodym Uniform Boundedness Theorem A£
recalled below. Hence £ is of bounded semivariation,

i.e., ||£||(0) < oo.

3.1. SEMIVARATIONS AND VARIATIONS

55

(2) For £ 6 ca(2t,j£) there exists A € ca(2l,R + ), called a control measure of £, such that lim ||£||(A) = 0, where R+ = 0,oo). Hence, ||£||(A n ) ->• 0 if An | 0. A(A)->0

A can be chosen so that X(A) < ||£||(A) for A g 21 (cf. Dunford and Schwartz [1, IV.10.5]). (3) (co(2t, 3£), | | | | ( 0 ) ) and (uca(a,3t), |-|(6)) are Banach spaces and the latter is a subspace of the former. The proof is similar to that of Proposition 7 below. We list three classical theorems in vector measure theory in the following since they will be used frequently hereafter. O r l i c z - P e t t i s T h e o r e m (cf. Diestel and Uhl [1, p.22]). / / £ : 21 -> X is weakly c.a., i.e., x*£ € ca(2l, C) for every x* g X*, then £ g ca(2l,X). V i t a l i - H a h n - S a k s - N i k o d y m T h e o r e m (cf. Diestel and Uhl [1, p. 23-25]). Let {^n}^Li C ca(2l,X) fee a sequence such that lim £B(-A) = £(A) exists m norm for n—foo

euen/ J4 6 21, tften £ 6 ca(2l, X) and the countable additivity of {^n}^Li is uniform, i.e., Ak 4,0 implies that lim (,n{Ak) = 0 uniformly in n. k—► oo

N i k o d y m Uniform B o u n d e d n e s s T h e o r e m (cf. Diestel and Uhl [1, p. 14]). Let {£,a}aex Q ca(%L,X) be a family such that sup||£a(4)||*
A 6 a,

i/ierc it follows that sup ||Ca ||(9)
equivalently, s u p { sup ||f 0 (A}||i} < o o .

Now let us return to our case and consider X-valued measures on (0,21). Definition 3. An X-valued measure £ g ca(2l, X ) is said to be orthogonally scattered (o.s.) if (£(A),£(B))X = 0 for every disjoint A,B e l £ is said to be gramian orthogonally scattered (g.o.s.) if [f(A),£(2?)] = 0 for every disjoint A, B g a . Put c a o s ( a , X ) = {£ g c a { a , X ) : £ is o.s.}, c a 3 o s ( a , X ) = {£ g c a ( a t X ) : £ is g.o.s.}. Clearly, cagos(^.,X)

C c a o s ( a , X ) , a proper set inclusion (cf. Example 24).

56

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

For £ e cagos(%,X)

define F^ e co(2l, T+(H)) by Fc(A)=[€(A),€(ii)],

Ae 3,

where T + ( H ) = {a e T ( # ) : a > 0}. Note that c a ( 2 l , T + ( # ) ) = w c a ( 2 l , r + ( / f ) ) . According to Theorem II.3.6 and Corollary II.3.7 it follows that X = L20(Q;H)

* H®L20(tl)

2 S(L20(U),H).

S(H, L Q ( O ) ) and X are isomorphic as Hilbert spaces, in symbols S{H,LQ(CI))

(1.1) ~ X,

by the identification x G X with T^, where ^

= {<J),X)H,

4> eH.

(1.2)

Hence, if we define V : X ->■ S ( # , L § ( Q ) ) by (7x = T£ for a; G X, then [/ gives the isomorphism in (1.1) (cf. (1.3.4), (1.3.5) and p. 12). We shall frequently use this identification later. The semivariation and the variation for X-valued measures are defined as in Definition 1. We introduce several other notions in the following. Definition 4. Let £ E ca(%X). (1) The operator semivariation ||£|| 0 (A) of £ at A e 2t is defined by

ll£llo(A)=sup

£ «AC(A)

aA e £(#}, ||aA|| < 1, A e n e U{A) } .

A£TT

6 c a ( a , X ) denotes the set of all £ € ca(2l, X) of bounded operator semivariation, i.e., ||€|U(e) < 00. (2) The strong semivariation ||£|| S (A) of £ at A £ 21 is defined by

na(A) = suP.

Yl Te(A)*A

feefl, II0AIIH
AGTT

where ||-|| 2 is the norm in L§(fi) (cf. (1.2)). (3) The weak semivariation \\£\\W(A) of £ at yl e a is defined by

||ClU(A)=sup{||^||(A) : 0GH,||^||„ e H, £*(•) = (£(-),4>)H = % ) ? € ca(a,Lg(fi)) and | | ^ | | ( A ) is the semivariation of £4, at A.

3.1. SEMIVARATIONS AND VARIATIONS

57

(4) The weak variation \£.\W{A) of £ at A G 21 is defined by |fU(A) = sup {|&|(A) : 4, e H, | H | „ < l } , where | ^ | ( A ) is the variation of ^ at A. If X is a Banach space and is also a left 5(.ff)-module, then for £ € ca(2l,£) we can define the operator semivariation ||£||0(") as in Definition 4 (1). If, in particular, F G ca(2t, T(H)), then we see that for A G 21 |F|| 0 (A) = |F|(A) = sup

(1.3) A6TT

where the supremum is taken over 6A G B ( # ) with ||6A|| < 1 for A 6 TT e 11(A). This can be seen from the proof of Theorem 5 (2) below. T h e o r e m 5. (1) Let £ G ca(2t,X) and A G 21. Tfeen ii holds that \\aA)\\x<sup{U(AnB)\\x:Be%}

< U\U) =

sup{\(a-),x)x\(A):xeX,\\x\\x
<4sup{||e(An5)||x:B62t}, where !(£(•)> ^ ) A : | ( A ) is the variation of the C-valued measure (£(■),x)x at A G 21. (2) / / £ G 6ca(2l, X), then for any x G X the T(H)-valued measure ( o i defined by ( £ o x ) ( - ) = [£(•),x] is of bounded variation, i.e., £ o x 6 vca(%.,T(H)), and it holds that ||e|| 0 (A) = s u p { | £ o x | ( A ) : i a , ||x|U < 1}. (1.4) (3) / / ( £ ca#os(2l,X), then ||£|| 0 (A) = U(A)\\X for every A G 21. ca£os(2t,X) C 6 c a ( 2 l , X ) . (4) / / £ G caos(2l,X), then ||f||(A) = ||£(A)|| X for every A G 21. (5) Let £ G ca(2t, X) and A e 21. Then it holds that llelU(A) < ||e||(A) < UUA)

< UWM)

tfence,

< KRA).

Proof. (1) is clear from Definition 1 and Remark 2 (1). (2) Let £ G 6ca(2t, X), x £ X and A G 21. Let TT G Il(.4) be fixed. For each A e i r choose a partial isometry a& G B{H) such that j | ( £ o x ) ( A ) | | T = | | [ £ ( A ) , x ] | | T = tr(a A [£(A),x]).

58

III. S T O C H A S T I C M E A S U R E S A N D O P E R A T O R VALUED B I M E A S U R E S

Then we see that E

||(S° X ) ( A ) | | T = Y, tr(a A [£(A),x]) = t r ( [ £

Agir AG

A€TT

a A £(A),:r )

^LA67I.

E a^^A)

\\x\\x
A6TT

Thus it holds that |J o i | ( A ) < ||f||o(^)||x||x < oo, which implies that £ox £ vca(%T(H}). Let a be the RHS (= right hand side) of (1.4). Then the above computation shows that ||f ||0(.A) > a. To see the opposite inequality, let n e U(A) and a A e B{H) be such that [|aA|| < 1 for A e it. Then we have by Lemma II.1.2 (5) that

E a^w

EaA^(A),x

sup

AgTT

: x g X, \\x\\x < 1

A£TT

s u p { E l | a A | | | | K ( A ) , x ] | | T : i 6 X , ||x

< n

< a.

A£7T

This implies that ||£||0(i4.) < a. Therefore the equality (1.4) holds. (3) Let A e 21, it e U(A) and a A e B(H) be such that ||a A || < 1 for A e it. Then we have that EaA?(A)

E°A£(A),

A6TT

Aevr

E<*A<e(A') A'Gir

E«A[aA),aA)]aA

,

since £ is g.o.s.,

A€TT

= E IM£(AU(A)]aA||r AgTT

< ^ | | a A ) | | 2 Y = ||a^)Hx>

since £ is g.o.s.

A6TT

This implies that ||£||0(-<4) < ||£(^)||x- Since the opposite inequality is shown in (1), we have the equality ||£|| 0 (A) = ||£(A)||x(4) is proved in a similar manner as in the proof of (3). (5) | | £ | U M < U\\{A): Observe that He|U(A)=8up{||^||(A):|H|H
by Remark 2 ( 1 ) ,

59

3.1. SEMIVARATIONS AND VARIATIONS

=

™p{m-)J4>)x\(A):\\4>\\H
< sup

{!(?(•),x)x\(A):\\x\\x
= U\\(A),

2<1}

by(l).

llfll(A) < UWM)- Observe that

iieiuc^) = sup

r

E

«(A)A

AGTT

sup

E

.e(A))H

AgTT

sup E ( £ ( A ) , / ^ A ) :

sup A£7T

A6T

(1.5)

= SUPE|(^(A),70A); AgTT

where the first two suprema are taken over ||<£A||H < 1 for A 6 IT € n ( A ) and the last three ones over ||/|| 2 < 1, \\4>A\\ < 1 for A e 7r e 11(A), and the last equality holds since, for each A e TT, we can choose an Q A 6 C with \a&\ = 1 such that | ( € ( A ) , / ^ A ) X | = O A ( ^ ( A ) , / < ^ A ) X and replace A by 5AA- By (1) we see that

IKH(A) = s u p { E

| ( e ( A ) , a ; ) x | : | | i | | x < 1, TT € 11(A)).

(1.6)

Comparing (1.5) and (1.6) gives the conclusion. U\\s(A) < U\\o(A): Let TT e 11(A) and {^A : A 6 TT} C H with | | 0 A | | H < 1 be given. Choose A 0 e 7r such that ||A0IIH = niax{||0A||/f : A G 7r} > 0 and put Choose, for each A 6 TT, OA 6 B ( / / ) so that a&c, I1
E% A )0A| ASvr

AA and ||a A || < 1.

= EK^A))^ = (^E^A)) A6TT

AGvr

| 7 £ a ^ ( A ) ^ | | 2 < ||T S a ^(A)|| < | | ? E 0 ^ ( A ) L E

«A£(A)

< ||flU(A),

A£TT

where II-l^ is the Hilbert-Schmidt norm. This is enough to show the conclusion. The last inequality is clear from the definition. Recall that X = S ( I § ( 0 ) , # ) C B(Lg(Q),if). Then we see that the weak semivariation of £ e ca(2l, X) is the ordinary semivariation of £ considered as an element of ca(2l, B(L%(Q),H)). In general, the following assertion is true:

60

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

L e m m a 6. Let K be a Hilbert space and £ € ca(2l, B(K, H)). Then it holds that for every A 6 21 |K||(A) = s u p { | | a - M I ( A ) : V € # , I W k < l } = sup {||f(OVIKA) : 4> e H, U\\H < 1} < oo, iwftere ||f (-)V'II(A) and \\£(-)*<j>\\(A) are the semivariations £,{■)* 6 ca{%,K), respectively.

of i{-)4> 6 ca{%H)

and

Proof. Observe that

[ieii(A) = su P -

£

Q

A£(A)

| a A | < 1 , A eir e H ( A )

A£TT

sup A£TT

I«AI < I , A e TT e n(A), | H | H < 1, ||^|| K < 1

sup

sup ■

|H
EQA(Wi)H

: | Q A | < 1, A GTT e n ( A )

A£7T

= sup{||(£(-)<M) H ||(yl) : I^IIH < 1, HVlk < 1}so we get s u p { s u p { | | ( £ ( - ) r M ) f l | | ( A ) : U\\g < 1} : U\\K < 1} snp{\\a-)MA):\W\K
sup{ne(r^ii(A):

< 1} : | | 0 | | H < 1}

H0||H
Note that c a ( 2 l , * ) , uco(2l,X) and bca{%X) are left £ ( # )-modules. ca(2l,X) is a Banach space with the total semivariation norm, while vca(%,X) is a Banach space with the total variation norm. Furthermore, we have: P r o p o s i t i o n 7. {bca(%,X),

(ca(a,x),|HI(e)).

|| • ||o(9)) w « Banach space and is a submodule of

61

3.1. SEMIVARATIONS AND VARIATIONS

Proof. Clearly the total operator semivariation ||-|| o (0) is a norm in bca(^.,X) and 6ca(2l, X) is a normed linear space. Since all other statements are obvious, we only need to show the completeness of 6ca(2l, X). Let {£n}£Li C 6ca(2l, X) be a Cauchy sequence, i.e., ||£„ — £ w ||o(©) - > 0 a s n , m - > o o . Then for any A e 21

\\UA) - U{A)\\X < ||& - U\o(A) < Un - UU&) -> 0 as n,m —> oo, so that there exists the limit lim £„(A) = £.{A), say. Hence by the n—foo

Vitali-Hahn-Saks-Nikodym Theorem we have £ e ca(2l,X). We now assert that ||£||o(e) < oo and ||£„ - i\\0{e) -* 0 as n -+ oo. Let 7T = { A x , . . . , ,4fc} e n ( 0 ) and tjj € B(H)

be such that ||OJ|| < 1 for 1 < j <

k. Then for any e > 0 we can choose an n 0 = n0(e,ir) € N such that \\Zn{Aj) - Z{Aj)\\x

< -,

1 < j < k,

n>n0.

Hence we have that for n > n0

Yl ai^Ai

x

j

<

j

£Mi||tf-W(^)L + llUo(e)

<e+[ie»iue). This implies that ||£|| o (0) < lim ||£n|U(Q) < oo, where we note that the sequence n—►oo

{ | | f n | | . ( e ) } ~ ! is convergent since |||£ n ||°(©) - ||£m|UC©)| < ||£„ - U l U © ) -> 0 as n, m —> oo. Moreover, we have 0=

lim lim | K „ - U | | 0 ( © ) =

Tl—+00 m—►oo

lim ||£„ - £||„(e).

Corollary 8. Let {£n}%Li Q ca (21, X ) 6e a sequence such that £(A) — lim £ n (A) n—>-oo

/or even/ A € 21. Then, £ € ca(2l, X ) and 2/ie following holds: ||£||(A)
A e 21;

n—>oo

||£||0(A)
A €21;

n—>-oo

|4|(A) < l i m m f | £ „ | ( A ) ,

Ae2l.

62

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

We now consider X-valued bimeasures, where X is a Banach space. We put 21 x 21 = {A x B : A, B 6 21}. For any function M defined on 21 x 21 we denote the value of M at A x B exchangeably by M(A x B) or M(A, B). Definition 9. Let X be a Banach space and X* be its dual. (1) OT(2t x 2t;X) denotes the set of all functions M : 21 x 21 -> X such that M(A,-), M(-,A) g ca(2l,X) for every A £ 21, i.e., M is separately c.a. Every element M £ 9Jt(2t x 21; X) is called an X-valued Mmeasure. If X = C, then we put M = 9Jt(2t x 21; C), the set of all scalar bimeasures. Note that, if M g SOT(21 x 21; X), then x*M 6 M for x* g X*, where x*M is defined by x*M(,4,B) = x*(M(A,B)) for A,B e 2 t . (2) Let M £ SK(Sa x 2t; X) and m 6 M . The variations \M\(A,B) of M and |m|(j4, B) of m at A x B € St x 21 are respectively denned by |M|(A,B) = s u p £

]T

||M(A,A')||x,

A£7r A ' S T T '

|m|(AB) = s u p ^

£

|m(A,A')|,

A£7r A ' 6 i r '

where the suprema are taken over 7r g 11(^4) and ir' g n ( B ) . OTt,(2l x 21; X) and M„ = Wlv (21 x 21; C) denote the sets of all M £ SOT(21 x 21; X) and m G M of bounded variation, i.e., | M | ( 0 , O ) < oo and | m | ( 0 , 0 ) < oo, respectively. (3) The semivariatwns \\M\\(A,B) of M and ||m||(A,B) of m at A x B g 21 x 21 are respectively defined by !|M||(>l,B) = sup

Y, £

aA&yM(A,A';

AgTT A'GTr'

||m||(A,B)=sup

£

^

' A€TT A'STT'

aA£A,m(A,A') '

where the suprema are taken over 0 4 , /3/y 6 C with |QA|, |/?A' | < 1 for A g 7r g 11(A) and A' g n' g 11(B). The terminology of Vita/z variation is sometimes used to stand for variation of bimeasures. The terminology of "semivariation" has usually been used for a vector measure and not for a bimeasure, and Frechet variation is used for bimeasures. However, we adopt the terminology of semivariation for both vector measures and bimeasures. Note that for M G OH (21 x 21; X), ||M||(-, ■) is separately monotone and countably subadditive, and \M\(-, ■) is separately c.a. iff M g 9K„(2t x 21; X).

3.1. SEMIVARATIONS AND VARIATIONS

63

Immediate consequences of the Orlicz-Pettis Theorem and the Vitali-Hahn-SaksNikodym Theorem are: T h e o r e m 10 (Orlicz-Pettis T h e o r e m for B i m e a s u r e s ) . Let X be a Banach space and X* be its dual. Then M : 2t x 21 —> X is an X-valued bimeasure iff M is separately weakly c.a., i.e., x* M is a scalar bimeasure for every x* € X*. T h e o r e m 11 ( V i t a l i - H a h n - S a k s - N i k o d y m T h e o r e m for B i m e a s u r e s ) . Let {Mn}^L1 C 9Jl(2l x 21; X) be a sequence of X-valued bimeasures such that lim Mn{A,B) = M(A,B) exists for every A,B €21. Then, M e 3Jt(2l x 2t; 3£) and n—yoo

are separately uniformly {M n }™ =1 In Theorem 11 it holds that

c.a.

||Af„||(i4,B)
A,B e%

(1.7)

A,B<=%

n—too

where the second inequality in (1.7) follows from Theorem 12 and Proposition 14 below. Also the Nikodym Uniform Boundedness Theorem holds. T h e o r e m 12 ( N i k o d y m Uniform B o u n d e d n e s s T h e o r e m for B i m e a s u r e s ) . Let { M a } Q g i C 2R(2l x 21; X) be a family of X-valued bimeasures such that sup\\Ma{A,B)\\x
A,B€

21.

Then it holds that sup{

sup ||M Q (,4,-B)lbe} < °o.

Proof. Consider the family {Ma(A,),Ma(-, B) : A, B e 21, a g 1} of X-valued measures and apply the Nikodym Uniform Boundedness Theorem. Then the desired result follows. T h e o r e m 13. Let m € M be a scalar bimeasure and put mi(A) = m(A,-) and m 2 (A) = m(-,A) for A e 21. Then, mi, m2 € ca(2l,ca(2l, C)), where ca(2t,C) is equipped with the total variation norm, and it holds that

HI(e,e) = |K||(e) = ||ma||(e). If m £ M U 7 then m%, 7n2 6 vca(21, ca(2t, C)) and it holds that | m | [ e , e ) = [m 1 |(6) = | m 2 | ( e ) .

64

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

Proof. To show that mi is c.a., let {An} C 21 be a sequence of pairwise disjoint sets. Then, lim ml U Ak,B)

exists for every B e 21. Hence, \mi{

n—>-oo

fc=l

U Afcjj fe=l

is a bounded sequence in ca(2l, C) by the Nikodym Uniform Boundedness Theorem. By Dunford and Schwartz [1, Theorem IV.9.5] this sequence converges weakly to ™i( U .4*:) in ca(2t,C). Thus m x is weakly c.a. and the Orlicz-Pettis Theorem implies that m i 6 ca(2l, ca(2t, C)) . Similarly, m 2 6 ca (2t, ca(2t,C)). To prove the equality ||m||(©,0) = ||mi||(0) consider the equation: £

£

aA^A,m(A,A')

=

A6TT A ' S T T '

£ A'STT'

/? A , ( £

aAmx(A))(A')

Aeir

where a A ,/? A < e C with | a A | , |/? A -| < 1 for A G TT 6 11(0) and A' e 7r' 6 11(0). Taking the suprema appropriately gives the desired equality. Similarly, we can prove

IH|(e,e) = |K||(e).

The rest of the theorem is not hard to see. P r o p o s i t i o n 1 4 . Let M € OT(2l x 21; X). Then, for A,B eX it holds that sup \\M(AnC,BnD)\L<\\M\\{A,B)<16 c,£>ea

sup \\M(AnC,B c.cea

n

D)\\r.

Proof. The first inequality is clear. To prove the second inequality, let x* 6 X* be such that ||x*||x- < 1. Then, x*M e M (cf. Definition 9) and observe that | | x * M | | ( 0 , 0 ) = ||(x*M) 1 ||(6),

by Theorem 13,

< 4 sup $x'M)i(A)\(0), Aea

by Remark 2 (1),

< 4 sup vU sup \(x*M)(A,B)$, Aea sea '/ < 1 6 sup \\M(A,B)\\X. A,sea

by Remark 2 (1),

Taking the supremum over j|ar* 11^e- < 1 gives the inequality: ||M||(0,0)<16

sup A,sea

\\M(A,B)\\X.

For A, B e 21 the desired inequality follows by considering M\AxB, of M to A x B. The proof of the following proposition is routine and we omit it.

the restriction

65

3.1. SEMIVARATIONS AND VARIATIONS

P r o p o s i t i o n 15. (£01(3 X 3 ; £ ) , || • | | ( 9 , 6 ) ) and, m particular, ( M , || • | | ( 9 , 9 ) ) are Banach spaces. We now turn to T(i/)-valued bimeasures. Definition 16. LetOT= QJt(3 x 3 ;T(JET)) be the Banach space of all T(H)-valued bimeasures on 3 x 3 , which is also a left and right B(H)-mod\i\e, called a B(H)bimodule. Let M G 9Jt. The variation and the semivariation of M are defined as in Definition 9 (2), (3). The operator semivariation \\M\\0(A,B) of M at A x B 6 3 x 3 is defined by ||M|| 0 (A,B) = sup Y, E

«AM(A,A>*A,

A 6 T A'g?r'

where the supremum is taken over a&,b&i 6 B{H) with \\a&\\, ||&A'|| < 1forA G Ti- € 11(A) and A ' e TT' € 11(B). OTb = £OT6(3 x 3 ; T ( t f ) ) denotes the set of all M G 9Jt of bounded operator semivariation, i.e., | | M | | O ( 9 , 0 ) < oo. An M e Tt is said to be positive definite if M : 3 x 3 —> T ( i / ) is a p.d.k. in the sense of Definition II.4.1. An m G M is said to be positive definite if Y^, ctjakrn(Aj,Ak) > 0 for any n G N, ax,... , an G C and A x , . . . , A„ G 3 . Clearly 7 fc

«mb (resp. Wlv = ( 3 x 3 ; T ( f f ) ) ) is a (left and right) submodule of Wl (resp. £Utb). The following lemma is easily verified [cf. (1.3) and the proof of Theorem 5 (2)]. L e m m a 1 7 . If M £ Ttb, then M(A,-),M{-,A) G vca(%,T{H)) for A G 3 . In fact, it holds that | M L 4 , - ) | ( e ) < ||M||„(i4,0) and \M(-,A)\(Q) < \\M\\0(B,A). For £,77 G ca(3,^C) define M^v and m^n by Af€,(A,B)=[e(A),»/(B)])

A,BG3,

miv(A,B)=(aA)MB))x^trM^(A,B),

(1.8)

A,Bg3.

(1.9)

We abbreviate as M j = M ^ and m^ = m ^ . Then M ^ G 9ft and m ^ G M and, in particular, M^ and m^ are positive definite. Conversely, if M G 9Jt is positive definite, then by Theorem II.4.13 there exist a r.k. normal Hubert B(H)-module XM of M and £ G c a ( 3 , X M ) such that M = Mv Moreover, for G H we define

mu(A,B) = fa(A)MB))r ^ B e a -

( L 1 °)

Recall that £^(-) = (£(■),<£)H £ c a ( 3 , Lg(fi)). The meafc variation \m^\w(A,B) 77i£ at .4 x 5 G 3 x 3 is defined by k | . ( A , B ) - s u p { | m € , | ( A , B ) : ^ G ff, ||4>||H < l } ,

of

66

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

where | m ^ | ( j 4 , B) is the variation of m^. at A x B. If £ e ca 5 os(2t,X), then M ? (-,-) = Fj(-) 6 c a ( 2 t , T + ( / / ) ) C vca(%T{H)). Hence, £ e ca(2t,X) is g.o.s. iff the support of M^ is contained in the diagonal A(2t x VL) = {A x A : A e 21}. Another characterization of gramian orthogonal scatteredness is obtained as follows. P r o p o s i t i o n 18. Let £ € ca(2l,X). Then it is g.o.s. iff £ 6 ca(21, Lg(ft)) is o.s. for every 6 H. Proof. Observe that for A,B 6 2t and <j> £ if the following holds:

(^(^),^(s)) 2 = ((a^)»H,(as),,i{B))Hd^ = /" ( ( £ ( A ) ® £ ( B ) ) < M ) „ < ^ = ( K ( A ) , £ ( B ) ] 0 , 0 ) H . Thus we have that ( ^ ( A ) , ^ ( S ) ) 2 = 0 for every 4> E H «=» ( K ( A ) , £ ( f l ) ] ^ ) „ = 0 for every cf> e H <=> (K(A),C(B)]0, V ) H = 0 for every < M e tf

<=>[t{A),z(B)]

= a.

Therefore, the conclusion holds. The following lemma gives interrelations among semivariations and variations of ^,T],M^v,M^,m^v and m^. The proof is routine and so is omitted. L e m m a 19. Let (,,n E ca(2t,X) and A,B 6 21. Then: (1) | K „ | | ( A , i ? ) < | | M ^ | | ( A , B ) < | | £ | P ) | M | ( / 3 ) . (2) | m 4 , | ( A , B ) < \Mir,\(A,B)

<

\Z\(A)\V\(B).

&)\\M&lUA,B)
(6) |KIU(A) 2 <|m^U(A,^)<|eU(A) 2 . Since vca (21, T ( H ) ) is a J3(#)-bimodule, we can define the operator semivariation for an M € ca(%.,vca{%T(H))) as in Definition 4(1), where vca{%T{H)) is equipped with the total variation norm | • | ( 9 ) . As before ca(2l,T(if)) is equipped with the total semivariation norm ||-||(@).

3.1. SEMIVARATIONS AND VARIATIONS

67

T h e o r e m 20. Let £,77 € ca(%X) and consider M = Min M{A, ■) and M2{A) = M(-,A) for A e 21. Then:

£ 9Jt. Put MX(A)

=

(1) M i , M 2 € c a ( a , c o ( 2 t , r ( / f ) ) ) and if ZioZds £/ia£ | | M | | ( 0 , 0 ) = ||Mi||(0) = ||M 2 ||(e). (2) / / £,r? e bca{%X),

(1.11)

t/iera M i , M 2 € bca (21, vca{%T{H)))

and it holds that

| | M | | 0 ( e , e ) = ||Mi|| 0 (9) = ||M 2 || O (0). (3) If M eWlv,

then MltM2e

vca(%vca{%T{H)))

(1.12)

and it holds that

|M|(e,e) = |M1|(e) = |M2|(e).

(1.13)

Proof. (1) Clearly Mi and M 2 are f.a. (= finitely additive). To see that Mi is c.a., let {Ak}kxLl C 2t be a sequence such that Ak I $. Then we have that

||Ml(Afc)||(e)=sup

J2 0MAk){A)

I/9AI
Ae7ren(©) \

AGTT

sup

Z(Ak), *£ MA)

:|/?AI
Ae7ren(0)l

Agvr

E ^(A)

<sup<M|£(A f c )||x

: \pA\<

1, AGTT

en(e)

Ae?r

= IK(40IUNI(e)->o (fc-voo). Thus M i ia c.a. Similarly, M 2 is c.a. too. ( I l l ) can be verified in a similar manner as in the proof of Theorem 13. (2) Let f,77 e 6ca(2t,X). Then by Lemma 19(3) Mir) e Ttb. It follows from Lemma 17 that M i (A), M2{A) 6 vca(2t, T ( F ) ) for A e 21. We have to show that M i is c.a. Let {Ak)kxL1 C 21 be such that Afc 1 0. Then it holds that

|Mi(A f c )|(0) = sup | E

||Mi(A f c )(A)|| T : TT e n ( 6 ) 1

A6TT

sup

£

[£(Afc),T7(A)]^

A£TI

= sup

£(Afc), E 6^»/(A) A£TT

,

by (1.3),

68

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

E &A7?(A)

«>),

where the last three suprema are taken over 6A g B(H) with ||&A|| < 1 for A g 7r g 11(0). Thus M i is c.a. and similarly M 2 is c.a. Now (1.12) can be seen as follows: | M | | o ( e , 0 ) = sup

J2 E aAM(A,A')6A, A€ir A ' e V

= sup

E A'Eir'

sup

(E a A M i( A ))( A ')&AAe^r

Y ^ O A M I ( A ) (9),

by (1.3),

ASTT

M x j| 0 (e), where the suprema are taken over a&, 6A* g # ( # ) with \\a^\\, ||6A' || < 1 for A g % g n ( 9 ) and A' g n' g 11(9). ||M 2 || O (0) = | | M | | O ( 0 , 0 ) is proved in the same fashion as above. (3) can be verified easily. R e m a r k 2 1 . For an M g M define Mi(A) = M(A,-) and M2{A) = M(-,A) for A € 21. Then, M i and M 2 are ca(2l,T(i?))-valued f.a. measures on 21 and satisfiy (1.11). Similarly, if M g OTb (resp. M g Wlv), then M x and M 2 are vca(%,T(H))valued f.a. measures and satisfy (1.12) (resp. (1.13)). The following theorem is useful in computing semivariations and variations of Xvalued measures and T(J?)-valued bimeasures, which in turn are used in classifying the processes. T h e o r e m 22. Let £ g ca(%,X). (1) The following statements are equivalent: a) \\ttA)\\x = MUA)foreveryAeZ; b) M^A, B)>0 for every A, B g 21; c) \Mt\{A,A) = U(A)\\2X for every A g 21. (2) The following statements are equivalent: d) H(A)\\x = U\\(A) for every A e 21; e) m^(A,B) > 0 for every A,B g 21,f) \rns\(A, A) = U(A)\\X for every A g 21.

69

3.1. SEMIVARATIONS AND VARIATIONS

(3) The following statements are equivalent: g) U(A)\\x = \t\(A)foreveryAG%; h) £(*) = KOz for some x e X and v € ca(2t,R + ). Proof. (1) a) => b): Assume that a) holds. It suffices to show that M^(A, B) > 0 for disjoint A, B € 2t. So let A, 5 e 21 be disjoint. Then we have that by a) U(A) + £{B)\\x = U(A u B)\\x = UUA U B) > \\u ■ f (A) + f{B)|U for every unitary u e B(H). Hence ||£(A) + £ ( £ ) | | ! - ||« • €(A) + £(B)[& = 2Re{(C(A),e(B))x - (u • £(A),£(B)) X } = 2Re{tr[(l-u)M c (A,S)]} > 0, where Re{- ■ •} means the real part and 1 is the identity in B(H). It follows from Schatten [2, p. 43, Theorem 6] that M^B) > 0. b) => c): Assume that b) holds. Let A e 2t and {Alt... , Am}, {B1:... , Bn} € 11(0). Putting Cjk = Aj n Bk ioi 1 < j < m, 1 < k < n, we see that

E ||M€(Ay,fl*)||T = X) H(v Cj *'V Cpfc)ll = E E M « ( ^ . ^ : < E

\\MdCjq,Cpk)\\T=

E

trM 5 (Q„C p f c ),

byb),

j,k,P,q

i,k,P,q

E (ac3q),acPk))x = \J2t(c]q) iM,p,q

=

U(A)\\2X.

},Q

Thus |M^|(A, A) < ||C(A)||^- = ||M^(A, A)\\T. Since the converse inequality always holds, we obtain c). c) => a) is clear from Lemma 19 (5). (2) follows from (1) by taking H = C. (3) Observe that the following equality holds for every A G 21: |e|(A)=sup

E

5A

^A)

(1.14)

Ag7T

where the supremum is taken over 5 A € B{X) with ||5A|| < 1 for A g -K e n(A). If we identify X with S(X, C), then the gramian in this case is [x, y}0 = x ® y, i.e., [x,y]02 = (z,y)xx for x,y,2 G X. Note that [x,y]Q > 0 iff (a:,y)x = tr[x,y] 0 = ||rc ® y||T = ||x||x||y||x iff there exists some a > 0 such that x = ay oi y = ax.

70

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

Also note that the variation |£|(-) of £ £ ca(2t,X) equals the operator semivariation HCIU-) of £ e ca(Sl, S(X,€)), which is seen from (1.14). Now since h) => g) is obvious we prove g) => h). Without loss of generality we may assume that there is some B £ 21 such that £(B) ^ 0. Put x = £(B). By (1) we have that [£(A), £(B)]o = £(;4) ®x" > 0 for yl £ 21. Hence there exists some I/(J4) > 0 such that £(A) = v{A)x for A £ 21. It is immediate that v 6 ca(2l,R+). Therefore h) holds. R e m a r k 23. (1) Theorem 5(3) and (4) follow from Theorem 22(1) and (2), re spectively. (2) It follows from the proof of Theorem 22(1) that, for x,y £ X, [x,y] > 0 iff \\x + y\\x > ||u ■ x + y\\x for every unitary u £ B(H). Hence, [x,y] = 0 iff ||x ± y\\x > ||w • x + y\\x for every unitary u £ B(H). Moreover, if H = C, we have that, for f,g £ Lg(O), (/,ff) 3 > 0 iff ||/ + |1«/ + ffUa for every a e C with |a| = 1, and (f,g)a = 0 iff ||/ ± 5 | | 2 > \\af + g\\2 for every a £ C with [or] = 1. Thus we have obtained a characterization of orthogonality in a Hilbert space and of gramian orthogonality in a normal Hilbert B(H)-modu\e. Now we give some examples of X-valued measures for which the strict inequalities holds in Theorem 5 (5) and Lemma 19. E x a m p l e 24. Consider the measurable space (R,*B) where K is the real line and 5B is its Borel a-algebra. We can and do assume that dim LQ(Q.) = oo. (1) ||ClU(R) < ||C||(K): Let n > 2 be arbitrary but fixed. Let {/ fc }£ =1 C L§(0) and {4>k)l=i CHbe ONS's. Define

£(A) = Yl Mk>

A £58.

keAnn

Then clearly £ £ cagosffi,

X) and it holds that by Theorem 5 (4)

Ham = um\x --

^fk
On the other hand, for <j> £ H it holds that

sup

^

otk{fk4>k,4>)a

\ak\ < 1, 1

k=\ n = sup ■

}

J

k=\

ak{4>kA)Hfk

\ak\ < 1, 1 < k
3.1. SEMIVARATIONS AND VARIATIONS

supH£K|2|(^»H|2

71

|a*l
fc=l

Yl \{<S>k,4>)H\

m-

fc=l

Thus we see that U\\W{R) = s u p { j | ^ | | ( R ) : \\4>\\H < 1} = 1- Hence ||f|| w (R) < ||£||(R). In this case we have |£| M (R) = \/n. oo: Assume that Lg(il) and H are real Hilbert (2) < Kl spaces, so that X is also real. Put Cnr:

n e N,

2n{log(n+l)}2' 7r(m —n)

sm (m — n)^/mn log(m + 1) log(n + 1)'

m ^ n, m,n 6 N.

Then it holds that (cf. Hardy, Littlewood and Polya [1, p. 214]) oo

oo

EE

7 7 1 = 1 71 =

OO,

< OO,

a = sup j=lk=l

1

where the supremum is taken over —1 < otj,Pk < 1,1 < j < m, 1 < fc < n and 77i, n € N. Since { c m n } ^ n = 1 is positive definite, we can find a system {gk}kLi £ L Q ( O ) such that {.9h9k]2 = cjk, j,keK Take a 0 ( ^ 0) € if and define £ by

t(A)=

Yl

9k
keAnn Then we have that ||£H(R)2=sup

=SUp^2Y;a3ak(9j,9k)2\\\\H

Y^Oik9k4> fc=l n n

X

j=lk-l

= S U P $ Z H aJakCjkW\\2H < a|Mlff> 3=1

fc=l

where the suprema are taken over -1 < ak < I, I < k < n and n e N. Hence we see that £ e ca(*B,X). On the other hand, it holds that oo

oo

Ki(R,R)=^^|(e({i}),?(W)) x | j=i

fc=i

72

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

j=ik=i

J=ifc=i

In this case we have |£L(R) = oo. (3) |m € |(R,R) < ||£|| S (R) 2 = oo: Take an / ( # 0) 6 Ig(fi) and let {0 fc }*Li CONS in H. Define £ by

be

a

^e<8-

£(>«)= E £/**• fceAnN

Then, £ e caos(
|m4|(R,R) = | K ( R ) l & = E I2II/H2 < TO fc=i

On the other hand, we see that

E(^-^) H

Haw > Eit=ir «(W)^ fc

fc=i

00

,

2 = 00. fc=i

In this case we have |£|u,(R) =

nth

2

(4) | M ? | ( R , R ) < |£|(R) = 0 0 : Let { / j , } ^ be an ONS in £g(fi) and {<j>k}%L 1 a CONS in H, and define £ by

^)=

E r/***.

^s.

fc€AnN

Then, £ G ca^os(!B,X) and it follows from Theorem 22 that 00

1

fc=i

On the other hand, we have that 00

*:=i

HI

11

°° 1 fc=l

2

be

73

3.1. SEMIVARATIONS AND VARIATIONS

In this case we have |f|„,(I (5) |m 4 |„,(R,R) < ||£|| S (R) 2 = oo: Let £ be as in (3). Then we see that

K U R . R ) =sup{|m € J(R,R) : \\
EEK^W)l:Wff^

p

I;=lfc=l (

= suP

su

oo

oo

^^-|(^^)HW,^)H|II/!IMI0||H

I. J = l

p|(£ll(^)"l) 111/13

SU

sup

< 1

fc=l

P{(EF)(EK^*)« r'lWlIrll/llf .

\B<1

:M

"IM|^IIH
H\\B

* II/II1

< oo.

VG °° i If we take $ = — Y\ —0fc, then actually the equality holds in (1.15) and hence ^

fe=l

r2|| f ||3

«

= oo was shown in (3). But |m4|„(R,R) = 2 (6) ||?||S(R) < uiy,£H», jtfcj = oo: Let f be as in (2). Then we see that 2

OO

||4|| s (R) 2 =sup

)H

a* = 0 , 1 , \\M\B < 1 , fceN

fc=l oo

sup

2

2 J ak(gk4>,4>k)H : a = 0 , 1 , ||0fc||H < 1 , * € N 2

fc=i oo

-1

= sup fc=l

2

= s u p j 5Za>afc(ffi,flfc)2||0||H : - 1 < Q f c < 1, fceNi < a\\4>\\2H < oo. Clearly |M C |(R,R) > |m 4 |

oo by (2).

R e m a r k 2 5 . As will be seen in Section 3, ||£|| 8 (A) < oo iff ||£|| 0 (.4) < oo (cf. Theo rem 3.15). Hence Example 24(6) serves as one for which ||£|| 0 (R) 2 < |M £ |(R,R) = oo. But it is possible to construct a £ 6 ca(2l„ X) such that ||£|| g (R) < ||£|| 0 (R) < oo.

74

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

3.2. Orthogonally scattered dilations In this section we discuss the orthogonally scattered dilation of Hilbert space valued measures. In doing this, we use the Grothendieck inequality and the notion of absolutely 2-summing operators. As in the previous section, let (0,21) be a measurable space. We begin with the integration of scalar valued functions w.r.t. a Banach space valued measure. By a C-valued ^-simple (or, merely, simple) function we mean a n

function / of the form f = YL ctk^Ak for cti, ■ ■ ■ , an 6 C and { A l s . .. , An} 6 11(9), fc=i

where 1A denotes the indicator function of A. L°{&) denotes the set of all C-valued 2l-simple functions on 9 . If / is a C-valued function on 9 , then the sup norm, ||/||oo> of / is defined by \\fl\oo = sup | / ( t ) | . tee Definition 1. Let X be a Banach space and £ 6 ca(2l, X). A G 21 is said to be £-null if ||f||(j4.) = 0. Thus £-a.e. (= ^-almost everywhere) refers to the complement of a £-null set. The ^-essential sup norm ||/||oo,£ of a C-valued 2l-measurable function / on 9 is defined by ||/||oo^=inf{o>0:[|/|>a]ise-null}1 where [|/| > a] = {( e 9 : 1/(01 S <*}• It is easy to check that ||-||oo,£ is actually a norm, where we identify two functions if they agree £-a.e. For a simple function n

f = J2 aklAk

e L°(&), the integral of f over A e 21 w.r.t. f is defined by

fc=i

(2.1) JA

fc=i

In this case we have

L

fd£

< ll/l|oo,«||£||(A).

(2.2)

L ° ° ( 0 = L°°(£;C) denotes the set of all C-valued functions / on 9 such that ||/n - f\\oo,( -> 0 for some sequence {/„}£° =1 C L ° ( 9 ) . It is not hard to see that L ° ° ( 0 is a Banach space with the norm ||-||oo,«- For / g L ° ° ( 0 let {/„}£° =1 C L ° ( 9 ) be as above. Then, { JA fn d £ } n = 1 C X is a Cauchy sequence in view of the inequality (2.2) since {/„}£° =1 is Cauchy in £°°(f). Thus we can define the integral off over A w.r.t. £ by / JA

fdi=

lirn ( "-+007A

/„#.

75

3.2. ORTHOGONALLY SCATTERED DILATIONS

A more sophisticated definition of the integral of C-valued functions w.r.t. a Ba nach space valued measure is given as follows: Definition 2. Let X be a Banach space and £ 6 ca(2t, X). A C-valued 21-measurable function / on O is said to be ^-integrable (or, integrable in the sense of DunfordSchwartz) if there is a sequence {fn}%Li £ £ ° ( 9 ) of simple functions such that a) fn(t)^f(t) £-a.e.; b) { J ^ / „ d £ } n = 1 C X is a Cauchy sequence for every A € 21. In this case the integral of f over A w.r.t. £ is defined by / / d £ = lim / / „

(2.3)

d£,

which is also termed the DS-integral. L x (£) = Z/ 1 (£;C) denotes the set of all de valued 2t-measurable functions on 0 which are ^-integrable. R e m a r k 3. Let £ 6 ca(2t,£), X being a Banach space. As was shown in Dunford and Schwartz [1, IV. 10.7] we have: (1) The DS-integral (2.3) is well-defined, i.e., it is independent of the determining sequences of simple functions. (2) For / € L x ( 0 and A € 21 it holds that

/.

fd£\

< \\f\umu).

(3) i 1 ( 0 is a linear space and the DS-integral is linear. (4) L°°(£) is a subspace of L 1 ^ ) . (5) If / G L : ( 0 and £f{A) = / A / d£ for A e 21, then £f e ca{% X). Moreover, we have: (6) For A e 21 it holds that

||£||(A) = sup

fd£

:/€l°(6),

sup

fdi

■■feLl(0,

iioo,« <^

lloo,£

l

< 1

This follows from Definition 1.1 and (2) above. (7) For / € L 1 ^ ) , HOIK 0 ) = 0 iff / = 0 £-a.e. Hence, if we identify f,ge when f = g £-a.e., then

11/11. e = HOIK©)

L l (£)

76

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

( V for semivariation) defines a norm in L 1 ^ ) , which may be termed the "semivariation norm" of / . In fact, assume that ||£/||(0) = 0. Put An =

0 < |/| < — I =

It e 0 : 0 < | / ( t ) | < - } for n £ N. Then, ||£||(A n ) -4 0 by Remark 1.2 (3) since An I 0. If we set B = [f ^ 0], then it holds that for n e N

||£||(fl\A,) = sup < sup

III/, III/-

9 e L 1 ^ ) , ||
5^

^ e L 1 ^ ) , lbllool€ < i

gnfd£

< n | | ^ | | ( J B \ A n ) < n | | e / | | ( G ) = 0. Thus, |K||(5) < | | £ | P V W + | | £ | P „ ) = llflKA.) -► 0, so that ||£||(fl) = 0. Therefore / = 0 £-a.e. T h e o r e m 4. If X is a Banach space and £ 6 ca(2l, X), then (L 1 (^), ||-|| s ,f) is a Banach space and the set of all simple functions L ° ( 0 ) is dense in it. Moreover, the set {£f : f e L1^)} is a closed subspace of ca(2l,X). Proof. Clearly (L 1 (^), || • ||s>g) is a normed linear space. We first prove that L°(Q) is dense in Ll(£). Let / € £*(£) a Q d £ > 0 be given. Define ,4 n = [|/| > n] for n £ N. Then ||C/||(A») -> 0 by Remark 1.2(2) since An | 0. Choose n 0 £ N such that ||£/||(J4.„ 0 ) < - and choose # e L°(Q) such that 5 = 0 on A„ 0 and

'3 _ / l

<

2||ei|(>lc ) °Q A"°' Th6n ** h ° ldS

that

\\g - /Il.,« = HC, - Oll(e) < ll£s-/ll«,) + UMPno) 2||e||Un / r W £ J + l = * 0

x

Thus L°{Q) is dense in L ( 0 To see that L 1 ^ ) is a Banach space we only have to show the completeness of x L ( 0 - So let {/ n }£° =1 C L x ( 0 be a Cauchy sequence. Since L ° ( 0 ) is dense in L 1 ^ ) we can find a sequence {#n}£Li £ L ° ( 0 ) such that \\gn - f„\\s^ -> 0 as n —> oo. Note that {ffnJ^Li is also a Cauchy sequence in L1^). We can assume that ^2hk+i

-9k\U,i

= Q < oo,

fc=i

since, otherwise we can consider a suitable subsequence of {
B = it £ 0 : JT l5fc+1(t) - Sfc(t)| = oo] *-

fc=i

J

77

3.2. ORTHOGONALLY SCATTERED DILATIONS

and we shall show that |[f||(J?) = 0. So suppose that U\\{B) = e > 0. Define Bn = It e 9 : ] £ \Sk+i{t) -9k(t)\ ^

> — j,

fc=i

neN.

'

oo

Putting B 0 =

U Bn, we see that Bn | S 0 and U\\(B0\B„)

-» 0. Hence we can

71=1

find n 0 e N such that ||£||(£„ 0 ) > | since ||£||(fl 0 ) > U\\(B) = e. Thus we have oo

no

fc=i

fc=i

2a, >||^oi|gk+i_3fc|||(Sno)>-||^||(i?no)>Q a contradiction. Consequently ||£||(.B) = 0. Define a function / on 0 by

{

oo

9i(t)+

£

(gk+i(t)-gk(t)),

dteB-

fc=i

0,

iiteB

and observe that / £-a.e. and, for ,4 6 21, { / A 5 n d £ } n _ ! is a Cauchy sequence in X, since for n, m £ N / £n+m d£,JA

I gn d£ JA

< ll^(9„+m-9„)ll(0) oo

< J2\\9k+i

-9k\U,i

->0

(n->oo),

k=n

which implies that / £ £*(£)• Since ca(2t,X) is complete (cf. Remark 1.2 (3)) there exists an n € ca(2t,3£) such that ||£ 9n - n||(0) -» 0. But since £ 9 „(A) -> f/(A) for A £ 21, we see that n = £/. Therefore \\gn - f\\3£ —> 0 and hence ||/„ — f\\s^ -» 0. The last assertion is obvious. Let us now consider i/-valued measures. If£ 6 caos(2l,i/), then ^ ( ) = |l£(-)||jj» £ ca(2t,R+). Note that ^ is a control measure of £ (cf. Remark 1.2 (2)). F o r / £ L ° ( 0 ) we can define the integral of f over A £ 21 w.r.t £ by (2.1). Observe that for / = t

ocklAk

£ L°{&)

fc=i -

2

/ / «*£ J A

,

= H

n

E Vi=1

n a

> ^ ( ^ n A), £ fc=1

,

<*kZ(Ak n A) /H

78

III.

STOCHASTIC MEASURES AND OPERATOR VALUED B1MEASURES

Y^

ajak(((AjnA),^AknA))H

j,k=\

Yj\ak\2vi{AkKA)=

I |/| 2 d^ = ||l,

fc=i

JA

l2,i/f'

(2.4)

where || • || 2 ,„, is the norm in L2{v^) = L 2 ( 0 , / / ? ) . For / € L2(v^) we can find a sequence {/„}£° =1 C L ° ( 0 ) such that ||/„ - /|| 2 ,„ 4 -> 0 and / „ -> / £-a.e., since (,-a.e. and v^-a.e. are equivalent because of ||£||(^4)2 = ||£(-4)HH = v^{A) for A 6 21 by Theorem 1.5 (4). By (2.4) we have for A 6 21

[ fn<%- [ JA

fmdi

IM/n-/,

771,1 | | 2 , l / £

JA

< ll/n - / m l b , ^ -> 0

(n,m->oo),

i,e-> { IA fn ^£} _i is a Cauchy sequence in if. Thus / is £-integrable. Moreover, (2.4) holds for / e L2{^) and A 6 21. Consequently L2(
iis.e

IK/IK©) = ll€/(©)IU

|2,i/f

Je

since £/ S caos{%, H), which can be easily verified, and we can apply Theorem 1.5 (4) again. Thus the closure of L°(9) w.r.t. || ■ ]| Si5and ||2,i/{ are the same. Therefore we have: P r o p o s i t i o n 5. If i €

raos(2l,if),

then ( L 2 ( J / ^ ) , ||-1| 2

Vi

(LHO.II-lk), »A ere

Definition 6. £ € ea(2l, i/) is said to have an orthogonally scattered dilation {o.s.d .) if there exist a Hilbert space ^ containing if as a closed subspace and an 77 £ caos(21, fj) such that £ = J77, where J : f) —► H is the orthogonal projection. The triple {77,^3, J } is also called an o.s.d. of £. £ is said to have a spectral dilation if there exist a Hilbert space .f), a spectral measure E(-) in f>, an operator 5 € B(S),H) and a vector 4> € f) such that

£(-) = S£(-WNow the question is: Does every measure £ 6 ca(2l, if) have an o.s.d.? affirmative answer is given after some considerations.

An

3.2. ORTHOGONALLY SCATTERED DILATIONS

79

Definition 7. Let X and 2) be two normed linear spaces and p > 0. A linear operator T : X —> 2) is said to be absolutely p-summing if 7rp(T) < oo, where TTP(T) = inf{C > 0 : (2.5) holds}:

[fl\\T'A\%y
:x*eX*,\\x*\\x,
for any n 6 N and xx,... ,xn 6 X. 7rp(T) is called the absolutely p-summing norm of T. We denote by Ylp(X, 2J) the set of all absolutely p-summing operators from X into 2JR e m a r k 8. Let 20, X, 2) and 3 be Banach spaces. (1) (n p (3T,2)), 7Tp(-)) is a Banach space for 1 < p < oo. In particular, n o o ( X , 2 ) ) = B(X,2J). (2) Let 1 < p < oo. If T € n p (3e,2J), V e B{W,X) and [/ 6 B(2J,3), then t / T V 6 n p (2U,3) and np(UTV) < ||l/||7r p (T)||V||. (3) If Sj and 8. are Hilbert spaces, then U2(fl,M) = S{f),M) and 7r2(T) = \\T\\a for T £ S(f», M) (cf. Wojtaszczyk [1, pp. 243-244]). Note that ( l Q ( 6 ) , ||- (loo) is a normed linear space and (£°{9), ||-||<»)* = /a(2l,C), where /a(2l, C) is the Banach space of all C-valued f.a. measures on 21 with the total variation norm | • | ( 0 ) and the isometric isomorphism U : /a(2l, C) —> L ° ( 0 ) * is given by {Uv){}) = j Q f dv (f e L°(Q)) for v g /a(2l,C) (cf. Lemma 3.3 in the next section). Here, the variation and the semivariation of a (scalar or vector valued) f.a. measure are defined in the same manner as in Definition 1.1, and the integral w.r.t. an f.a. measure is in Dunford and Schwartz [1, Chapter III]. We need one technical lemma. L e m m a 9. Let vQ 6 /a(2l,]R + ) and define v on 21 by

v{A) = inf < y^ i>o(Ak) : {j4fc}fcLi is a countable ^-measurable I fc=i

partition of A >. J (2.6)

Then v e ca(2t,R+) and v{A) < v0{A) for every A € 21. Proof. For A G 21 denote by I P (A) the set of all countable 2t-measurable partitions of A. v is f.a. In fact, let A, B 6 21 be disjoint. Then we have (

oo

c V{A u B) = inf t ]T MCk) ■ {Ck)T=i e n (A u B) V fc=i

80

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

= inf I J {MA n Ck) + MB n ck)} : {Cfc}r=i e nc(A U 5) I >KA) + ^(B). On the other hand, let e > 0 be given and choose {Ak}%L n c ( B ) such that oo

1

e I1C(A) and { S k } ^ e

oo

i/(A) > £ n,(Afc) - |2,'

HB) > Yl MBk) " v " ' " f-" " J V ~ " "

fc=i fc=i

2

Then we have that oo

v(A) + v{B) > J2 {MAk) + MBk)} - e fc=i oo

= ^i/0(4uBt)

-e>v{AuB)-e.

fc=l

Thus v{A) + 2/(1?) > !/(A U B). Consequently v is f.a. To show that v is c.a., let {,4„}™=1 C 21 and An | 0. For each n € N, A n OO

U (A fc \A fc+1 ) and {A fc \A fc+1 }£L n e n c ( A „ ) . Hence

fc=n

J%o(Afc\A f e + i) < MAi)

< oo-

fc=i

Thus 0 < v{An)

< £3 MAk\Ak+i)

-> 0 as n —> oo. Therefore v is c.a.

fc=n

Now we can give some necessary and sufficient conditions for o.s.d. T h e o r e m 10. Let £ € ca(%,H). (1) £ ftas an o.s.d. (2) £ /ias a spectral dilation.

Tften the following conditions are equivalent:

(3) Tftere m s i s a constant C > 0 sucft £/m£ /or any i i e N and f\,... n

2

n

<

/i« H

, / n € Z/°(©)

2 £i/*i i=i

(4) T/ie operator S^ : ( i 0 ( 6 ) , ||-||oo) -> H is absolutely 2-summing, defined bySif = fef d£ for f e L ° ( 0 ) .

(2.7)

where Se is

3.2. ORTHOGONALLY SCATTERED DILATIONS

81

(5) There exists a positive finite measure v E ca(2l,R + ) such that ii 2

< [ |/| 2 ^,

[fold J&

/ e L c (9).

(2.8)

J®

\\H

In this case v is called a 2-majorant

of £.

Proof. (1) => (2): Suppose that £ has an o.s.d. {T],F>, J}, SO that £ = J-q where J : ij —> H is the orthogonal projection. We can assume that

b =

e0{r](A):Ae
the closed subspace of fj generated by the set {r){A) : A E 21}. For A E 21 let E(A) be the orthognal projection of fj onto f)(>l) = &o{r/(A C\ B) : B E 21}. Then it is not hard to check that E{-) is a spectral measure in Sj such that n(A)

= E(A)rt(0),

Ae%.

Therefore, £(•) = JE(-)ri(Q), i.e., £ has a spectral dilation. (2) =>- (3): Suppose that (2) holds, so that £(■) = SE(-)ip as stated in Definition 6. Note that T?(-) = E{-)tp E caos{%?)) and j/(-) = \\T}(-)\\% E ca(2l,R+) with the total mass v{&) = \\il>\\%- Now for any n E N and / i , . . . , / n € £ ° ( 0 ) we have

E

/ /H

=E

5

/* /■<

<\\s\\2jr[

\f3 2 ^<||s|| 2 w| Ei/, i 3=1

Thus (2.7) holds with C = | | S | | 2 | M | | . (3) O (4): Since L ° ( 9 ) * = /a(2l,C) it is sufficient to show that for any n E N and/!,... ,/n€i°(G)

Ei/ii J=I

sup

{gl/e

/ ,
:i/6/o(a,q > H ( e ) < i

The inequality " < " is seen from n

LHS of (2.9) = s u p V \fj(t)\2

r

n

= sup / V | / j | 2 d 5 ( < RHS of (2.9),

(2.9)

82

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

where 5t is the point mass at t G 8 , i.e., 5t{A) = 1 if t e A and = 0 otherwise. To see the converse inequality ">" let v € /a(2l,C) be such that \v\(&) < 1 and / i , ■ ■ • , /n € L ° ( 0 ) be arbitrary. We can write them as i

where {J4I, . . . ,Aq} € 11(0). Then we see that

2 = ^ n ^ ]q

i/(Ajfc)i/(i4£)aj,fca^£ = £

j = lfc,£=l

= 5>(Afc )x fc fc=l

^(A fc )i/( J 4£)(x fc ,X£)c"

fc,£=l

<(J2 W(Ak)\\\ XfcHc V

C"

7

fc=l

< max llxkil^f ^K^oi) < ,?» £>i,*l3He)a fc=l

3=1

where Xfc = (Qj,fc)j =1 € C" for 1 < fc < g, and {-,-)o> and || • | | o are the inner product and the norm in C", respectively. (3) => (5): Assume that (3) holds. Let L g ( 0 ) = {/ e L ° ( 9 ) : / is R-valued} and define p0(f) for / e L ° ( 9 ) by Po(/) = i n f { s u p ( c / ( t ) + C £ | / f c ( i ) | 2 - £

ft,...

/"/fcdf

}:

,/„6L°(6), n e N

.

Then we see that po is a sublinear (i.e., positive homogeneous and subadditive) functional on L R ( 9 ) such that Cinf f{t)
Csup f(t). tee

It follows from the Hahn-Banach Theorem that there exists a linear functional p on Ljjj(6) such that p(f) < p0(f) for / 6 £ R ( 9 ) . Hence we have that C i n fe fit) *6

< -po[-f)

< p(f) < p0(f)

< Csup/(t). tee

83

3.2. ORTHOGONALLY SCATTERED DILATIONS

Moreover, (2.7) implies that for / € L°{@) A > H / | 2 ) < s u p < ! - C ] / ( i ) | 2 + C|/(i)|2 tee

Je

H)

Jo

fdi

and hence by - p ( | / | 2 ) = p ( - | / p ) < p 0 ( - | / | 2 )

Je

(2.10)

Define VQ on 21 by VQ{A) = P ( 1 A ) for A g 2t. Then it is easily verified that ^ 0 £ / o ( 2 l , R + ) since ^o(0) = p ( l e ) = C and p is linear. Furthermore, define v on 21 by (2.6). Then, Lemma 9 implies that v is c.a. and u(A) < VQ(A) for A 6 2t. n

Now we shall show that (2.8) holds. So let / = £ e ^ l ^ . e I ° ( 6 ) . For j = i=l 1 , . . . , n and q € N let {Aq- k}^=1 C 21 be a countable partition of ^4^ such that oo

lim J 3 vo{-^\ k)

=

"(-^j)) which can be a nonincreasing limit. Define for q,N 6 N n

N

A^E"^ j=i^=1

and observe that f'L —> / , pointwise, as iV —> oo for q 6 N and that by (2.10) 2

Je

H

n j

=

/V i

fc=i

Since for q € N we have that

it follows that

Je

fc=l

Letting g —► oo, we see that the RHS of the above inequality converges in an nonn

increasing manner to ]T lar^ | 2 ^ ( J 4 J ) = Je \f\2di/ 3=1

obtained.

and the desired inequality (2.8) is

84

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

(5) => (1): Suppose that (5) holds. Define m : 21 x 21 -> C by m{A,B) = v{AnB)-mi{A,B),

A,Be%

where m^A^B) = ((,{A), ((B))H. By (2.7) we see that m is a p.d.k. and hence there exist the RKHS $jm of m and a ( e ca(2t, f} m ) such that m = m c , i.e., m{A, B) = (C(A), C(B)) fi for 4 , B e 21. Put F) = H © f) m , the direct sum, and let J : .fj —> H be the orthogonal projection, where i / is identified with H @ {0} C 9). Now define 7/ € ca(2t,ij) by 77(A) = (£(A),0) + (0, <(>!)) = (f(A),C(A)),

A £ 21.

Then we see taht 77 is o.s. since for disjoint A, B € 21 it holds that

(v(A)MB))a = {(aA),c(A))AaB),aB)))f) = (^(A),a5))H+(C(A),C(B))^ =ro€(J4,J3)+ m(A,B) = i / ( A n B ) = 0. Moreover, it holds that £ = J77. Therefore {77, f}, J} is an o.s.d. of £. The condition (2.7) is of particular interest and is closely related to the following Grothendieck inequality (cf. Grothendieck [1, p. 59]). The proof may be found in sev eral books such as Diestel [1, pp. 174-180], Jameson [1, pp. 104-108], Lindenstrauss and Tzafriri [1, pp. 68-69] and Wojtaszczyk [1, pp. 210-211]. T h e o r e m 11 (Grothendieck inequality). Let n € N be arbitrary and f) be any Hilbert space. If [ajk)'jk=1 is an n x n matrix such that

/

J

j,fc=i

for every s l f . holds that

sn,t!,...,tn

ajkSjtk

max \sA\tk\

(2.11)

l<3,k
6 C, then, for every xi,...

, x„, yu ... , yn 6 fy it

Y] ajk{xj,yk)?, < KGa max \\xj\\f,\\yk\\g„ j,fc=i

(2-12)

1<J,k
where KG > 0 is the Grothendieck universal constant, independent of n and 9), and {-,-)f, and | | | | f t are the inner product and the norm in Sj, respectively. Using the above inequality we can derive the condition (2.7) as follows.

3.2. ORTHOGONALLY SCATTERED DILATIONS

L e m m a 1 2 . Let £ e ca(Qi,H).

85

Then there exists a C > 0 such that for any q 6 N

and A,... ,u /,6l°(e) ? II f 1=1 II • ' e

/<#

5>

(2.13)

«=i

Proof. Let / = ]T a y l ^ , # = £ /3 fc l Al € L°{@). Then we have that j-i

fc=i

E °j/?fc(£(^U(4t))„ = ( f /d£, / 5 d ^ We

jk=1

Je

In

< [ fd£ J&

f gd£

H ■>&

< IKII(©) 2 ||/llooll5lloc = IKII(O) 2 , max K-||/3 fc |. l<j,fc
Let / i , •. . , / , 6 £°(@) and write them as

where { ^ i , . . . , A n } 6 11(0). Then it holds that

E //«« £=1

H

£=lj,fc=l 9

= E (Ea^"<.*)(^Ai).^*))Hj,fc=l \ £=1

Considering the nxn matrix ({^(A^J^A^H)

(2-15)

' fc=1

and the vectors x;- = [ai^)ql_l

6

9

C , 1 < j < n and noting that (xj,Xfc)oi = J^ mjat,k, e=i so that we have by (2.11), (2.12), (2.14) and (2.15) *

II r

six

ftdt

we can apply Theorem 11,

< K G | | £ | | ( e ) 2 max ||x,-|||, l<j
KGU\\(&)2

max £

\atJ\2

=

e=i

which implies that (2.13) holds with C =

tfG||£||(9)2

E l/« £=1

«^2 KGU\\(B)

86

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

It follows from Theorem 10 and Lemma 12 that: T h e o r e m 13. Every £ 6 cai%,H)

has an o.s.d.

In this chapter we assumed that H is separable. However, Theorem 13 is true for arbitrary Hilbert space H. It follows from Lemma 12 and the proof of Theorem 10 that, if £ 6 ca(2t, H) has a spectral dilation £ = SEtp, then we can take S and ip to satisfy IISIIIMIa < ^ | | £ | | ( e ) .

3.3. Gramian orthogonally scattered dilations In this section we consider the gramian orthogonally scattered dilation of Xvalued measures. Although every Hilbert space valued (bounded) measures has an o.s.d., not every AT-valued measure has a gramian o.s.d. So we shall give some necessary and sufficient conditions for that. As before let (O, 21) be a measurable space. Definition 1. An ^C-valued measure £ € ca(2l, X) is said to have a gramian or thogonally scattered dilation (g.o.s.d.) if there exist a normal Hilbert B(H)-mod\i\e Y containing X as a closed submodule and an rj € cagos (21, Y) such that £ = PT], where P : Y —> X is the gramian orthogonal projection. The triple {77, Y, P} is also called a g.o.s.d. of £.

Let £ € ca(2l,X) and consider Lg(f2)-valued measures £0 6 ca(2l, Lg(S~2)) for (/> £ H, i.e., £$(•) = (£(•), )H- It follows from Theorem 2.13 that, for each £ H, has an o.s.d. {77^,, ^ , , J^,}, i.e., there exist a Hilbert space &$ a n d ^ G coos(91,&$) such that £0 = J^rjj,, where J^ : 8$ —*• Z/g(Q) is the orthogonal projection. If d i m # = 9 < 00, then, letting {(/>i,... ,<j>q} be a CONS in # , we put M.~ ® ^ ^ = ( ^ j )?=i

and

^ — © Jj ■ S o

we see t h a t

{??> ^ . ^ }

is a

g-o.s.d. of f. If dim H =

3=1

co, we have to work a bit more. Let £ e ca(%X). L°(Q;B{H)) denotes the set of all S(#)-valued 2t-simple functions on 0 . The sup morm H^H^ of $ 6 L°(Q;B(H)) is defined by p H ^ = s u p | | $ ( i ) | | . The inie^ra/ 0/ $ S L°(Q;B{H)) w.r.t. £ over A e 21 is defined by tee »

n

/ $ ^ = ^0^(^1-1^), •IA

•_,

(3.1)

,

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

where $ = £ ajlAf,

ax,...

87

,o„ e £ ( # ) and { A t , . . . , An} g 11(9). It holds that

for A 6 21 and $ 6 L ° ( 6 ; 5 ( H ) )

/,

< ll*iUKIU(A),

$d£

III'

|£||0(A) = sup{

/ $d£

$GZ,°(e;B(iJ)).p>|!co
LetFGco(a,T+(fl')) a n d $ = £ a j l

A j

,*=

j=i

E M B , e L°(Q ; B(H)).

Then

fc=i

the integral of (<E\\&) ro.r.t F o?;er A £ 21 is denned by „

7n

n

/ $ dFtf*= Y, Y. aJF(Aj n Sfc n A)b£.

(3.2)

We can give some necessary and sufficient conditions of g.o.s.d. T h e o r e m 2. Let £ € ca(2t,X). TTien the following statements (1) £ has a g.o.s.d.

are equivalent:

(2) T/iere exists an F e c a ( 2 t , T + ( / / ) ) swc/i tfcai / $ d £ , / $ d £ < [ <S>dF
(3.3)

$eL°(0;fl(fl)).

of £.

(3) For .some CONS {4>k}'kLi *« # ttere exists a family {i}f.,Rk, Jjjfcgpj o/ o.s.d. 's oo

o/ {£0fc}fceN sacft that Yl \\Vk{Q)\\2fik < oo, ||-||AJ, teiwo *^e norm in £fc /or A; 6 N. fc=i

Proo/. (1) => (2): Let {v,Y,P}

be a g.o.s.d. of £ and put F = Fv g ca(21, T + ( H ) ) ,

i.e., F ( - ) = [i7(-),»7(0]y Then we have for $ = f ) a i l A . g L°(0;B(H))

that

j=i

P [ $drt= J/ ee

j $ d(P7?) = /" $ d£ J9 Je e J e J

since P commutes with the module action (cf. Proposition II.2.4) and hence

f *d£, /$rf£

Je

Je

P j <&dri,P [ <$>dii

Je

Je

(3.4)

88

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

by Proposition II.2.4,

$dT]

Je

L

Je

$dF$*,

by a simple computation (cf. (3.1) and (3.2)), i.e., (3.3) holds. (2) => (1): This part is similar to that of (5) => (1) of Theorem 2.10. Suppose (2) holds and define M : 2t x 21 -»■ T ( # ) by M(J4,B) = f ( J 4 n 5 ) - M £ ( A B ) ,

4 , B e 21,

where M € ( A , S ) = [£(A),£(B)] for A, B e 21. By (3.3) we see that M is a p.d.k. and hence by Theorem II.4.13 there exist the r.k. normal Hilbert B(H)-modu\e XM and a C € ca{%XM) such that M = M0 i.e., M(A,B) = [((A),((B)]X for A,B e 21. P u t y = X © X M , the direct sum, and let F : F —> X = X © {0} be the gramian orthogonal projection.

Now we define 77 e ca(2l, K) by T?(J4) =

( £ ( J 4 ) , C(J4)) for

yl e 21. Then we see that 77 is g.o.s. and £ = P77, i.e., {77, V, B} is a g.o.s.d. of £. (1) => (3): Suppose that £ has a g.o.s.d. {77, F, P} and write V = H ® if, where iC is a Hilbert space. Then, for each <j> € H, {T],p,K,J} is a o.s.d. of £4,, where % ( ' ) — ( 7 ?(')i^) i f a n 

V(Q)\\2Y = E IIW e ).**)X = E H^(0)H2iCfc=l

fc=l

(3) => (1): Suppose that, for some CONS {0fc}£°=i in H, {r]k,&k, Jk}km

is a

00

family of o.s.d.'s of { f ^ U e N such that J2 ll»7*(©)IIL < 00. Consider the direct fc=i 00

sum Hilbert space £ = ® M.k and put fc=i 00

v{-) = E ^ &%(■)• Then, we see that 77 e ca(2l, H®8) by the natural embedding of £fc into 8. forfce N. Moreover, 77 is g.o.s. since for disjoint A,B e 21 it holds that

[u(A),i?(B)]

^^®77,(J4),^^fc®77fc(B) L

7=l

fc=l

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

89

oo

because TJ^S are o.s. Now define P : H ®R^

X = H Ljj(ft) by

P=i®(§4 and observe that since JkVk = £<#>, for A: e N oo

oo

k=l

fc=l

Therefore {n, H ® &,P} is a g.o.s.d. of £. Now we want to obtain other characterizations of g.o.s.d. which are more compre hensive and applicable. To do this we are going to take a detour to consider a more general case, i.e., the quasi-isometric dilation of B(H,K)-val\ied measures, where K is a Hilbert space and B(H1K) is the Banach space of all bounded linear operators from H into K. We begin with some preliminary results. Let X be a normed linear space and X* be its dual. The duality pair of X and X' is denoted by (x,x*) = x*(x) for x 6 X and x* e X*. /a(2t,5E) denotes the set of all X-valued f.a. measures £ on 21 with finite total semivariation ||£||(0) < oo. It can be seen that (/a(2l, X), ||-||(0)) is a Banach space. L°(Q;X) denotes the set of all X-valued 2l-simple functions on Q. n

For ip = J2 xj^A, £ L°(Q;X), where x%,... ,zn 6 X and {Au ... , An) 6 11(9), i=i define the sup norm by ||v>||oo = SU P HvWIU- If £ € /a(2l, X*), then the integral of tee
ml* = sup- Je

defa(%X'),U\\(Q)
("*" for dual). Then we have the following. L e m m a 3. Let X be a normed linear space. Then:

(3.5)

90

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

(1) (L°(Q ; X), || ■ ||,) is a normed linear space. (2) ( L ° ( 0 ; X ) , | | ■ | | , ) * and (/a(2t,X*), || ■ ||(0)) are isometncally isomorphic as Banach spaces, written as ( L ° ( 0 ; X ) , | | ■ ||»)* ~ (/a(2l, X*), || ■ ||(©)), where the isometric isomorphism U : /a(2l, X*) —» L ° ( 0 ; X)* is gzuen 6?/ ( t / 0 ( v ) = (¥>

Je

Proof. (1) For p = £ x 3 l A j € L ° ( 9 ; X ) and £ e /a(2l,X*) with ||£||(9) < 1 it holds that

/Vdo = f;<^,€(^)> J e

<£IMUII«^)IU3=1

j=l

<^iiviuneiic©) <«i]v]uHence \\tp\\* < n||v>j|oo < oo. That ||-||, is a norm is clear. Thus ( L ° ( 0 ; X ) , ||-||«) is a normed linear space. (2) Let ( € / a ( 2 t , £ * ) . Then, £ denned by

./e is a linear functional on L ° ( 0 ;X). Moreover, it is bounded since

|
d£

lieil(e)

/ >

< ||^|U||^||(0),

^eL°(9;X),

so that Hill < ||f ||(9). n

Conversely, let p 6 L ° ( 9 ; X)*. First note that for

Q j , . . . , a „ 6 C and {^Ij,... , ,4^} € fl(0) we have

\\
-•M"s

[ {
Je

l|-r|ix

^ a ^ ^ j

:ZeM%X*),

||£||(0) < 1

3=1

< ||a;||imax{|ai|,... , |a„|}

(3-6)

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

91

Fix A 6 21 and observe that for x E X \(xlA,p)\<\\p\\\\xlA\\,<\\p\\\\xU by (3.6) above. This implies that (xlA,p) hence there is a £(A) e X" such that (xlA,p) Obviously £ e fa(%X*)

= (x,Z(A)),

xeX.

and

||£||(9) = sup { (x, £> A £(A)

: \aA\ < 1, A 6 it e 11(6), ||ar||i < 1

Q

sup

\Y1

< sup '

[(
Je = sup{\(tp,p)\

is a bounded linear functional on X and

AZ,A;

: \aA\ < 1, A e n e Yl(&), \\x\\x < 1

ip E L ° ( 0 ; X ) , \\
: tp e L°(Q;X),

y\\t

by (3.5) and (3.6),

< 1} = ||p||.

Consequently ||e|l(6) = [|p||. We should remark that if X = C, then H/H^ = | | / | | , for / e L ° ( 0 ) (cf. (2.9)). If X= V {q e N), then H-H^ and ||-||» are equivalent norms in L ° ( 0 ; C ) . In fact, it holds that

^iG(8;C«).

M|oo
If dim X = co, these two norms are not equivalent in general. Definition 4. Let A" be a Hilbert space and consider the Banach space Let £ e fa(X,B(H,K))

and
B(H,K).

Then the mtejro/ o/ y>

iu.r.£. * is defined by

/ d^ = ^ a ^ ) 0 r Define an operator 5 ? : L ° ( 0 ; H) -> K by S ^ = /* d£(p,

Je

ip

An easy consequence of the above definition is:

eL°{Q;H).

(3.7)

92

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

L e m m a 5. Let £ € fa(%B{H,K)). Then, the operator 5 ? : ( L ° ( 9 ; # ) , | | ■ ||„) if defined by (3.7) is bounded and it holds that \\S^\\ = ||£||(9) and

L

\SM\K

Proof. Let ip — j ^ 0 j l x

d£

K

£ £ ° ( 9 ; # ) and observe that

3=1

E ^)^

llsHk

sup IIV-lk
= if

E?(^J>*

sup M/c
3=1

sup na-)^ll(e) IMk
Hence ||5 4 || < ||£||(9). To show the converse inequality, let e > 0 be given. Again by Lemma 1.6 there exist cf> e H with \\4>\\H < L a i > - - - > a n £ C with \ctj\ < 1 for 1 < j < n and {Ai,. .. , A n } e 11(0) such that

neii(9) <

+ e.

If we define tp = Yl aj4>^A4, then we see that \\
|5^k

Y. Q>f(Ai)* j=l

K

>||f||(e)-e.

Thus | | 5 ? | | > ||£||(9) as desired. D e f i n i t i o n 6. (1) £ 6 fa(%,B(H,K)) is said to be weakly countably additive (w.c.a.) if (£(•)#) i>)K 6 ca(2l, C) for <> / 6 H and V e A', i.e., c.a. in the weak operator topology, wcafa, B{H, K)) denotes the set of all w.c.a. measures in /a(2l, B(H, A')). (2) By a f.a. spectral measure in H we mean an E £ fa(%,B{H)) whose values are orthogonal projections and which satisfies that E(Q) = 1, the identity operator on H, and E(A)E(B) = E(A n B) for A, B e 21. If £ is w.c.a., we call £ a w.c.a. spectral measure or simply a spectral measure in H.

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

93

(3) £ € /o(2l, B(H,K)) is said to have a. f.a. (resp. w.c.a.) spectral dilation if there exist a Hilbert space &, a f.a. (resp. w.c.a.) spectral measure E in R and operators R € B(H, fi.) and 5 € B(R, K) such that

i{A) = s-E^fl, In this case the quadruple {S,M,E,R)

v4 e a.

is called a f.a. (resp. w.c.a.) spectral dilation

oft(4) £ g fa{^,B(H,K)) is said to be quasi-isometric + F e fa(%B (H)) such that

f(fln(A) = F(iinB),

(q.i.) if there exists an

i4,flea,

where £ + ( # ) = {a € B ( # ) : a > 0}. (5) £ 6 / a ( 2 t , B(H,K)) (resp. wca(%,B(H, K))) is said to have a quasi-isometric dilation if there exist a Hilbert space £, a q.i. measure 77 6 fa[<2i,B(H,K)) (resp. wca(2l, £?(//, £ ) ) ) and an isometry J 6 B(K,&) such that £(A) = J*T,(A),

Ae%.

(6) £ € fa(%,B(H,K)) is said to have a f.a. (resp. w.c.a.) 2-majorant F if i*1 6 / a (21, £ + ( # ) ) (resp. tuca(21,£+(#))) is such that for any n e N, $ 1 , . . . ,<£„ 6 H

and {Ai,... ,^ n } en(6) n

£ ^)*i i=l

2 K

n

< E CWfc, ^)„-

(3-8)

j=l

Then we have the following theorem. T h e o r e m 7. Let £ € fa{%,B{H,K)) (resp. wca{%,B(H,K))). Then the following conditions are equivalent: (1) £ /ias a f.a. (resp. w.c.a.) 2-majorant. (2) £ /ias a / . a . ('resp. w.c.a.) q.i. dilation. (3) £ /ias a f.a. (resp. w.c.a.) spectral dilation. (4) £* feas a f.a. (resp. w.c.a.) spectral dilation, where £*(•) = £(-)* : 21 —> £(A\tf). Proof. We only prove the f.a. case since the w.c.a. case is similar. (1) =*> (2): Suppose £ has a f.a. 2-majorant F 6 /a(2l, £ + ( # ) ) and define T :

a x a -> 5(ir) by r(A,B) = F(i4nB)-e(4)*f(J0),

A,Bd%.

94

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

Then F is a 5(H)-valued p.d.k. in the sense of Definition II.4.F From Proposition II.4.23 we get the RKHS 9) of T (cf. the note following the proof of the proposition). Foe each A 6 21, C(A) = T(-, A) can be regarded as an operator from H into 9) and it holds that for 6,6' e H and A, B e 21 (T(A, B)4>, 4>')H = (r(-, B)6, r ( - , A ) ^ ) f l ,

by Definition II.4.22,

= (c(/irc(B)<^v which implies that Y{A,B) fa(%B{H)). Putfi=K®9)

= ({A)*({B). Now it is easily verified that C £ and define 17 € / a ( 2 l , # ( # , £ ) ) by »;(■)= € ( - ) © C ( 0 .

i.e., n(A)4> = (f(A) © ({A))4> = {£{A)) 6 ft for A , B € 21 and Then we have that for A, B £ 21 and 6,6' £ H

€ H.

= (£(A)^(B)^ K +(<(#£, C W ) S = {{aByaA) + r(B,A)}6,6')H = (F(AnB)6,6')H, from which we conclude that r\ is q.i. Moreover, if we define J : if —> .ft by «/?/> = (•0,0) for tp € K, then J is clearly an isometry and J* = J~1Jy\(j)i where J^j) is the orthogonal projection of .ft onto the range £ft( J) of J, which is a closed subspace. So J*/? = £. Therefore (2) holds. (2) => (3): This part is similar to that of (1) => (2) of Theorem 2.10. £ 6 fa(pi,B(H,K)) have a q.i. dilation {?/, ft, J } . Here we can assume that

Let

. « = © o { 7 ? ( A ) 0 : 0 € J t f , .4G21}, where the RHS is the closed subspace generated by the set {rj(A)6 : 6 e H, A £ 21}. For A e 21 let E{A) be the orthogonal projection of .ft onto K(A) = &o{ri(A n 5)<^ : 6 e H, B e 21}. Then it is not hard to check that E(-) is a f.a. spectral measure in ft such that 77(A) = B(A)r,{e), AeK. Therefore £(•) = J*r}{) = J*E(-)T/(Q),

i.e., £ has a spectral dilation.

(3) =>• (1): Suppose that £ has a f.a. spectral dilation {5, ft, E, R}. For any n e N, { A i , . . . , A n } e n ( 0 ) and 6X,... ,6n e H it holds that

£$(A-j)*,i=i

=

< l|5||:

S^EiAjW, j=i

;<

j=i

95

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

\\S\\2Y,(R*E(AJ)WJ,4>3)H-

=

Putting F(-) = \\S\\2R*E{-)R £ fa(%B+{H)), is a 2-majorant of £. (3) 4=> (4) is clear from the definition.

we see tha (3.8) holds and hence F

There is another characterization of (f.a. or w.c.a.) spectral dilation as follows. It is also useful. T h e o r e m 8. Let £ 6 fa(Vi,B(H,K)). Then £ has a f.a. spectral dilation iff there exists a constant C > 0 such that for any m,n £ N, {4>j,k : 1 < J < w, 1 < fc < n} C H and {Au .. . , Am} g 11(6) n

m

£ £W<^

< Csup

Fefa{%B(H)),

||F||(e)
/ / £ «s w.c.a., then £ ftas a w.c.a. spectral dilation iff (3.9) holds. Before we proceed to prove the theorem we need some preparation. The condition (3.9) is similar to (2.7) replacing ||-||oo by ||-||*- To see this let H © K* denote the algebraic tensor product. The greatest crossnorm (or the protective crossnorm) 7 is defined by 7 ($)

=inf I £

UJWHUJWK- ■■<$> = YJ

4>s®1>j> h e ff, Vy e«r*. 1 < i <

3=1

I 3=1

)

for $ g / / 0 fC*. M H ®1 K* denotes the completion of H © K* w.r.t. 7, then it is isomorphic to T{K,H), the Banach space of all trace class operators from K into H, denoted H ® 7 K* ~ T{K,H) (cf. Schatten [1, p. 120] and Takesaki [1, p. 190]). Thus it holds that (H ® 7 K*)* ~ B(H, K), so that it follows from Lemma 3 that (L°{e;H0^K*),\\-\Uy ~ (fa{% B{H,K)),\\\\(0)). More fully, for each p E L°(Q ; H ® 7 K*)* there corresponds a unique £ £ /a(2l, B ( H , K ) ) such that p($) = ($,£)=

[($,<%)=

f tT($d£),

$GL°(9;ff®

For t/Ji, (^2 e L°(Q ; # ) write 771

m

j=l

3=1

7

n

(3.10)

III.

96

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

where fo,... ,tf>m,i/>i,.. ■ , i/>m € H and {Au... ,Am} e 11(9). Then the algebraic tensor product tpi ® Tp2 is a T ( H ) ~ /f ® 7 //'-valued function on 0 defined by m

where as before 4>j®i>j is defined by {(t>j®ipj) = (,ipj)Hjfor £ H- Now sin (if 7 if*)* = T{H)' = B{H) we have by (3.5) and (3.10)

: F e / a ( 2 t , 5 ( / f ) ) , ||*1|(e) < 1

|
sup

3=1

^tr(F(^)(^®^-))

Fefa(%B(H)),

\\F\\(e)

7=1

:F6/a(a,B(H)), ||F||(e)
sup< J=I

Hence (3.9) can be rewritten as

six

rfCv&

(3.11)

"Pk

k=l

for any n 6 N and ^>i> ...,
I ^2
»Vn e L ° ( e ; F ) , J ] fc=i

/ d£^ f c •'e

= l,neN K

Observe that | | $ | | , > — for $ e W and W is a convex set in L°(Q;H

®7 if*).

Then it follows from the Hahn-Banach Theorem that there exist a e R and Fo £ fa(%,B{H)) =L°(G;H®7H*y with ||F O ||(0) = 1 such that Re($,F0) > a > R e ( * , F 0 ) for $ , * e L ° ( 0 ; H®yH*)

with $ e W and | | * | | , < 1 where ( $ , F 0 ) = / Q ( $ , dF 0 )

and Re{- ■ ■ } is the real part. Hence

a>8up{Re(*,F 0 >:||*||.
97

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

= S up{K$,F 0 )| ; ||$|U
there is (3 G C with \(3\ = 1 such that | ( * , F 0 ) | =

Re(/?$,F 0 ) and | | * | | . = ||/7¥||*. This implies that R e ( $ , F 0 ) > - for $ e W. P u t

o Fo(A) + Fo(A)*

F(A)

Ae2t.

Clearly F e /a(2l, B(H)) and ||F||(0) < 1. n

We shall show that F is F + ( # ) - v a l u e d . For $ = J2 Vfc ® ^fc

e

Wwe

nave

fc=i

y ( x ) ^ ® vi»dF)=/_ so that

n

X

„

<$' d F > = R e / <*> di?°) ^ ^

2

/ #¥>k

for any
.

n

<*®? f c ,dF) such that ^

l/g^Vk

(3.12)

»■ > 0- Suppose that

( F ^ ) . / * , ^ = -5 < 0 for some A e a and <> / € # with ||||H = 1. We may assusme that ||^(5)V'll/c = 1 for some B £ 21 and ip e H. Define ipi = (3(j>lA ((3 e C) and ip2 = ipl-B- If || J e ^ VI||K-

=

I I / ^ C ^ M I K > °> t n e n by (3.12) we see that

2

0<

/ #¥>i
a contradiction. Thus || Je d£tpi\\K

= C(F{A)0tP4>) =-C5\(3\2

= 0. Again by (3.12) we see that

/ d(,Vi + d£ip2 ,P4>)H + C(F(B)i>,ip)H = -C5\(3\2 + C ( F ( B ) V , V ) H < 0

1 =

for large enough 0 e K, a contradiction. Consequently F is B+(H)-valued. More over, (3.11) holds for any ipi,... ,
Vi — X] ^jl/4'i ^ holds that 3=1

E tf^to

K

/" dfVi
98

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

= cY,{F{Aj)jA3)H3=1

Therefore, CF is a 2-majorant of £ (cf. (3.7)). The w.c.a. case is easily verified. In order to obtain our main theorem (Theorem 15) in this section we have to work a little more. Recall that X = S(L§(ft), i i ) by the identification x = T*, where Tx e S{H,L2Q(n)) is defined by Txcj> = (<j>,x)H e L§(ft). L e m m a 9. Let £ € ca{%X) and % } <E ca(2l, S{H, Lg(fi))) be defined by %) = (0>^('))jf / o r 4> S -ff- / / 2£(-) ^ a 5 a W-co. spectral dilation, then £ has a g.o.s.d. Proof. For the sake of simplicity we write K = L§(fi). By hypothesis there exist a Hilbert space &, a w.c.a. spectral measure E(-) in R and operators 7? 6 B(H,&) and 5 € i?(-ft, K) such that 2 V \ = SE(-)R. Here 5* can be taken as an isometry by Theorem 7. If we identify S*(K) C £ with if, then S :R-^ K

~S"{K)

can be regarded as an orthogonal projection and £(•) = R*E(-)S*. Since dim if = oo and £ is Hilbert-Schmidt class operator valued, R* must be of Hilbert-Schmidt class. Put r/(-) = R*E(-), then n e capo.s(2l, S ( £ , if)) since for A, 5 e 21 it holds that [ri(A),r,(B)]=T,{A)T,(By

=

R*E{AnB)R.

Put P = 1® S* : H ® Si^ H ® K = X, the gramian orthogonal projection. Then £(.) = Pr ? (.) = T?(-)5*, i.e., £ has a g.o.s.d.

L e m m a 10. Let K be a complex Hilbert space. Then there exists a constant C > 0 such that for any n,m e N, x i , . .. , x m e C™ and ,„ 6 A' ^

(x J -,x fc ) C "(^-,(/'fc)K < C /

^sgn(x,x;)Cn||x||1I>^

B

j,fc=l

"

M n (dx),

(3.13)

7=1

w/iere £?n is £/ie unit sphere of C™, / i n is i/ie rotationally invariant measure on Bn ct

and, for a € C, s g n a = •,—r i / a ^ 0 and 0 otherwise. \a\ Proof. For u, v e C it holds that

/ .B

u,x)t> ( w . x ) ^ , nn(dx)

1 = -(u,w)c« n

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

99

||w||c» / (u,x) C nSgn(w,x) c „ /x n (dx) = — — - ^ - ( u , w)j> , JBn 2i [n + j ) where T(-) is the usual gamma function. This can be seen by integration in polar coordinates. Consider the Hilbert space L2(Bn,(in) and define operators Pn and Q„ on 2 L {Bn,(in) by (P„/)(x) = n / (<3„/)(x)=/

/ 6 L2{Bn,(xn),

/(y)(x,y)c./in(dy),

f E L2(Bn,

/(y)sgn(x,y)CnMn(dy),

fin).

(3.14) (3.15)

■/B„

Then we can show that (QlfJh,^

> Ct(Pnf,f)2^

L2{Bnifin),

f e

1

/— r / ^

where Cn = ——. p - and (■, -)2,/t„ is the inner product in L2(Bn, (in). 21 (n + 2) Assume that K is separable and let {k}'^Li be a CONS in K. Consider the Hilbert space Y = L2(Bn, (i„ ; K) of all K-valued strong random variables which are /i„-square integrable and define operators Pn and Qn on Y in the same manner as in (3.14) and (3.15), respectively, where / € L2(Bn,fin) is replaced by ip E Y. Then we can verify that (QinCZ(Pn
veY

(3.16)

2

by using tp(x) = Y, (
\K).

fc=i

Let n,m

E N, x ; E C" and <j>j E K,\

< j < m be fixed. We can assume that

||x,||c" = 1 for 1 < j < m since otherwise we may consider x'- = -— 3 -— and [Ixjllo (j)'. = ||xj||j>j, 1 < j < m. Since we shall work on K0 = 60{<j>i,... ,m} C K, we can actually assume that K is separable. Now, for k E N and j = 1 , . . . , m, let ^fc(x) = { y e 5 „ : ||y - x||Cn < - J, / f c ) i (x) =

/^

(x

N ) 1 A . ( X J ) ( X ) , and define

m

v?fc(x) = 53/ f c ,j(x)^j,

fe

e N.

Then y?fc € F for A: e N and, by applying (3.16) to
(x,y)c„/fcj(y)/fc,«(x)/in(dy)/^l((ix)(^i,^)K

<—p^Y\{jAt)KJ nC

n

^

/

/

sgn(x,u)c~sgn(x,y)Cl/fcj(y)/fc]£(u)

JBJBJB«

/x„(dx)/i n (dy)/i 7 l (du).

100

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

If g is a C-valued bounded measurable function on Bn and is continuous at Xj, j — 1,.. . , m, then g(x,-) = lim /

g(x)fkJ(x)nn{dx),

l<j<m.

Since g(-) = sgn(x, -)c* is continuous at x^, 1 < j < m for p,„-a.e. x g £?„, it follows from the Bounded Convergence Theorem that Tfl

E

771

(xj,xe)c»(4>j,4>k)K

< -7^2 E n

jf.fcl

r.

/

sgn(x,x £ )c.sgn(x,Xj) c „(i/>j, (/><)/£• p„(dx)

j,£=l

I

r

—y

2

m

x^*-^^

/j„(dx)

1 4 4 Since ——5- —> — (n —> 00), we conclude that (3.13) holds with C = —. In the above lemma, if we consider the real Hilbert space K and R n instead of 7T TV C™, then it is known that C = —. Hence, in either case we can take C = —. 2 2 Absolutely 2-summing operators are of special interest and we need some results on them. At first Pietsch Factorization Theorem is stated as follows (cf. Diestel, Jarchow and Tonge [1, pp. 45-47], Jameson [1, p. 60], Pietsch [3, p. 58] and Wojtaszczyk [1, P-203]): T h e o r e m 11 ( P i e t s c h Factorization T h e o r e m ) . Let X and 2J be Banach spaces and T g Il2(3C, 2J). Then there exist a Hilbert space F) and operators Ti g Y[2(X,Sj) and T2 € B(£,2J) such that T = T2TX with n2{T) = TT2(TI) and \\T2\\ = 1. Note that the absolutely 2-summing norm of T 6 B(3E,2J) can be written as TT 2 (T) = sup U

f ] \\Txk\%J

where, for a sequence x = {xk}kxL1

: e2{{Xk}^=1)

<

y ||, !|(

.

l|,

C X,

e 2 (x)=SUp|(^f;|x*(x f c )|

2

T h e o r e m 12. Let X be a normed linear space and 9) be a Hilbert space. Consider an operator T g B(f),X). For an ON sequence e = {ek}'k-1 in S) define \\T\\e by 00

E nTefciix

fc=i

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

101

Then, T 6 n 2 ( f | , X ) iff \\T\\e < oo for any ON sequence e in 9). In this case it is true that 7T2(T) = sup {||T|| e : e is an ON sequence in f)j. (3-17) Proof. Assume that T <= U2{9),X). Since for an ON sequence e = {ek}kKLl in f) we have e 2 (e) = 1 and hence 7r2(T) > sup ||T|| e . e

Conversely, suppose that ||T|| e < oo for any ON sequence e in ^ . Let $ = {^fclitLi Q ^ be such that c-2($) < 1 and f) 0 = &o{4>k ■' k £ N}, the closed subspace of f) spanned by $ . Take a CONS e = { e f c } ^ in f)o, being clearly separable, and write each 4>k as 00

4>k = ^2 ak,tet,

keN

£=1

for some {ctkie}'^=1 Q C. Then there exists an operator S 6 -B(-fJ) such that Sek = 4>k for k e N and | | 5 | | = e 2 ($) < 1. In fact, define 5 on f) by

Then we have that 1 > e 2 ( $ ) = sup<j f 52|(«^fe,^)j)

< 1

00

sup

^(^,^)flQfc

= Uplift < 1, £

fc=l

l»fc|2 < 1

fc=i

00 (

s u p ■ X ] ( Pk,ip)f,('P,ek)f, fc=i OO

sup

y

OO

is.imis

v

5 ^ ( ^2oik,eet,ip

j {4>,ek)f,

fc=i ^ £=1

'«

00

< 1

ft. HVIIft < 1

00

< 1

sup ■

implying the well-definedness of 5, OO

y

OO

sup fc=l

^£=1

s.imis < 1

102

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

= sup {|(50, V) a |:|Wla,|HI*
(3.18)

ke

-ejj,

In fact, by the polar decomposition of the operator S obtained above we can find a unitary operator U € B(9)0) and a self-adjoint operator Q 6 B(9)Q) with ||Q|| < 1 such that 5 = UQ. Putting V = Q + i(l-Q)$ (i = i/-L), it is seen that V e B(f) 0 ) V + V* is unitary and Q = ■ Furthermore, putting Wj = UV and W2 = UV , we see that W\ and W 2 are unitary on JOo and hence isometries on 9), and satisfy (3.18). Now put a = sup||T|| e . It suffices to prove that 7r2(T) < a since the opposite e

inequality was noted at the first stage of the proof. If Sj0 is a closed subspace of 9), U : f)o —>■ 9) is an isometry and e = {efcj-^Lj C i^o is an ONS in Jo, then ||Tl/|| e = | | r | | ( / e , where Ue = {Uek}'jfL1 is an ONS in 9). Hence, for any isometries WuWa :9)0^9)we have \\T(W1+W2)\\e < ||'TI^ 1 || e +||TH^ 2 || e = | | T | | W i e + | | T | | W a e . Let $ = {<j>k}kLi C 9) be such that £ 2 ($) < 1 and find two isometries Wx,W2 : f,0 = 6 o {0 f c : fc e N} -> £ and an ONS e = H } ^ in f) 0 such that (3.18) holds. Then we have:

Hm 'kill

£

r(W! + w2;

T{W! + w2) 2

by assumption.

efc

fc=l

fc=l

It follows that T 6 Yl2(9),X)

\\T\

\\T\\w> "

2

W2e

<

OO

and 7r2(T) < s u p | | T | | e .

Therefore

(3.17) holds. P r o p o s i t i o n 13. Let X be a normed linear space, 9) be a Hilbert space and T G B(X,9)). Then, T admits a factorization through a Hilbert-Schmidt class operator iff
7n frazs case it holds that a ( T ) = sup a e ( T ) = 7 r 2 ( T )

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

103

and there exist a Hilbert space A, Tx E 5 ( J , . S ) with \\Ti\\ = 1 and a Hilbert-Schmidt class operator T2 e S(R,fi) such that T - T2TX and \\T2\\a = a(T). Proof. Assume that ae(T) < oo for every ON sequence e = {ek}^=1 that oo

in S) and observe

oo

a e ( T ) 2 = J2 sup{|(T:r, e f c ) f l | 2 : ||ar|U < l } = £

\\T'ek\\\.

= ||T'||e.

fc=i fc=i

Taking the supremum w.r.t. e in both sides of the above equality and applying Theo rem 12 give us 7r2(T*) = o(T) < oo. By Pietsch Facorization Theorem (Theorem 11) (cf. also Remark 2.8 (3)) there exist a Hilbert spcace f)i, a Hilbert-Schxnidt class op erator S i e S(Sj,SJi) and a bounded operator S 2 € B(f)i,X*) such that T* = S2Si, 7r 2(5'i) = ||5i||CT = TT2(T*) = o~(T) and ||S 2 || = 1- Passing to the dual relation and restricting to X C X " , we obtain T = S\S\ on % ||52*|| = 1 and \S\\B = a{T). Since 7r2(T) > cr(T), we have 7r2(T) = a(T). Thus we get the desired factorization. The converse is almost obvious. P r o p o s i t i o n 14. Let £ e fa(%B(H,K)) (resp. wcaC2k,B{H,K))) and consider the operator S j : (.L (&;H),\\ ■ ||») —> K defined by (3.7). / / S^ is absolutely 2-summing, then £ has a f.a. (resp. w.c.a.) spectral dilation. Proof. We only prove the f.a. case. Since S^ : L°(Q;H) —> K is absolutely 2summing there exists a constant C > 0 such that for any n 6 N and ifix,... ,
L°(e;H) E l l % > f c l L * < C s u p j ] C /(Wfc.dO : C e / a ( 2 t , / / ) , HCII(O) < l l . fc=i I fe=i Je J Let C € fa{%H) with ||C||(6) < 1. Since B(C,H) ~ H we can apply Theorem 7 to obtain a f.a. spectral dilation of £, i.e., £(■) = J*E(-)ip where E(-) is a f.a. spectral measure in some Hilbert space 8., J € B(H, 8.) is an isometry and ip g M.. It follows from the remark after Theorem 2.13 that \\ip\\& < Co, CQ being a constant ( < — ) . Define F{) = J*E{)J.

Then F E fa(%,B+{H))

|F||(e)=supj {

^QAJ*E(A)J AGTT

3lA] € L°{S; # ) it holds that

L{ ,dQ v

and ||F||(6) < 1 since | a A | < 1, A € 7T e n ( 6 )

104

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES 2

n

J

J

A

,

Y,{ to> £( i))

since J*J = 1 on H,

3=1

Yd{E{Ai)Hi^)i

since C =

J*Eip,

3=1

£

E(Aj)J3

3=1

< C02^(F(^)^,
since ||V|U < C0,

3=1

Je Consequently, it holds that for (pi,...,ipn

E L°(Q;H)

and C, E }a{&,H)

with

llcil(e) < i n fc=i

.

2

p

JB

n

Je

n fc=i

fc=i

since | | F | | ( 9 ) < 1. Therefore (3.11) holds and we can apply Theorem 8 to obtain the conclusion. Now we are in a position to state and prove our main theorem in this section which completely characterizes the g.o.s.d. of ^"-valued measures and extends the result of Theorem 2. T h e o r e m 15. Let £ 6 ca{%,X) and put ((■) = % ) € ca(%S(H,Ll{il))), C{-)4> = {4>,£(-))H for <[> E H and £*(•) = CO* € c o ( a , 5 ( L g ( n ) , f f ) ) . following statements are equivalent:

i.e., Then the

(1) £ has a g.o.s.d. (2) £ has a T+(H)-valued (3) ||f|| 0 (e) < o c , i.e., (4) \\£U&) < co. (5) For every CONS

2-majorant F E £Ebca(%X).

{(p^l'Li

ca(%,T+{H)).

in H there exists a family

{nk,fi.k, J f c } f c e N of

oo

o.s.d.'s

of {^JfceN such that £

||r? fc (e)||| 4 < oo, where £0fc(-) = (£(■),
ca(2t,L§(ft)) for k E N. (6) For some CONS { ( M ^ in H there exists a family {r)k, &k, Jk}keN oo

of {^JfceN such that £

fc=i

ll%(0)lli t < °°-

of o.s.d.'s

105

3.3. GRAMIAN ORTHOGONALLY SCATTERED DILATIONS

(7) E if*.IK©) 2 < °o for every ONS {4>k}T=i ™ Hk=l

(8) The operator Sc : ( i ° ( 0 ; Lg(fi)), ||-||.) -> tf de/med 6y Sc

/a(2t, Lg(f2)) is absolutely 2-summing, defined in (8).

where S^- is

Proof. The equivalence (1) <^> (2) 4=> (6) was shown in Theorem 2. The implications (1) => (5) => (6) and (3) => (4) are obvious. We put K — L§(Q) for simplicity in the proof below. (1) => (3): Let {TJ,Y,P}

be a g.o.s.d. of £. Then by Theorem 1.5(3)

lieiU(e) = \\PT,U&) < ||P|||hiU(e) < \\v(@)\W < °°(4) => (1): Assume that ||f|| s (0) < oo. We shall prove that ( satisfies (3.9) by invoking Theorem 8 and Lemma 9. Let m,n 6 N, {4>j,k '■ 1 < j < "»> 1 < k < n} C i / and {.Ai,... ,y4 m } e 11(0). Choose a CONS {>!,..'. ,i/>r} in H 0 = S0{^j,fc : 1 < j < m, 1 < fc < n}. Then we have that n

n

m

m

£ Ea^-,* = E EwE^.*^)*^

(3.19)

*:=i' j = i m r

E

E

(C^WP-CW^IKE

;,£=lp,<J=l m

r

= E

E

(t,k)M

fc=l

(C(J4J)V'P,C(^)V,?)A:(XJ,P,X£,,)«>J

j,£=lp,9=l

where x J ) P = ((4'j,k,'lPp))k=.1 € C 1 for 1 < j < m and 1 

LHS of (3.19)
^ ^ s g n ( x , x , , p ) 0 l ||x i i P || c -C(^)V' P JB„

r =c

r

Hn (dx)

J=IP=I m

( T E ' ^ H Esgn(x'x^)c"iixJ,piic"'/'p

for some constant C > 0. By putting

ViW = E sgQ(x'xJ'.p)c" II^.PIIOV'P, P=I

x € C"

Mn(dx) K

106

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

for 1 < j < m , we see that

|^-(x)||2H < Y,IIX^II?, = E E K ^ - ^ I p=l m

2

= E II^MIIIT,

p=lfc=l

for 1 < j < m2 and hence

xGe

fc=l

n

5 ] C(^)^(x)

< ||£|| 3 (9)

2

||^, fc || 2 H

max £

,

x e

C

Consequently it holds that LHS of (3.19) < C | | ^ | | s(e)

n

2

max Y ) ||Mla fc=l

Kj<m

*—*

j=ifc=i

where j 0 satisfies that max Y^ ll$i,klllr that 5 A ( A j J = 1 and §io(Aj0)

=

X) ll^io.fclla

an(

^ ^jo

e

ca(2l,R+) is such

= 0. Thus

LHS of (3.19) =ifc=i

F e fa(%B{H)),

\\F||(9)

< 1

Therefore (3.9) holds. (5) => (7): Let {Vfc}^i be an ON sequence in H and {(pk}^ be a CONS in H including {^fc/fcLi- Let {%, itfc, Jfc}fc€N be a family of o.s.d.'s of { ^ t } f c e N such that oo

£

11^(9)11^ < oo, which exists by assumption. Then, for each k € N

fc=i

llf*.IK©) = l|J*%||(6) < ||J f c ||||%||(9) < | k ( 9 ) | k , so that we have OO

CO

2

oo

2

E H€**IK©) S E IK*J(©) < Yl h*m\%. < oo. fc=i

fc=i

fc=i

3.4. THE SPACES Ll(F)

AND L2(F)

107

(7) => (9): Suppose (7) holds. Then, for any ON sequence e = {(pk}k
= J2^P{\(^S^k)\2:V,eL°(e;K),

\\
fc=i oo

= 5 2 sup {\(Sctp,4>k)H\2 ■

(3.20)

^eL 0 (e;K), |M|, <1

S sup fc=l

V ) sup^ k=i I

/ (p,dC
: v eL°(e;if), W , <1

< £ sup {MSllCWkIKe)3 : V e L°(0;^), IMI. < 1} oo

«Eii^Ji( e ) a
by assumption. Thus by Theorem 12 ST. is absolutely 2-summing. (9) => (8): For any ON sequence e = { (1): Since S c - : (L°{B;K), ||-||») -> H is absolutely 2-summing it follows from Proposition 14 that £* has a w.c.a. spectral dilation. Then Theorem 7 implies that C has a w.c.a. spectral dilation, so that Lemma 9 concludes the proof.

3.4. T h e s p a c e s L^F)

and

L2{F)

In this section we construct an L 1 -space LX(F) for a T(H)-valued measure F of bounded variation consisting of operator valued functions. If, in particular, the measure F is T+{H)-valued, then we can construct an L 2 -space L2(F) which turns out to be a normal Hilbert 5(H)-module.

108

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

Let (6,21) be a measurable space and F 6 vca(%T(H)). Since H is assumed to be separable, T(H) is a separable dual space. According to Diestel and Uhl [1, pp. 218-219] T(H) has the RN- (= Radon-Nikodym) property. Hence, putting u(-) = |F|(-), the variation of F, F has an RN-derivative F'v = — G L1 ( 9 , v; T ( # ) ) , the Banach space of all T(H)-valued Bochner integrable functions $ on 0 with the norm 11*111 = /© \\^{t)\\rv{dt) < oo. Thus, we have F(A)=

I Fl{t)u{dt), JA

Ae%.

We need measurability concepts which are measure free and slightly different from those in Hille and Phillips [1, 3.5] and Diestel and Uhl [1, p. 41], Definition 1. (1) As before let us denote by L ° ( 6 ; H) the set of all //-valued 21simple functions on 0 . An //-valued function 0 on 0 is said to be strongly measurable if there exists a sequence {$n}£Li £ L°(Q;H) such that \\4>n{t) — 4>{t)\\H —> 0 for every t € 0 . <^> is said to be weakly measurable if the scalar valued function [4>{')i'/')« is 2l-measurable for every ip £ H. Since H is supposed to be separable, it holds that is strongly measurable iff it is weakly measurable. A B(//)-valued function $ on 0 is said to be ^.-measurable if it is strongly mea surable, i.e., for every (f> 6 H the //-valued function $(■),ip)H is 2l-measurable for every ,xp 6 H. Denote by O(H) the set of all linear operators a with domains 2)(a) C H and ranges SK(a) C H. An O(H)-valued function $ on 0 is said to be %-measurable if there exists a sequence {^n}'^L1 of F(//)-valued 2l-measurable functions on 0 such that ||*„(t)0-*(t)tf||H-*O (n->oo) for every t G 0 and 0 g 3)($(«)). (2) Let F € vca(%,T(H))

dF with the RN-derivative F' = — e / ^ ( Q , i/;

T(H)),

where i/ — \F\ is the variation of F. An 0(//)-valued 2l-measurable function $ on G is said to be F-integrable if $ F ' € Ll(@, v ; T(Z/)). In this case, the integral of $ w.r.t. F over A 6 21 is defined by / $ d F = / $ F ' di/. Let us put L^F)

= { $ : G -> 0 ( H ) , 2l-measurable and F-integrable}.

3.4. THE SPACES Ll(F)

AND

L2(F)

109

We identify $ , * <E L X (F) if $ f =
||$il 1)F =||$F'||x= [ W^Wrdv.

(4.1)

Je

For $ € L1{F)

put F$(-) = J.^dF.

Then we see that F * 6

vca(%T(H)).

Now the question is: Is L 1 ( F ) complete? Of course the answer is yes. To prove this we need some preparation. For a E B(H) we put \a\ = (a*a)s, the absolute value of o, and define U>Q on the range £H.(jo,]) of \a\ by wo(|a|^>) = aip for 4> E H. Then it can be seen that w0 is isometric on 5K(|a|) and can be extended uniquely to a partial isometry w with the initial set iH(|a|) and the final set !H(a) and that a — w\a\, called the polar decomposition of a (cf. Schatten [2, pp. 4-5]). For a closed subspace Hi of H denote by JHY the orthogonal projection of H onto H1. For a E B(H) denote by 71(a) the null space of a. L e m m a 2. Let $(•) and $(•) be B(H)-valued (1) $ * , $ + * and $\t are ^-measurable. (2) / / $ is B+(H)-valued, (3) | $ | is ^.-measurable.

then $? is

%-measurable functions on 0 .

Then:

^-measurable.

Proof. (1) is almost obvious. (2) There exists a sequence { p n } ^ = 1 of polynomials such that {p n ($(i))}^L 1 con verges strongly to $(£)? for every t E 0 (cf. Riesz and Sz.-Nagy [1, pp. 263-264]). Thus $2 is 3-measurable. (3) follows from (1) and (2). Note that O(H) is a linear space if we define, for a, 6 6 0(H) and a E C, 33(aa) = 53(a) and 5)(a + b) = 53(a) n 53(b). Moreover, 0 ( # ) is a B(H)-bimodule if we define, for c E B(-ff), 53(ac) = {

j> E 53(a)} and 53(ca) = 53(a). Then the following lemma is easily verified: L e m m a 3 . Let $(•) and <&(-) 6e 0(H)-valued %-measurable functions on 0 and a(-) be a B(H)-valued ^-measurable function on 0 . Then, $ + * , a $ and $ a are 0(H)-valued ^.-measurable functions on 0 . D e f i n i t i o n 4 . The generalized inverse a~ of an operator a E B(H) is defined by

a~ = J ^ j i a

Jm{ay

where a - 1 is the (multivalued) inverse relation to a. Note that a~ is densely defined. In fact, 53(a") = !R(a) + f R ( a ) x = {

E IR(a), V £ ^ ( a ) X } , which is a direct

110

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

sum. The well-definedness of the generalized inverse can be seen as follows. Let € S)(a~) and write it as <j> = <j>i + 4>2 with 1 e 9\(a) and <j>2 £ IH(a)-1. 0i = aip for some ip e H and a " 1 ^ ^ ^ ^ = a^^atp) = {4>' e H : atp' = atl>}. If ipui>2 6 91(a)-1 and atpj = ai> for j = 1, 2, then a{ipi - if>i) = 0, implying that Vi - V'2 e 91(a) n 91(a)-1-. Hence Vi = i s well-defined for every <j> £ 2 ) ( a - ) . Let C(i?) be the space of all compact (= completely continuous) operators on H. Each operator a e C+(H) = {b e C(H) : b > 0} can be written as

a=J2XJEJ

= '^2akk®~k'

3=1

(4-2)

fc=l

where, in the first expression, Aj's are distinct nonnegative eigenvalues of a such that Xi > A2 > • ■ • > 0 and Ej's are orthogonal projections on H onto the eigenspaces corresponding to A^-'s, and, in the second, cafe's are nonnegative eigenvalues of a counted as often as their multiplicities such that a\ > a2 > ■ ■ ■ > 0 and 0fe's the corresponding ON sequence of eigenvectors. Moreover, if Xj > 0, then 9t(£j) oo

is finite dimensional, and Y, Ej = 1, the identity operator on H.

If a 6

C{H),

; =1

then by the polar decomposition a = w\a\, where w is a partial isometry, we have oo

oo

__

a = Yl XJWEJ = Y ctk{wk) ® (jik if |o| is expressed as in (4.2). j=i

fc=i

Then we have the following lemma. L e m m a 5. Let a € B(H). (1) / / a* = a, then a" =

Then: J^-)a~1J=r^j

(2) a " a = J
(3) The closure of aa~ = J-^r\ ■ (4) a " =

(a*a)-a*. OO

(5) If a=

Y

X E

3 3 € C + ( i 7 ) , then a~

Y A J?J, tuaere A" = A" 1 if X ^ 0

3=1

and = 0 otherwise. Proof. (1), (2), (3) and (5) are easily verified. (4) We first show that £>(a~) = £ ( ( a * a ) - a * ) . Note that D ( ( a * a ) - a * ) = {0 e H : a*0 6 3D((a*o)~) = 91(a*a) + ^ ( a ' a ) - 1 } = { E H : a*(j> € 9
3.4. THE SPACES Ll(F) AND L2(F)

111

since lK(a*a)-L = ^{0*0) = 91(a). If <j> E 3)(a~), then 0 = a0i + 0 2 for some 4>i £ H and (/>2 e D^a) 1 - = = a*a0! £ V\{a*a). Thus 0 e JD((a*a) _ a*). Conversely, if 0 e S)(a") c , then 0 = ai + 02 + ^3 for some 4>i E H, 4>2 & W(a*) and (/)3 G {£K(a) U91(a*)} c . Hence, a*0 = a*a0i + a*0 3 g £)((a*a)~) c since a*<j>3 E { E H we have (a*a)"a*(a0) = J ^ ( o . o ) ± ( a * a ) _ 1 < % ^ a * ( a $ ) = •An(a)- L (a*ar 1 "'9i{^)a*a^ = J< n ( a ) j-(a*a) _ 1 (a*a0) = a~a,

by (2),

so that (a"a)~a* = a~ on 9t(o). L e m m a 6. Let a E C+(H) and a% > a2 > ■ ■ • > 0 be its eigenvalues counted as often as their multiplicities. Then it holds that for any n E N

Y^ctj j-l

= max i ^2 (<#i> i>3)H ■ {V'l, • • • , 4'n) is ON in H \. I 3=1 J

(4.3)

Proof. Let {(pjjjeN be the ON set of eigenvectors of a, i.e., acj>j = cxjtpj and {4>j,4>k)H = ^jfc for j,k E N. Let n E N be arbitrary and {?/'i,..- ,i/'n} C i f be any ON set. For each j = I , . . . , n we have that

(aipj,ipj)H

= I ^a f c (0fc,^)/fe,i'3)H(, ^fc=i

£=1

= y^afcH^fc.V'j)//! fc=i OO

/

= an^|(^fc,^i)w| k-1 n

R

+ ( 53+ k—1

< ctn + ^ i a k - an)\(
OO

*.

5 1 ) ( a f c ~ an)\{
,

112

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES oo

2

since {4>k}ken is ON and hence £

|(^,^)H|

 fe - £>^,tfj)* > E K - ^ l 1 " E |(^.V'3)H|2}fc=l

3= 1

fc=l

*•

Since {^i,... ,V>„} is ON, we have £ |(<£ fc ,^)fl|

(4.4)

3= 1

< H ^ I I H = 1 forfc= 1 , . . . , n,

3=1

so that the RHS of (4.4) > 0. If ipj = 4>j,\ < j < n, then the LHS of (4.4) = 0. Therefore (4.3) holds. L e m m a 7. Let $ be an ^.-measurable C+ {H) -valued function

on 0 and write it as

oo

$(t) = Y/\J(t)Ej(t),

tee,

3=1

where, for each ( € 0 , ^j(t) 's are the distinct eigenvalues of $(£) such that Xi(t) > X2(t) > ■• - > 0 ane( .&,(() is the orthogonal projection onto the eigenspace of \j(t) for j 6 N. Then: (1) For each j 6 N, Xj(-) is an ^-measurable function on O. (2) For each j 6 N, fij(-) is %-measurable on @j = {t € 9 : A7(t) > 0}. Proof. (1) For £ € 9 let cti(t) > aa(i) > ■ • • be the eigenvalues of $(£) counted as n

many as their multiplicities. It follows from Lemma 6 that J ] aj(-) is 2t-measurable 3=1

for every n 6 N and hence ctj(-) is 21-measurable for every j e N. P u t rii(t) = 1. Then, Ai(t) = a n i ( £ )(t) = Qi(i) and hence Aj(-) is 21-measurable. Next let n2(t) be the first n € N such that a n i ( t ) (f) = (^(t), j < n - 1 and ani(t)(t) > an{t), and observe that n 2 (-) and hence A2(-) = a B2 (.)(') is 21-measurable. In the same manner we define rij(-) for j > 3 and see that Aj(-) = a „ (.)(•) is 21-measurable. (2) We have the following equalities: £?i(t) = lim n—t-CC

l + $(t) 1 + Aj(t)

(4.5)

3.4. THE SPACES L*(F) AND

L2(F)

113

Ej(t) = lim

3 > 2-

(4-6)

fc=l

In fact, observe that since Yl Ej{t) — 1

an

d ^i{t) > Aj(t) > 0 for j > 2

RHS of (4.5) = lim n—»oc

= lim <{ £ i ( i )

fil(«).

Hence (4.5) holds. (4.6) is proved similarly. The 2t-measurability of E3(-) on &j for j e N follows from (4.5), (4.6) and (1). Now we can prove the completeness of Ll(F)

as follows:

the space Ll{F)

T h e o r e m 8. For any F 6 vca(21,T(H)) the norm | | - | | I , F given by (4.1).

is a Banach space with

Proof. Clearly L 1 ( J F ) is a normed linear space. To see the completeness let { $ n } ~ = 1 C L\F) be aCauchy sequence, i.e., | | $ „ - $ m | | i , F = | | * n F ' - $ m F ' | t i -> 0 dF a s n , m - > oo, where as before F' = -r— and v = \F\. Then, {^„F'}^L1 is a Cauchy dv sequence in L 1 ( 6 , i / ; T(H)). Hence, there exists $o £ Ll{Q,v ;T(H)) such that | | $ „ F ' - $ 0 | | i -»• 0. Put $ = $ 0 F ' " , where F'"(<) = (F'(4))~ is the generalized inverse of F'(t) for t E © and is 2t-measurable by Lemmas 2, 5 (4), (5) and 7. Then, by Lemma 5 (2) we have | | $ F ' | | T = \\$0F'-F'\\T

= \\9aJmlFI)±\\T

< ||*o||r,

so that $ 6 2/ 1 (F). Moreover, it holds that P „ - * | | l t F = | | * » F ' - $F*||i = / | | $ „ F ' - $ 0 F ' ~ F ' | | T d^ / \\$nF'Jnnx f

-$>0Jm{F,}±\\

dv

p>„F'-$o||Tdi/=||$„F'-$o||i->0

as n -> oo. Thus L X (F) is complete.

114

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

u X L°(Q;B(H)) L\F) vca(K,T(H)). We do not know whether Clearly L (0;B(H)) C7 ( F ) for F € vca(2l,T(77)). l l B(H)) is dense in L (F) or not. L°(e ; F(77)) Now let F e ca(2l,T+(77)) ca(%T+(H)) C vca(%T(H)). vca(%,T(H)). As before we put i/ v = |F| \F\ and

F'=^fe

Ll(0,v;T(H)).

Clearly i/(-) = ||F(-)|| \\F(-)\\TT = = trF(-) e € ca(2l,R+).

0(77)-valued functions on 9 6 . Then D e f i n i t i o n 9 . Let $ and * be ^-measurable 0(E)-valued 5(77)-valued (and a pair ( $ , * ) is said to be F-mtegrable if $ F ' * and # F ' * are 5(7Fj-valued r(77)-valued function ( $ F ' * ) (( **FF"' * ) * is i/-Bochner hence 2l-measurable) and the T(#)-valued integrable, i.e., ( $ F ' * ) ( * F ' * ) * € Z ,,1 1^ ,^*;; F(/7)). T(H)). Here, 5 ( H ) is the Hilbert space 77. In this case we write of all Hilbert-Schmidt class operators on H. [ * , * ] , = / » d F * * = / (W*)(*f*r*. Je Je

(4.7)

Let us define L2(F) by L22(F) ( F ) = { $ : 0 -» 0(77), a-measurable and (#, * ) is F-integrable}. For $ , * e L 2 ( F ) we identify $ with * if $ F ' ^ = <5F'' 4 ^ * v-a.e. y-a.e. P u t

i e t F g co(a,T+(if)), ||F(-)LT and F L e m m a 1 0 . Let ea(a,F+(77)), i/(.) = ||F(-)|| F'1 = — . \

(1) yin An 0(H)-valued

7 2 (e,^;5(77)).

\

if,

u

II WIIT

^-measurable %-measurable function

Then:

di/

$ belongs be/ongs to L 7 22(F) ( F ) iff z# $ F ' * €

(2) /7// * $ , * e 7 2 ( F ) , ifeera ( * $ , **)) is F-mtegrable. (3) [-,-]F ffiwcn 6?/ (4.7) is a gratman tfrwntan zra L 7 22(F), ( F ) , so that L2(F) is a normal preHilbert B(H)-module. (4) L 7 22(F) ( F ) C L\F). Ll(F). 7J(.) ( - )=s ll..

More fully, | | $ | | 1 F < | | * | | FF ||||7/ | | F for /or $ e 7 2 ( F ) where

Froo/. (1) is straightforward from the definition of L2(F) and (2) follows from: Proof.

| ||($F'*)(*f*)\e&, < ^ |||<3>F'*||J*F'*|| | $ F ^ | | J | * F ' - | | ff^ ^ < < |11*11^11*11, | * | | | | * | | <
C T

F

F

(3) is almost clear. As to (4) let $ g € L2(F). Since /(•) /(■) g L 7 22(F) ( F ) ( * I7)) is F integrable and | | * | | l i F < ||*||F||7||F by the above inequality. This means $
AND L2(F)

3.4. THE SPACES L^F)

115

That L2(F) is complete, i.e., it is a normal Hilbert B{H)-moduie larly as in Theorem 8. In fact,

is proved simi

T h e o r e m 1 1 . For any F £ c a ( 2 l , T + ( # ) ) , L2(F) is a normal Hilbert with the gramian [■, -\p. Proof. We need only show the completeness of L2(F).

B{H)-module

Let {$n}%>=1 C L2(F) be

a Cauchy sequence. Then { $ n F ' T } £ ° = 1 is a Cauchy sequence in L2{Q,v; 2

Hence there exists * e L ( 0 , v; S{H)) such that | | $ F ' 2

ty{F'*)~F'*

S(H)).

- * | | 2 l / -> 0 as n - > oo,

Put $ = ^ ( F ' 1 ) - and observe that $

where ||-|| 2 ,., is the norm in L ( 0 , v ; S{H)). is ^-measurable. Moreover,

T

= *J

j , and 11*J

i ,\\

< ||$|L

2

v-a.e. Thus we have $ 6 L (F) and j | >$n n_- $$| || |2 2 == /j |||$„F'* | $ n F ' 5 - ** JJ , / ||$„F'*./

,. ,II | | 2z d i /

i , - * J

i , \\2 dv

2

< / ||$ n F' 2 - * | f di/= | | $ n F ' J - "*" 2 , K e"

a s n - > oo. Therefore $ n —> $ in L2{F),

and L2(F) is complete.

What is more important is that B(//)-valued 2l-simple functions are dense in L2(F), which is proved in the following. T h e o r e m 12. For any F e ca(%T+{H)),

L°(Q ; B(H)) is dense in

L2(F).

Proof. We present this in several steps. Claim 1. Let $ £ L2(F) and e > 0 be given. Then there exists a C(if)-valued function * e L2{F) such that dimiH(^'(-)) < oo v-a.e. and | - <S\\F < e, where K-)=|F!(-). For, write F ' ( t ) = J2 a*(<)#*(£) ® ^fc(t) for ( e 9 , where a/t's are nonnegative fc=i

eigenvalues of F'(-) counted as often as their multiplicities such that «].(•) > a2(0 > ••• > 0 and i?!ifc(-)'s a r e corresponding ON sequence of eigenvectors. $ 6 L2(F) implies that $ F ' * 6 L 2 ( 0 , ^ ; £ ( # ) ) • So we have that $ F ' 7 is bounded and

'**(•)

116

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

for n > 1. Define a B(H)-valued

function $ n by

n

fc=l

Then we see that d i m ? t ( $ n ( - ) ) < n v-a.e. for n > 1. Now it holds that OO

OO

W^'Hl = E II

$F

'"^IIH

fc=i fc=i

= E IMto® W*IIH

OO

= ^

a

k\\^k\\2H < oo v-a.e.

k=\

Since | | * „ F ' * | £ = £ ak\\^kfH

< | | # ^ * | | * we get $ n G L 2 ( F ) and

fc=i

/ \\*F,*-*nF"sffo=

||*-*»IIF =

j Q

/ ^ Je

'

afc|]$0fc|||fdi/-^O

k=n+1

as n —> oo by the Bounded Convergence Theorem. This proves Claim 1. Claim 2. Let * e L2(F) be a C(i?)-valued function and e > 0. Then there exists a C(H)-valued function ^>0 on 0 such that

/ll* 0 || 9 ||f*||>

Je

||*-*OIIF<£-

For, define for each n > 1 *„(£) = *(*) if ||*(t)ll < n and = 0 otherwise. Then \!/ n 's are 2l-measurable and C(ff)-valued. Moreover, we have

L |tf„ll ||i^||;><»V(e)
||*-*„||F = /

||(*-*f,)F'»||*di/=

•'e

/

||¥F'*||2di/-K)

"'[||*ll>n] 2

as n -> oo since [|[*|| > n] | 0 and * 6 L ( F ) . This proves Claim 2. Let {V>fc}£Li be a fixed CONS in # and Jk : H -> 6 { ^ : 1 < j < fc} = Hk be the orthogonal projection for fc > 1. If a e C(H) and afc = J fc aJ fc for fc > 1, then we see that \\a — ak\\ —> 0 asfc—> oo. Claim 3. Let * be a C(.ff)-valued function on 6 such that fQ | | * | | | | F ' ^ || 2 dv < oo and ^fc = Jk^Jk forfc> 1, where Jk's are denned as above. Then we have:

3.4. THE SPACES ^(F)

L2(F)

AND

117

(a) / e ||*|| | | F ' ' \\l du 1. (b) | | * - * k | | F -> 0 as fc->oo. (c) For each fc > 1, * fc g L 2 (F fc ), where Ffc 6 ca(2t,T+(i/)) is defined by Fk = JkFJk and is actually T + (if f c )-valued. (d) Foe e > 0 and fc > 1 there exists a £(#)-valued 2l-simple function <&k on 9 such that $ fc g L2(Fk) and ||* fc - $fc||Ft < £• Moreover, * fc J fc e L 2 (.F) and ||* fc - $ fc Jfe||F < e(a) and (b) are clear. (c) Obviously Fk g ca(%T+(H)). (a) implies that * fc g L2{F). Observe that for fc > 1 [*fc,*fc]F= [ VkF'%dv= Je

[ Je

VkJkF'Jk%dv

= f ykFftkdu=l*k,yk]Fk,

(4.8)

Je where F'k = JkF'Jk. It follows that * fc g L 2 (F fc ) and ||* f c || F = \\Stlk\\Fk for fc > 1. (d) Since ||*fc|| F = / e l^feF^"5 II df < oo, it is not hard to see that there exists a B(Hk)-valued 2t-simple function $ fc € L2(Fk) such that ||* fc -$k\\Fk < £• By (4.8) we get $fcJfc € L2(F) and from the above inequality it follows that \\tyk — 0. By Claim 1 we can find a C(#)-valued function * 0 g L2(F) such that ||$ - * 0 | | F < £• By Claims 2 and 3 there exists a £?(#)-valued 2t-simple function * i g L2(F) such that ||*o - * I | | F < e. Therefore ||# - * x | | F < 2e. Let £ g capos(2t,X) where as before X = L§(fi;tf). Then F 5 (-) = [£(•),£(•)] € ca(2t, T+(H)) and we can construct L2(F^). We want to define integrals of elements in L 2 ( F ? ) w.r.t. £. As in (3.1) for $ = £ ajlAj

g L°(Q;B(H))

we can define the

integral of $ w.r.t £ ower A g 21 by

/ j^^^ojpj-ni) m

n

For $ = X) M ^ , and * = £ J

j=i .

L Je

gx

-

-

6fclBfc g L ° ( 9 ; £ ( # ) ) we have that

fc=i i

r

-

m

j=i

n

fc=i

- i T n n

;=lfc=l

118

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

,-=i

(4-9)

7 e

fc=i

and hence

/.

(4.10)

$d£ .Y

For a general $ e L 2 (F £ ) there exists a sequence {$n}£° = i Q L°(Q;B(H)) that ||$„ - $ | | F -> 0 by Theorem 12. It follows from (4.10) that l*it-*n

Ff

=

7e

ie

since { $ n . } ^ = 1 is Cauchy in L2(F^). of $ w.r.t. £ ower 0 by

* n ^

(k,n

such

oo J

Thus we can define unambiguously the integral

/$d£=

lim / $ n d £ ,

(4.11)

where the limit is in \\-\\x- It is easily seen that (4.11) is well-defined, i.e., it is independent of the choice of {$n}£°=i Q LP{®\B(H)). For A € 21 we have $ 1 A £ L2(F^) and we can define

[ <S>d£ = / $ l A d £ = lim f $ n d£. The integrals defined by (4.11) and (4.12) are sometimes referred to as integrals since £ is a stochastic measure. Let us denote, for each A € 21, 6t(A)

(4.12) stochastic

= 6 { £ ( A n B ) : B e 21},

the closed submodule of X generated by the set {t;{A n B) : B e 21}. Then the isomorphism between L2(F^) and ©^(0) is stated as follows: P r o p o s i t i o n 13. For £ 6 cagos(%X) it holds that L 2 (F C ) ^ 6 j ( 0 ) , where the 2 isomorphism U : L (F^) —> ©^(0) is given by U(
L

$d£,

$ei2(F{).

(4.13)

Proof. Let U0 be defined by C/0$ = / Q $ d£ for $ 6 L°(Q;B(H)). Then [70 is gramian preserving on L°(Q;B(H)) by (4.9) and its image is dense in 6 ^ ( 0 ) . By Theorem 12 L°(@;B(H)) is dense in L2{F^). Hence U0 can be extended to a gramian unitary operator U from L2(F^) onto 6 ^ ( 0 ) , which is given by (4.13). Therefore U is an isomorphism.

3.5. THE SPACES £ x ( 0 AND

£ 2 (M)

119

3.5. T h e s p a c e s £*(£) and £ 2 ( M ) As before we put X = LQ(CI;H). In this section we shall study integration of operator valued functions w.r.t. an X-valued measure f and a T(H)-valued positive definite bimeasure M, and define a kind of L 1 -space £ * ( 0 and that of L 2 -space £ 2 ( M ) consisting of operator valued functions. The latter is an extention of the scalar valued bimeasure case, including MT-integration theory which was mentioned in Section 1.2. So let (9,21) be a measurable space and L°(Q;B(H)) denote the set of all B(.ff')-valued 2l-simple functions on 0 . Definition 1. Let £ £ 6ca(2l,X). If $ is a B(H)-va\ued uniformly measurable function on 0 , i.e., $ is strongly measurable when B(H) is considered as a Banach space, then the ^-essential sup norm ||#||oo,£ is defined by inf {a > 0 : [||$|| > a] is£-null}.

11*11

L°°(£; B(H)) denotes the set of all B(H)-va\ued functions $ on 0 for which there exists a sequence {$ n }£° = i C L°(Q;B{H)) such that | | $ n -$||oo,£ -» 0. Let £ e bca{%,X). It is not hard to verify that (L°°(£; B{H)), \\ • ||oo,«) is a Banach space and a left B(H)-mod\i\e. For a function $ £ L ° ( 0 ; B{H)) its integral w.r.t. £ over A 6 21 was defined by (3.1), i.e., $ * ; = 'Y^a.jHAj

/.

n A),

(5.1)

?=i

yhere $ = ]T % T A • And we noted that 3=1

$d^

UUA)

{I/, $de

= sup ■

< ll*iUieiU(A), :$eL°(0;5(H)),||$|

(5.2)

< 1

(5.3)

where ||$||oo = sup ||$(t)||, the sup norm. In (5.2) and (5.3) we can replace ||5>||oo tee by p||oo,€- For $ e L ° ° ( £ ; B{H)) let { $ n } ~ = 1 C L°(© ; 5 ( H ) ) be such that ||*„ *l|oo,€ ^ 0- Then, since | | $ n -<E>p||oo,£ -> 0 as n,p -> oo, we see that { / A $ n ^ } ™ , is a Cauchy sequence in X for every A 6 2t by the inequality (5.2). Hence, we can define the integral of $ w.r.t. £ over A G 21 by / $ d£ = lim / $ n d£. 7.4

"^°°JA

(5.4)

120

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

It is easy to see that the above integral is well-defined, i.e., it does not depend on the choice of the sequence {$„}n=1 C L ° ( 0 ; B{H)). We begin with consideration on integration w.r.t. a scalar valued bimeasure. So let M = 9K(2t x 21; C) be the set of all C-valued bimeasures defined on 21 x 21. As to definitions and basic properties of bimeasures we refer to Section 1. Definition 2. Let m € M and / , g be C-valued 2l-measurable functions on 0 . (1) The pair (/,g) is said to be m-integrable if the following three conditions a), b) and c) hold: a) / is m(-,B)-integrable for every B e 21 and g is m(A, )-integrable for every Ae2C; b) mx{-)~ feg{t)m{-,dt),m2{-) = fe f(s)m(ds,-) e ca(H,C); c) / is mi-integrable and g is m2-integrable and it holds that / f{s)mi{ds) Je

= I g(t)m2{dt). Je

(5.5)

The common value in (5.5) is denoted by / / (f,g)dm JeJe

or

/

/ JeJe

f(s)g(t)m(ds,dt)

and is called the integral of (/,
m f (•) = JEf(s)m(ds,-)

G ca(2l,C) for every E,F £

c') / i s mf-integrable for every F € 21 and g is mf-integrable for every E E 21, and it holds that / f{s)mf(ds)=

[ 0(t)m?(dt),

JE

E,Fe%.

(5.6)

JF

The common value in (5.6) is denoted by /

/

JEJF

if, 9) dm

or

/

/

f{s)g{t)m(ds,dt)

JEJF

and is called the integral of (f,g) w.r.t. m over Ex F. Z2,{m) denotes the set of all C-valued 2l-measurable functions / on 0 such that ( / , / ) is strictly m-integrable.

3.5. THE SPACES &{£) AND £ 2 (M)

121

2 2 2 £^j^yilOJ (mm) ) £g) ~ ££2(m) m ) ,) and £<02 (m)_(£ fJ g, y £€ lNote i U b C that IjilCLU £*{ AJ ({lit')) d U U if 11 /,
E x a m p l e 3. Let 9 = Z and 2t 2( = C by

2 2 where [j {(?", fc) {(j, k) : j 22 + k2 > 0}. Let /f(s) (s) = 5g(s) (s) = s for s e Z. Then w. we we [j* + k > 0] = {(j, can see that I(f,g) ( / , 5 ) is m-integrable, but ((fl/ 1A.gl A.S A)1 A ) is not, if A = K In fact, fact, for In foi each B, E £€ 21 we see that

[ f(.s)m(ds,B)= Y,

JE

E?,

^ ?»

3ek{0}k^{\j\+\k\) jefi\{o}fces vui T I'M/

exists. g{t) ,FF ee 2t. Similarly, fJ/L go (f it )l m mm{A,dt) m{A, ( i .4r,f fdt)
Ir = g{t)m(A,dt). = E7 EV 4=== / SW g{t)m(A,dt). ?4 = V 7 m(,A,atj. k Jx 3EA 7eAfcez\{0}(lj'l lfcl) ^ keZ\{0}(\3\+\ \) + l*l) Hence (/, & / u £j h = flA,9x=glA,A then A, oi = 01,1, A = = N, N, then yj

r„

nWm

' •? '

dm = 0. On the other hand, however, if CO

J*m***~]&w?w-

B€

^

so that TO TO <7iW WMdt) Ati(dt)--=EEE EEE ^^^ 777 ^^^ 44 ===TO / 5i 5i(t) Ati(dt) - -/z J* 1*1) ./z . . . IOil+ \k\\ ■ fc=ii=i k=n=i O1i l + + 1*1) ■^ fc=l>=l (b'l + w)

Consequently (/ f/,,o,) {flA,glA) / 1 A , o U ) is not m-integrable. 1 ) f f l ) = ((flA,glA) We now restrict our attention to nositive positive definite bimea.siires bimeasures. Rpra.ll Recall that mm ce M is said to t.n be hp positive -nnsrUiw definite definite, ifif \ a,a ^, .-^7. ™ f d . 4 . t ^> Y > n0 j km{Aj,Ak)

122

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

for every n g N, a i , . . . , a„ € C and Ai,...,An£%. Then there exists a unique RKHS $jm of m. That is, f) m is a Hilbert space consisting of C-valued functions on 21 such that m(A, ■) € f j m and f(A) = (/(•), m(j4, •)) for each A e 21 and / G flm, where (-, - ) m is the inner product in Sjm. Since an abstract Hilbert space is isometrically isomorphic to a closed subspace of L Q ( 0 ) for some probability measure space (f2,5> I1) (cf. Rao [2, p. 414]), we may assume that Sjm C Lo( f i )- N o w d e f i n e $ : 21 -» Lg(O) by

f (.4) = m{A, •),

Aea.

Then we see that £ 6 ca(2l,Lo(f2)) since m is a bimeasure. Moreover, we have m{A,B)^{i{A)^{B))v

A,Be%

i.e., 77i = rri£. In Section 2, we defined the space L1^) of all C-valued 2l-measurable functions which are £-integrable, and proved that L 1 (£) is a Banach space with the norm ||.||, j £ which is denned by \\f\\.£ = \\St\\(Q) for / e ^ ( 0 , «"ere £,(•) /(.) f d£, E ca(2l,Lo(0)). Now we have the following proposition. P r o p o s i t i o n 4. Let £ 6 ca(2l, L§(fi)) and m = m ? 6 M . / / / , # g L1^), f,g £ £j(m) and

[ [ (f,g)dm=( JAJB

[ fdZ, [ gdA , W/l

JB

A,Be*.

then

(5.7)

/2

/Yoo/. Let / , # 6 L J ( 0 - Choose sequences { / n } ^ , {ff„}£La Q L°{&) such that fn -* / , Pn ^ ff pointwise and | / „ | < | / | , |#„| < |g| for every n 6 N. For n £ N write / „ and gn as

3=1

fc=l

Let A, £? € 21. Then we have that

f fn (s) m(ds, 5 ) J A

= £

an,jrn(AnJ

7=1

nA,B) = ^

a„j(€(A„j

n A), £(B)) 2

7=1

-Us*!.*),* UJ«-!.*),• as n -» oo, while the left hand side of the above expression can be written as LHS = J fn{s){S{dsU(B))2

-> J f(s)(Z(ds),C(B))2

= J

f{s)m(ds,B).

3.5.

THE SPACES Z1^)

AND £ 2 (M)

123

Thus we get

Then it holds that

/ .'

gn(t)

m$(dt)

= J2 Pn,km2{Bn,k

n S)

fc=l

= ^2A,k

f(s)m(ds,Bn,knB)

= X] ^",fc „ lim 5Z atJm(Ae,j ■

n

^> - B ",t

n

-B )

t—► oo

fc=i

y=i

Y^ aijZ{Atj r\A),^2p„,kaBn,k n B) 1 ;=1

= fdi, 9rid u A L ^) a s m

fc=l

'

2

f , 9d

~* a

^ i ^)'

oo. Consequently, 3 is m^-integrable for each A € 21 and it holds that

f W)m*{dt)= ( [ fdi, I gdA ,

A,Be%.

JB \JA JB /2 Similarly, we can see that f is 771^-integrable for each B E 21 and it holds that

Jj(s)mf(ds)= [Jj^JB9^y This shows that [f,g) is strictly m-integrable. In the same manner we see that ( / , / ) and (g,g) are also strictly m-integrable. Therefore, / , g g £j(m) and (5.7) holds. It follows from Proposition 4 above that L 1 ( 0 C £ j ( m ) . To prove the converse inclusion relation we need the Dominated Convergence Thoerem for bimeasures. T h e o r e m 5 ( D o m i n a t e d Convergence T h o e o r e m for B i m e a s u r e s ) . Let m e M and {fn}%Li and {Pn}^Li be sequences of C-valued ^-measurable functions on 0 . Suppose that there exists a pair (h,h') of strictly m-integrable functions on 0 such that \f„\ < \h\, \gn\ < \h'\ for each n G N and fn —>• / , gn -> g pointwise. Then, (fn,gi) (n,£ € N) and {f,g) are strictly m-integrable and it holds that for A, B 6 21 /

/

(f,g)dm=

lim lim / /

(/„, gt) dm = lim lim / /

{fn,ge)dm.

(5.8)

124

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

Proof. It follows from Definition 1 that (fn,9e) We only have to verify the equality (5.8). For A, B £ 2t we have that f [ (f,g)dm= JAJB

f /(s)mf(ds),

is

m-integrable for each n,£ 6 N.

where mf{D)

JA

= /

= f

g{t)m{D,dt),

JB

lim

fn{s)mf(ds)

= lim /

fn{s)mf(ds),

n->oo JA

by the usual dominated convergence theorem, = lim / g(t)m£n{dt), where m ^ n ( £ ) = / f„(a) n ~>-°°JB ' JA = lim lim / ge{t)mf

m(ds,E),

Jdt)

n—>oo £—foo J £j

= lim lim /

/

n-Kx> £^oo JA

(fn,gt)dm.

JB

Similarly we can prove that / /

{f,g)dm=

lim lim /

JAJB

/

(fn,ge)dm.

e^oon->ooJAJB

Corollary 6 ( B o u n d e d Convergence T h e o r e m for B i m e a s u r e s ) . Let {/n}S?Li and {gn}^Li be sequences of C-valued ^-measurable functions on G such that |/n|,| 0 and fn —> f, gn —> g pointwise. Then, f,g G £,l(m), (f,g) is strictly m-integrable and / /

{f,g)dm=

JAJB

lim

/ / (fj,gk)dm,

A,Be%.

3,k^coJAJB

Now we can prove the following. T h e o r e m 7. Let £ 6 ca(2l,L§(ft)) and m ~ ras € M. T/ien LX(C) = -C»(m). Proo/. It follows from Proposition 4 that Ll{(,) C £j(m). To see the opposite inclusion relation, let / e £»(m) and set An = [|/„| < n] e 2t and / „ = flAn for n e N. Then, since / is bounded on An, we have

/ /'(/,/) dm =( j JA ' AnJA nJ n An

\JA \JAnn

fdtJ JJA, A„

fd(,\ , /2

neN.

3.5. THE SPACES £ x ( 0 AND £ 2 (M)

125

Moreover, we see that / „ —> / pointwise, | / n | < | / | for n 6 N, | / | G £ 2 ( w ) , and J A /n dC £ -^o(^) for n e N and .4 G 21. Furthermore, for every A G 21

/ (/j " A) dC = / f (/„/,) dm- f f (fjjk) dm - / / (fkjj)dm+ JAJA

[ [ (/*,/*) An->0 JAJA

as j,k —> oo by Corollary 6. Therefore / € i 1 ^ ) Let us give some remarks here. Suppose that m = m^ for some £ € ca(21, Lo(fi)) and define (•, - ) m and ||-|( m by

(f,g)m= f f(f,g)dm,

H/IU = ( / , / ) !

for / , £ G £ 2 ( m ) . If we identify / , j £ £ 2 (m) when ||/ - #|| m = 0, then £ 2 ( m ) or £ j ( m ) becomes a pre-Hilbert space. In Theorem 7 we proved that Ll(£,) = £ 2 ( m ) , the equality being as a set and note that

l«.e

He/IKG),

feL1®,

so that £ 2 ( m ) may not be a Hilbert space in general. Then, the question is: When is £ 2 ( m ) or £ 2 ( m ) a Hilbert space? We observed that if £ G caos(2l, Lg(fi)), then £ 2 ( m ) = £ 2 ( m ) = L x ( 0 = L2(y^) holds and hence £ 2 (m) becomes a Hilbert space (cf. Theorem 2.4). We shall discuss this problem later in Section 4.3 and see that £ 2 ( m ) is actually a Hilbert space. Recall that 0{H) denotes the set of all linear operators on H. We want to define integration for 2l-measurable 0(H)-valued functions on 0 w.r.t. a T(H)valued bimeasure M G 9tt = 9tt(2l x %;T(H)). We only consider positive definite bimeasures M, so that we can assume that M = M% for some £ G ca(2l,X) in view of Theorems II.4.13 and 24, where X = L ^ f i j t f ) . That is, M{A,B) = [£{A),£{B)] for A, B G 21. Suppose that M is of bounded operator semi variation, denoted M G 9ftb, which is equivalent to £ G bca(%i,X). Then, for each x £ X, the measure £ o x defined by (£ o x ) ( ) = [£(■),x] is of bounded variation by Theorem 1.5(2), i.e., ( o i e vca (21, T(H)). In particular, M(A, -),M{-, B) G vca(<&,T{H)) for every A, B G 21. Hence we can construct an L 1 -space L 1 (^ox) as was done in the previous section. An analogy of Definition 2 is as follows. D e f i n i t i o n 8. Let M G Tlb be a positive definite bimeasure, so that M = M^ for some £ G bca{^,X). Let $ , * be 2l-measurable Ofii")-valued functions on 6 .

126

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

(1) ( $ , * ) is said to be M-integrable if the following three conditions hold: a) $ , $ e L X (M(-,A)) for every A E 21. b) Mi(-) = J e * ( t ) M{; dt); M 2 (-) = / e *(«) M ( d s , ■) e u c a ( a , 1

c) $ e /^(AfJ), * € i ^ * )

aad

^

holds

T(H)).

that

J *(s) M*(ds) = (Jv(!)

Af,*(«ft)) ■

The common value in c) is denoted by / e / e $ dM **, called the integral of ( $ , * ) w.r.t. M oner 9 x 0 . £ 2 ( M ) denotes the set of all 0(H)-valued 2l-measurable functions $ on 9 such that ( $ , $ ) is M-integrable. (2) ( $ , * ) is said to be strictly M-integrable if the conditions a) above and b') and c') below hold: b') Aff (•) = JDV{t)M(;dt)*, M f ( - ) = / c $ ( s ) M ( d s , - ) G «ca(SX,r(H)) for ev ery C, £> G 21. c') |0 6 L 1 ( ( M f ) * ) for every D E 21 and $ € L 1 ((M 2 C )*) for every C 6 21, and, for every C, D E 2t it holds that

(s)(M1L)r(ds)

J *(t) (M,c)*(dt)) .

c The common value in c') is denoted by fcJD$dM #*, called the integral of ($,"!&) w.r.t. M over C x D. £%[M) denotes the set of all $ 6 £ 2 ( M ) for which ( $ , $ ) is strictly M-integrable. It is easily seen that for every £ 6 6ca(2t, X) L°{Q;B(H))

C L°° ( £ ; £ ( / / ) ) C £*(*/«) C £ 2 ( M ? ) .

Note that for $ G L°°(£ ; fl(i/)) and A G 21, the integral / A $ d£ was defined by (5.4) and it holds that for $ , * E L°°(£; B{H)), C, D E 21

/ / $dM?** =

f $d£, / *d£

(5.8)

Let us examine a special case where £ is of bounded variation, i.e., £ G vea(2t, X ) . = |||(-), there exists an RN-derivative f = — G Ll(Q,v\X) dv such that £(A) = J A £' dv for A e 21. As in Definition 4.1 an 0(tf)-valued 21measurable function $ on 9 is said to be (,-integrable if $ £ ' G Ll(Q,v\ X), where we identify X = S(LQ(Q), H). In this case we write fA $ d£ = J A # f di/ for A G 21 Then, putting v()

3.5. THE SPACES fi1^) AND £ 2 ( M )

127

and define the norm ||$|| € = H ^ ' l l i = / e | | * C I U * ' - L l ( 0 = ^ ( © . f ; W ) ) denotes the set of all 0(if)-valued £-integrable functions on 6 . P r o p o s i t i o n 9. Let ( e vea(%X). Then (L x (£), ||-|| e ) is a left B{H)-module a Banach space. Moreover, L 1 (^) C £^(Mj).

and

Proof. The first part can be proved in a similar manner as in the proofs of Theorems 4.8 and 4.10. We only have to show the inclusion relation L 1 (^) C £?,(M), where M = Mf. Let $ e L*(£) and we shall check a), b') and c') of Definition 8. a): Clearly M(-,A) e vca(%T{H)) for A e 21. To see that $ 6 L1(M(-,A)) observe that

L

dM

['A^ d;/

=

[£',£(A)], $

dM(-,A) di>

T

Je

d M

jjA dz/

)

= [*£',£
m'\\x\\aA)\\xdv = \\nMM\x<™-

Hence $ e L 1 ( M ( - , A ) ) . b'): Let C, D e 21. Then we see that

dM? dy

M?(A)

JD

I *dt,t(A)

Ae%

and hence

|Mf |(6) < /.

Similarly, |M 2 C |(6) < oo. Therefore M?,M?

d(M?y

c'): Let C,D g 21. Observe that $ -

iei(e)
$d£

di/

E

vca(%T{H)).

$f, / $ d£

is well-defined and

f ^^MDl dv< [ m'Wxdv / $ ^ =n*ne /$d£ 7e

ai/

T

•/©

JD

X

< oo.

JD

Hence, $ 6 ^ ( ( M f ) * ) . Similarly, $ e L 1 ((M 2 C )*). Moreover, we see that

j Ms) (Mf )*(ds) = J$dt,J*dS=(J$(t)(M?y{dt)\.

(5.9)

Therefore ( $ , $ ) is strictly M-integrable, so that $ £ £»(M). (5.8) and (5.9) suggest that in order to define a bimeasure integral / „ J* $ dM^ \£* it is sufficient to define vector integrals Jc $ d£ and fD * d£. So let £ e 6ca(2l, X ) and

128

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

{77, Y, P) be its g.o.s.d. Note that Y need not contain the whole space X but 6 £ ( 0 ) , We say that {77, Y, P) is a mmimal g.o.s.d. of f provided that F = ©,(9) 2 6 , ( 0 ) and £ = Prj, and if {77',Y', P'} is such, then Y is regarded as a closed submodule of Y', The existence of a minimal g.o.s.d. of £ follows from Zorn's lemma. Let {T7,F,P} be a minimal g.o.s.d. of £. Recall that L2(F,) * 6 , ( 0 ) by the isomorphism C7 : L2(F,) -^ 6 , ( 0 ) given by (4.13), i.e., U9 = / e * * 7 for $ € L 2 (F,). Since 6 , ( 0 ) C 6 , ( 0 ) we can define £ 1 (O = t/" 1 6?(0) and the gramian orthogonal projection Q : L2{Fn) -> £*(£) by Q = / J ^ P t / (see the diagram below). L2(F,) ^ ^ 6,(9) = Y

£ i(0

<

6,(0)

a

Since £ x (£) = 6 , ( 0 ) , £ x ( 0 is a closed submodule of L2(F,), and hence a normal Hilbert B(H) -module with the gramian [*,*]= /#dF,**,

* , * e^CO-

(510)

For $ 6 L2(F,) define the mtesre/ 0/ $ w.r.t. £ by [$<% = p[$dr), (5.11) Je Je which is motivated by (3.4). To justify (5.11) choose a sequence {*„}~ =1 C L°(e;B(H)) such that ||* B - $|| F , -+ 0 since L°{Q;B(H)) is dense in L 2 (F,) by Theorem 4.12. Then observe that

I j[ * , « - j T •,*|[ c -||pjf(*,--« t j«6i|| jf <||* 1 ,-*,!!,,-^o as p, g -4 00 and that Hn^-P / $ndr) = P I $^77. Thus (5.11) is well-defined. For A € a we define / $d£EE / lA$d(, = P

lA$dr] = P / $d?7.

(5.12)

129

3.6. RIESZ TYPE THEOREMS

The integrals defined by (5.11) and (5.12) are also called stochastic integrals. Now we can rewrite (5.10) as (5.13) IJe

Je

J

[Je

Je

which may be denoted by fJQ2 $ d M ^ * * in view of (5.8) and (5.9). Moreover, for A,B 6 21 and <&^ e £X(C) we may "define" the integral of ( $ , * ) w.r.t. M$ by <$>dMc^* JAJB

[ *dt, f * % JA

(5.14)

JB

Well-definedness of (5.14) is noted in a similar manner as was done for (5.11). It will be interesting to find relations among £ 1 (0> £»(M<) a n d £?(M(:).

3.6. Riesz t y p e theorems In this section we assume that 0 is a locally compact Hausdorff space. We prove Riesz type integral representation theorems for operators from spaces of continuous functions on © to a Banach space and especially to X = LQ(Q ; H). C ( 0 ) denotes the Banach space of all C-valued continuous functions on 0 with the sup norm || • ||oo- Co(0) and Coo(©) denote the subspaces of C ( 0 ) consisting of functions in C ( 0 ) that vanish at infinity and have compact supports, respectively. For a Banach space J£ let C 0 (© ; X) be the Banach space of all X-valued continuous functions $ on 0 vanishing at infinity with the sup norm ||<&||oo = sup ||#(t)[|^. tee Let 21 be the Borel a-algebra of©, i.e., the 0 there exist an open set O and a compact set C such that C C A C O and ||£||(0\C) < e. rco(2l,X) denotes the set of all regular measures in ca (21, X). In view of Remark 1.2 (1), £ € ca(2l,3t) is regular iff for any A e 21 and e > 0 there exist a compact set C and an open set O such that C C A CO and ||£(£>)|| x < e for

130

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

every B e 21 with B C 0\C. For measures defined on 2t0 the regularity is defined as follows: f € ca(2t0,3C) is said to be regular if for any A € 2t0 and e > 0 there exist an open set O 6 2t0 and a compact set C € 2lo such that C C /I C O and | | £ | | ( 0 \ C ) < e. It is known that every scalar valued Baire measure is regular and has a unique extension to a regular Borel measure. R e m a r k 2. Let X be a Banach space. (1) Every measure f € ca(2t 0 , X) is regular. In fact, by Remark 1.2 (2) there exists a control measure A 6 ca(2l 0 ,R+) = rca(2t 0 ,R+) of £. Then it is easy to see that f is regular. (2) Let £ € ca(2l,X). Then, £ is regular iff the set {x*£ : x* € X*, ||x*||i- < 1} of scalar measures is uniformly regular, i.e., for any A £ 21 and e > 0 there exist a compact set C and an open set O such that C C A C O and |x*£| ( 0 \ C ) < e for every x" € X* with ||a;*||x- < 1. In fact, this follows from Remark 1.2 (1). (3) If £ g rwca(2l, X), i.e., £ is w.c.a. and x*f is regular for every x* 6 X*, then f € rca(2t,X). In fact, it follows from the Orlicz-Pettis Theorem that £ 6 ca(2l, X). Moreover, by Diestel and Uhl [1, Lemma 13, p. 157] we see that {x*(, : x* e X*, ||x*||x- < 1} is uniformly regular. Thus by (2) above we have that £ is regular. As is well-known we have Co(©)* = rca(2l, C), where the identification is made by the equality AM=

[
Co(e)cc(e)cL°°(f)cL 1 (o (cf. Definitions 2.1 and 2.2). In fact, every

^C(9).

3.6. RIESZ TYPE THEOREMS

131

In this case, £ is unique and \\T\\ = ||£||(0). The locally compact space case is also established as follows: Corollary 4. A bounded linear operator T : Co(0) —> X is weakly compact iff there exists a measure £ G rca(2t,3£) such that

T{v)= [ tpdt, Je

V 6 Co(0).

(6.1)

In this case, £ is unique and \\T\\ = |[£||(0). Proof. Suppose that T is weakly compact. Let 0 be "the one-point compactincation of 0 , i.e., 0 = 0 U {too}. Consider the space C ( 0 ) . For every tp G C ( 0 ) there exists a unique function ip G C*o(0) such that
{T(<^): H^Hoo < 1,

/ G C o (0)}, which is relatively weakly compact, we see that T is also weakly compact. Thus by Theorem 3 there exists a unique measure £ G rca(2l,X), 21 being the Borel cr-algebra of 0 , such that

'-&) = $dZ, I.

veC(Q).

Je Define £(A) = (,(A) for A G 21. It is not hard to see that £ G rca(2t,£) is the unique measure such that (6.1) holds. The converse direction is easy to see. The equality |[T|| = ||£||(0) is noted as follows:

||r|| = su P {||2>|| x :v>eQ ) (0),M| oo
{ll/e-

sup

{HL*«) :¥>eCo(e),|MU
sup

| j
■■vec0(e), Moo
= s u p { | x * £ | ( 0 ) : x * G X * ; ||x*|| x . < 1 }

= llfll(e), by Remark 1.2(1).

132

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

The unique measure £ obtained in Theorem 3 or Corollary 4 is called the repre senting measure of the operator T. Now let X = Ll(tt; H) as before. We consider an operator T : C 0 ( e ; B{H)) -> X. Note that C0(& ; B{H)) is a left £ ( # ) - m o d u l e with the natural action of B{H). Recall that T is a module map if T ( a $ + M-) = aT{<S>) + 6T(*) for every a, 6 € £ ( # ) and $ , * G C 0 ( 9 ; £ ( # ) ) . We put rbca(%X) = rca(%X) n &ca(»,X). Consider the algebraic tensor product C0{Q)(DB(H). Each element $ 6 Co(O) 0 n

B(H) is of the form $ = £ ] <^j ® a j" f° r
1 < j < n, and

n

It is known that the completion C o ( 0 ) ®A 5 ( H ) of C o ( 0 ) 0 B{H) w.r.t. the least crossnorm (or the injective crossnorm) A is identified with C o ( 0 ; B(H)), i.e., Co(Q) ®x B(H) = C0(Q

;B(H)),

n

where the least crossnorm A is defined for $ = ]P ipj ® a^ by

A(3>) = sup

^2H
A e Co(8)M|A|| < i, P e fl(JJ)MHI < l

i=J

sup ■

sa

9?j
£ g r o a ( a , C ) , 1*1(9) < 1, a £ W ,

||a|| T < 1 I

(see e.g. Diestel and Uhl [1, pp. 234-235]). Hence C o ( 0 ) 0 B{H) is dense in C0(G;B{H)). Now let $ 6 C o ( 0 ; £ ( # ) ) and e > 0 be given. Then we can n

find a $ 0 = E / ) ® » J ^ i=i we can find gi,...

C

o ( 0 ) © B(iJ) such that ||$ - $0Hoo < e.

Moreover

,g„ 6 £ ° ( 9 ) such that ||^- - ^ H ^ < — for 1 < j < n, where on n

(S = m a x { | | a i | | , . . . , | | o „ | | } . Putting $ a = J^9j®a}

6 L ° ( 0 ; B{H)),

j=i

| | $ " * l | | o o < 11$ - $0||co + ||*0 " * l | | o o < 2e.

we get

133

3.6. RIESZ TYPE THEOREMS

Therefore we proved: L e m m a 5. C0(@;B(H)) C L°°(£;B{H)) sion is a continuous embedding.

for every f e bca{%,X),

and the inclu

Now let £ 6 6ca(a, X ) and define an operator Tc : C o ( 0 ; B(if)) -+ X by

r € (*)= [ $d£,

$eC0(e;B(iO).

(6-2)

which is well-defined by Lemma 5. Clearly T{ is a module map. Since

l|T£($)|U<||eilo(e}||$||oo,e<||eiWe)||$||oc holds for every $ 6 C o ( 0 ; B{H)), we see that T5 is bounded with \\T^\\ < ||£|| 0 (9). If f is regular, then we have the equality as is proved in the following: L e m m a 6. If £ € rbca($L,X), then the operator T^ : Co(0 ; B(H)) (6.2) is a bounded module map with \\T^\\ = ||£|| o (0).

-4 A" defined by

Proof. Let us consider the operator 7 j , defined on L°° (£; B(.ff)) as an integral operator. Suppose first that 0 is compact. Let e > 0 be given. Choose a measurable finite partition {Alt... ,An} e 11(0) and O i , . . . ,an 6 B(H) with ||a,-|| < 1 for 1 < 3 < n such that

|r£(*0)

Xa^:

>lieilo(e)-e,

(6.3)

3=1

where $ 0 = £ OJIAJ S L°(Q;B(H))

with ||$o||oo < 1- Since £ is regular, we can

3= 1

find, for each j , a compact set Cj such that C 3 C Aj and |j^||(A,-\Cj) < —. Again n by the regularity of f, let { O i , . . . ,On} be a family of pairwise disjoint open sets such that Cj C O3 and ||£||(0,-\Cj) < — for 1 < j < n. Since 0 is normal, there exists a family {ipi,...

, ipn} C C ( 0 ) such that

0 <
^ = 1 on C,-,

ipj = 0 on O]

for 1 < j < n . P u t $ i = E V j ® a j £ C o ( 0 ; B ( # ) ) . Then ||$i||oo < 1 and j=i

n

p.

n

3= 1

■/e

3=1

|T€($i)-r4(*0

(6.4)

134

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

<E

a

i ( [ 'Pi dZ ~ t(Ai)

n

< }=1 E

r / Vjdt-ZiAj

<E

/

3=i

VfdZ-tiA&Cj)

by (6.4)

Jo^c,

< E{lfcll(<W,0 + U\Mi\Ci)) < 2e -

(6-5)

3=1

By (6.3) and (6.5) we get |[T f ($i)||x > ||£|la(©) ~ 3e. Since e > 0 is arbitrary and ||3>i||oo < 1, we have ||Tj|| > ||£||o(©)- The converse inequality was already seen, so the equality \\T^\\ = ||£|| o (0) holds. If 0 is not compact, let © = 0 U {tx} be the one-point compactification of 0 . Note that for any l> e C ( 0 ; B(H)) there is a unique $ 6 C o ( 0 ; B(H)) such that

*(*) = *(*) + *(*»),

*e©.

(6.6)

Let 21 be the Borel cr-algebra of 0 and define an X-valued measure 6, on 21 by i{A) = £{A) for A € 21 and £({*«,}) - 0. Then £ is regular and ||£|| o (0) = ||flU(0). Consider the operator T : C(© ; £ ( # ) ) -* X defined by T ( $ ) = / <£df = / $ d £ , 7e ie

$6C(9;B(I)).

Then the argument in the compact case can be directly applied to T and £ by modifying that the family of open sets {0\,... ,On} is chosen so that Oj C © with Oj compact for 1 < j < n and by noting that the restrictions of the function n

$ = ]T ^ ® aj to 0 belongs to C 0 ( © ; B ( H ) ) by (6.6). i=i The following Riesz type theorem can now be proved. T h e o r e m 7. Let T : C0(Q;B{H)) -^ X be a bounded module map. exists a unique £ 6 rfeca(2l, X) such that

T($)= / s d £ ,

with \\T\\ =

uue).

$eCo(0;B(H))

Then there

(6.7)

3.6. RIESZ TYPE THEOREMS

135

Proof. Consider the restriction of T to the algebraic tensor product space CQ(Q) © B(H). Since the space C 0 ( 6 ) 0 1 is identified with C o ( 0 ) and T is weakly compact because X is a Hilbert space, there exists a unique regular measure £ g rca(21, X) such that

T ( v ® i ) = [ tpdt, Je

by Corollary 4. Then we see that (6.7) holds for every $ 6 C o ( 0 ) O B(H) since T is a module map. If we can show that £ is of bounded operator semivariation, it follows that C o ( 0 ; B{H)) C L°°(£ ; B ( i ? ) ) . Since T is bounded and C0{Q)QB{H) is dense in C o ( 9 ; B(H)), (6.7) holds for every $ 6 C o ( 0 ; B{H)) and ||T|| = ||£|| o (0) follows from Lemma 6. Now we shall prove that £ is of bounded operator semivariation. Suppose the contrary. Then, for any large number 5 > 0 there exists a partition {Ay,. ■ . , An} g 11(0) and a family { a i , . . . , an} C B(H) with ||a,-|| < 1 for 1 < j < n such that

£>i€(4»

>S.

(6.8)

We may assume that 0 is compact, otherwise we can work with its one-point compactification space 0 = ©U{too}. Then, for any e > 0 we can find, by the regularity n

of £, a function $ = Yl
rW-E°M

< 2e,

(6.9)

where {5-2e, from which we conclude that ||T|| > 8 - 2e because ||$||oo < 1- This implies that T is unbounded, a contradiction. Therefore £ is of bounded operator semivariation and the theorem is proved. Now consider a mapping B : C o ( 0 ; B{H)) x C o ( 0 ; £ ( # ) ) -> T ( H ) such that for $ , * i , $ 2 , * e C o ( 6 ; B ( H ) ) and o e S ( H ) (1) £ ( $ , $ ) > 0 . (2) B ( $ i + $ 2 , * ) = -B($ ll >I') + B($2, , I')(3) S ( a $ , * ) = a B ( $ , * ) . (4) £ ( $ , # ) * = B ( * , $ ) . That is, !?(■, •) is a positive and conjugate bimodule map. If £ g rbca{%, X) and B^ is denned by B€(*,*)=[TS($),T{(*)]= j j

$dM5$*,

$,*eC0(e;B(H)),

(6.10)

136

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

where T( : C 0 ( 9 ; B(H)) -> X is defined by (6.2). Then we see that 5 £ is abounded, positive and conjugate bimodule map, where the norm is given by | | S € | | = s u p { | | B e ( * , t t ) | | T : \mU

Halloo < 1} = ||T € || 2 = | | e | | 0 ( e ) 2 .

Considering Theorem 7 and Theorem II.4.13, we have: P r o p o s i t i o n 8. Let B : C 0 ( 6 ; £ ( # ) ) x C 0 ( 6 ; £ ( # ) ) -> T ( # ) 6e a bounded, positive and conjugate bimodule map. Then there exist a normal Hilbert B(H)module Y and a regular measure £ e r&ca(2l, Y) such that B = B^ with \\B\\ = | | ^ | | 0 ( 6 ) 2 , where B^ is defined by (6.10).

3.7. Convergence In this section we consider some convergence and approximation properties of sequences of X-valued and T(H)-valued measures defined on the Borel <7-algebra 25 of the real line R. Since 25 is the Baire cr-algbera, every measure on 25 with values in any Banach space is regular, i.e., ea(25,3:) = rea(25,£) for any Banach space X by Remark 6.2. Recall that «ca(25,3E) denotes the set of all measures in ca(25,3E) of bounded variation. Also recall that, for f £ ca(21,X) and A £ 25, 6^(A) = &{£(AnB) :B £ 25} denotes the closed submodule of X generated by the set {£{A (1 B) : B 6 25}. T h e o r e m 1. (1) If £ e 6ca(25,X) and D i m 6 € ( K ) < oo, then £ £ t>ca(23,X). (2) For each £ £ 6ca(25, X) there is a sequence {^ n }^ = 1 C vca(25,X) such that ||£n(A)-f(A)||

->0,

A £23.

(7.1)

Proof. (1) Since Dim 6^(R) = q < oo, we can choose a gramian basis { x 1 ; . . . , xq} C S £ (R) (cf. Definition II.3.9), so that

f M = £[£(>*).**]**.

4e«8

fc=i

(cf. Theorem II.3.4). Let n £ II(R) and observe that

fc=l

A6irfc=l

3.7. CONVERGENCE

137

^ E E ||«oi fc )(A)|| r < £ |foit|(R) fc=l

A6TT

fc=l

an(

we use

^

^ (1-4). hence the total variation of £ is

||e(A)i|. Y : TT e n ( K ) l
(2) Since 03 has a countable generator, we have Dim6^(R) < N0- Hence there is a countable gramian basis {xi,x2, ■ ■ ■} Q 6^(R). Thus we get oo

fc=i

Putting £„(■) = £

[£(•),ifc]xfc for n > 1, we have that { £ „ } ~

x

C wca(«8,X) by

fc=i

(1) and \\£„(A) - £(A)\\X

-> 0 for every ^ g S3.

Let £ € ea(S3,X) and put
= 0}:

where dA is the boundary of A. Every A g 03^ is called a continuity set of £. L e m m a 2. 2$£ is an algebra for each £ 6 ea(S3,X).
If £ £ uca(93,X), i/ien

Proof. The first assertion is clear. As to the second, let O be an open set and put Aa = {t 6 R : dist(f,O c ) > a},

a > 0,

where dist(£,O c ) = inf {|£ - s\ : s e Oc). Since f is of bounded variation and dAa C { ( e l : dist(t,Oc) = a } , we can choose a sequence { a f c } ^ of positive oo

numbers such that a* I 0 and AQk g 93^ for k > 1. Hence O = U ^ 4 ^ g
=

{TT

g n(X) : A g 93? for A g

TT}.

138

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

The role of 5$£ is clarified in the following: T h e o r e m 3. For £ e vca^^X)

and open A, B e 25^ it holds that

|e|(A) = sup I £

|K(A)||x : TT 6 n € (A) 1,

|M € |( J 4,B) = s u p j ^ £ ||W<(A1A')||T:T€ne(^),jr'enc(fl)L I A£ir A'gjr' J

(7.2)

(7.3)

Proof. Since £ is of bounded variation, |£|(-) is a regular finite positive measure on 03. Assume that A = (a, /?), a bounded interval. [The case where a = — oo or (3 = oo is similarly proved.] Let e > 0 be given. We can find real numbers to,ti,. .. ,tn such that a = to < ti < ■ ■ ■ < tn = P and n

mA)-e<^2U{Ak)\\X} fc=l

where Ak = (t fc _i,t fc ] for 1 < k < n - 1 and An = {t„-i,tn). Suppose ||f||({t,-}) = IIC({*j'})l|x > 0- Such a point tj is called an atom of £. We remove this point tj as follows. Since |£|() is regular and the atoms of £ are at most countable, we can find non-atomic points sj and s | such that tj_i < s)
s) < tj+1,

|£|((s],s ; 2 )\{^}) < - .

Then it holds that UtAMx

+ U(AJ+1)\\x

= | | e ( ( * j - i , » J ] ) | | x + l|€((*},*i))L + Il^({*i})||x

+ llf((*i.*J))L + llc((^*i-»l)L ^llf((«J-i.-J])Hx + ||f((.il,^))||jr + ll«([^*>+il)ll, + 2iei((4,»f)\{*i})

\\mMx<\m)\\x+\m\{tMx

3.7. CONVERGENCE

139

Hence we have that

J2u(Ak)\\x< £ ii«^)iu+i;il«i*DL + | , fc=i

Mij+i

fc=i

so that we removed the atomic point tj. In at most (n — 1) steps as above we can get rid of all the atoms from among { t i , . . . , t „ _ i } and obtain a set of disjoint intervals { 5 i , . . . , f i m } e n s ( . 4 ) such that m

|£P)<£ilf(Bfc)IU + 3£. fc=i

Since e > 0 is arbitrary, (7.2) holds. For a general open set A g ?B^ it can be written as the union of disjoint intervals oo

as A =

U (a„,0„)

with relatively compact (an,(3n)

6 53j for n > 1. Since |£|(-) is

c.a. on 25, (7.2) also follows from the above argument. (7.3) is proved with the same idea of proving (7.2) although the details are little more complicated, and the proof is left to the reader. [Note that (7.3) holds if M^ is of bounded variation.] Definition 4. A sequence {^ n }^Li ^ ca(*B,X) is said to converge weakly to £ G ca( £, if ||f n (A) - f(-A)||x -► 0 for every A 6 25 ? . A sequence {^nJ^Li of finite positive measures on 23 is said to converge weakly to a finite positive measure v on 25, denoted vn ==> */, if v„{A) —> i^(^l) for every A 6 58 with I/(9J4) = 0. C(R) denotes the Banach space of all bounded continuous functions on R with the sup norm. For v € ca(23,R + ), where R + = [0,oo), let A„ be defined by \„((p) = / ipdv,

tpe C(R).

Then, the following theorem is known (cf. Billingsley [1, pp. 11-14]). T h e o r e m 5 (Weak C o n v e r g e n c e ) . Let 0 be a metric space and 21 be its Borel a-algebra. For a bounded sequence {^n}^Li S ca{% R + ) and v £ ca(2l,R+) the following conditions are equivalent: (1) vn => v. (2) A„„( A ^ ) for every tp E C ( 6 ) .

140

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

(3) A^n (ip) -4 t\.v(ip) for every uniformly continuous ip G C(©). (4) lim sup vn(C) < v(C) for every closed C G 21. n—too

(5) liminf vn(0) > v(0) for every open O 6 21. n—>oo

We want to consider weak convergence of X-valued measures. Let us first consider weak convergence of Hilbert space valued o.s. measures. T h e o r e m 6. Let £ n (n > 1), £ € eaos(Q3,ff). Then f„ => £ ijff \\Tin(ip)~Ti(ip)\\H^Q,

(7.4)

Proof. Suppose that £n => £. Let ip G C(R) and e > 0 be given. P u t 5 = sup{||Cn||(K), | | e | | ( M ) : n e N } = s u p { | | ^ ( R ) | | H ) | | £ ( R ) | | H : n e N } < co.

It suffices to show that (7.4) holds for all real valued ip e C(R). Let ip e C(R) be real valued and choose a finite interval (a, /3) such that a < ip < (3. Let a = to < t\ < ■ ■ ■ < tk = P be such that ti-ti_i<^,

u({tj})=0,

l<j
Put ^4j = yj — 1 ((tj_i,tj]) for 1 0 so that

j—l

40

llu^-)-^)L< 2 f c ( N c + | / 3 | r

i*.

Hence we have for n > jV

\\Titt(ip) -T^)\\H

< \\Tin(ip- vo||H + ||r € .(v) -r e (v>)L + ||T€(V> - V )|| H < h-

V-IIOO(II^IKR)

+ iieii(R)) + £ IMIM^) - W L

3.7. CONVERGENCE

141

Consequently (7.4) holds. Conversely suppose (7.4) is true. Then this implies that / \
-> / \
J«

tp€ C(R),

Jm

where ^ „ ( - ) = ||£n(-)llfl and U((-) = \\€(-)\\%v^n =>■ V£. Note that 5B^ = 3$„,, where ^

= { ^ 3 :

It follows from Theorem 5 that

i/ { (9A) = 0}.

Let A g 03 £ be fixed and e > 0 be given. Since i/^(A\A) exists n 0 > 1 such that 0<^„(A\A)<(|)2,

= 0 and v^a =>■ v^, there

n > n0.

For each a > 0 put A Q = {t e I : dist(i, A) < a } . We can choose {ak}'^=1 C R such that Qfc | 0 and A a i € 03^. Moreover, since A n A£fc = 0, there exists i/pfc g C(R) such that 1^4 < 1. Now choose fc > 1 so that 0<^(AQt\^)<(^)2Then, again by j / ^ n => v^ we can choose n j > 1 so that

0
< (|) ,

n>»i.

Furthermore, we can choose n 2 > 1 so that ||T^n(v?) — Tj(v?)|| H < - for n > «2 by assumption (7.4). Thus for n > max{no,n 1 ,7i2} it holds that

\\UA) ~ t{A)\\g < \\UA) ~ aA)\\H + \\UA) - £(A-)\\H + \\Zll)-Z(A)\\B < ^„{A\Ay + \\UA) + \\TdVk)-aA)\\H

-^.(V-OIIH

+ \\Tui
+ M\\{A\A)

< \ + vi« {AcMV + 1 + ^{Aa„\A)k < e. Therefore we conclude that fn => £

T^k)\\H

142

III.

STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

C(R;B(H)) denotes the Banach space of all B(H)-va\ued bounded continuous functions on K. Then, weak convergence of X-valued c.a.g.o.s. measures are charac terized as follows: T h e o r e m 7. For £,£„(n > 1) € ca#os(!B,A") the conditions (1),(2) and (3) are equivalent:

(1) £n => e (2) | | T € i » -Te(
-»- 0 /or euen/ y? € C(R).

(3) | | r u ( $ ) - T 4 ( $ ) | | x - > Q / o r every * e C(R;JBf(J5T)). / / one o/ i/ze above conditions is satisfied, then it holds that (4)

=> v€.

V(n

(6) ^n,0 =*■ £0 /or euen/ <j> e H. Proof. (1) <S> (2) is clear from Theorem 6 and (3) =>■ (2) is obvious. (2) => (3). First note that {||£n[|o(R)}SLi is a bounded sequence since ||f n || 0 (R) = \\U{R)\\x - » | | f ( R ) | | x = ll€Ho(R) (cf. Theorem 1.5 (3)), so that ||e„||o(R) < Q, n 6 N for some a > 0. Let C(R) 0 B(H) be the algebraic tensor product. For $ = p

£

J=I

|r£„(*) - r5($)||_Y = ]T

O,(T 5 ,>,)

- r£(^))

J=I I.Y

j= l

L e t e > 0 be given. Forageneral $ € C(R;2?(if)) we can choose a * e C(R)©S(i7) such that

| | $ - * |

U

< _

M O ( R )

since C(R) © £ ( # ) is dense in

C(R;B{H)).

Moreover, choose n0 > 1 such that ||T ? n (#) - T ? ( * ) | | Y < e for n > n 0 . Then, we have that for every n > n0

||r?„($)-rc($)||x < ||r?B($) - r t ( * ) | x + \\Tum-T^)\\x +

\\T^)-T^)\\X

(4) and (5). Recall that F 4 (-) = [£(•),?(•)] € ca(«B,T+(ff)). Since .4 e
3.7. CONVERGENCE

143

for every A g S ^ implies that vu (A) -> v${A) and ||F € n (A) -F${A)\\r A e ^ . Hence (4) and (5) follow.

-4 0 for every

(2) => (6). Recall that ^ ( - ) = (£(•), )„■ Observe that for £ H and

JR

JR

JR

in ||-|| 2 norm. Theorem 6.

)H

Since £ 0 , £„>(^ (n e N) G caos (58, Lj^fi)), we have fni0 => £$ by

Recall that, for £ e ca(5B,X), the scalar bimeasure m^ e M was defined as m { ( A , B ) = ( f ( A ) , f ( B ) ) x for A , B e 58. If |m ? |(R,R) < oo, then m c and |m € | can be extended to measures on (R 2 , 23 ® 58), where 58 ® 58 is the Borel cr-algebra of R 2 . The following proposition corresponds to Theorem 6. P r o p o s i t i o n 8. Let £,£„ (n 6 N) 6 ca(<8,X). (1) / / {||^ n ||(R)}~ = i is bounded and £„ => £, iften ||r«.(¥»)-T€(¥.)l|JC-+0,

P6C(R).

(7.5)

(2) / / | m ? | ( R , R ) , | m ? J ( R , R ) < oo for n e N, | m s J => |m 4 | and (7.5) /io/d.s, rTien fn => f• Proof. (1) The same proof of necessity part of Theorem 6 is applied. (2) In the proof of sufficiency part of Theorem 6, v^ and v^n are replaced by m^ and m £, then ||T 5 „($) - T ^ ( $ ) | | x -4 0 for every $ e C(R;B(H)). This follows from Theorems 6 and 7. For a sequence of measures of bounded variation we can give a necessary and sufficient condition for weak convergence. T h e o r e m 10. Let f , f „ ( n e N) € vca{*&,X) and suppose that {|f„|(R)}£° =1 bounded. Then £„ => £ iff |fn| => If I and (7-5) noZds. Proof. The "if part is proved in a similar manner as in the proof of Theorem 6.

is

144

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

As to the "only i f part, suppose £„ => £. It follows from Proposition 8 that (7.5) holds. To show that |£ n | => |£| it suffices to prove that \i\(0)

for every open set O € *B in view of Theorem 5. So at first let O € 93^ be open. By Theorem 3 for any e > 0 there exists a partition {Au . .. ,Ak} € n^(O) such that k

KI(0)<]T||^)||x+ £ . Since £ n => £, there is some no € N such that llUii,) " f (^i)||x n 0 .

Hence we have that for n > no fc

lei(O) < £ { | | U ^ ' ) I U + IK(^) - M^)IU} + £ < 16.1(0) + 2e. 3=1

Thus |£|(0) < liminf | f n | ( 0 ) . n—*oo

For a general open set O e B we can choose an oo

increasing sequence {Ok}kx>=1 Q 23e of open sets such that U Ok = 0 by Lemma 2. fc=l

Consequently |£|(0) = lim |€|(Ofc) < lim liminf |f„|(O fc fc—>oo

fc—>oo

n—>oo

n—>oo

The following is a consequence of Theorem 3 and the proof of the above theorem. Corollary 11. Let £,£„ (n e N) e t>ca(2},X) and suppose that £ n => £. T/ien it ) < liminf |£„|(0). |f |(B) < liminf |f„|(R), |M{](K,R) < liminf |M ? n

Definition 12. A set {fx : A G 1 } C ca(25,X) is said to be uniformly regular if for any A e 2? and e > 0 there exist a compact set C and an open set O such that C CACO and ||fA|| ( 0 \ C ) < e for every A € X.

BIBLIOGRAPHICAL NOTES

145

Uniform regularity of a set of measures is applied to obtain conditions for weak convergence. P r o p o s i t i o n 13. Let £ , £ n ( n g N) g ca(Q3,X) and suppose that {||£ n ||(R)}£° =1 is bounded. If {£„ : n g N} is uniformly regular and (7.5) holds, then (,n => £. Proof. Let A g 2}^ be fixed. Since {(,„ : n e N} is uniformly regular, for any e > 0 there exist a compact set C and an open set O such that C C A C O and ||f | | ( 0 \ C ) , \\U\\{0\C) < e for n g N. Define a continuous function

) ~~ ^ ( v ) l l x < e for n > n0. Then we have that for n > n0 ||e„(A) - £ ( A ) | | X < | | ^ ( A ) - % » | |

x

+ | | r

c

» - r € ( ^ | | x + ||T e ( v ) - £ ( A ) \ \ X

-+ 0. Therefore £ n => f.

Note that under the assumption of Proposition 13 we may conclude that ||£„(yl) — £(A)\\x -> 0 for every A g 58. C o r o l l a r y 14. Let £,£„(n g N) g «ca(<8, X ) and suppose that {|£„|(R)}£°=1 is bounded. Then, fn => f iff {£„ : n g N} is uniformly regular and (7.5) holds. Proof. The "if part follows from Proposition 13. As to the "only i f part, it follows from Theorem 10 that (7.5) and |£„| => |£| hold. Moreover, |£„| => |£| implies that {£n : n g N} is uniformly regular by Billingsley [1, p. 241] and Diestel and Uhl [1, p. 157],

B i b l i o g r a p h i c a l notes We have spent a considerable amount of space for vector measure and integra tion theory as well as the corresponding bimeasure theory. The general references for vector measures are Dunford and Schwartz [1, IV](1958), Dinculeanu [1](1967), Kluvanek and Knowles [1](1976) and Diestel and Uhl [1](1977). For bimeasure the ory we refer to Chang and Rao [2](1986) and there seems to be no monograph treating this subject. 3.1. Semivanations and variations. Hilbert space valued c.a.o.s. measures were introduced by Wiener [1](1923) and were extensively studied by Masani [3](1968). Masani also wrote a research expository paper [4](1970) on q.i. measures, which includes g.o.s. measures. (2), (3) and (5) of Theorem 1.5 are proved in Kakihara

146

III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES

[2](1982), [3](1983) and [15](1992), respectively. Lemma 1.6 is from Makagon and Salehi [1](1987). Proposition 1.7 and Corollary 1.8 are noted here. The first part of Theorem 1.13 is due to Ylinen [2](1978) and the second to Chang and Rao [2]. Proposition 1.14 is due to Ylinen [2]. Proposition 1.15 and Lemma 1.17 are new. Proposition 1.18 is proved in Kakihara [15]. Lemma 1.19 is noted here and some of the results are stated earlier in Kakihara [2]. Theorem 1.20 is new. (1) and (2) of Theorem 1.22 are proved in Kakihara [5](1983) and [6](1984), respectively and (3) in Kakihara [5, 15]. Example 1.24 is mostly from Kakihara [15] and, in particular, (2) is essentially due to Edwards [1](1955) and (6) is newly noted. A more about c.a.g.o.s. measures is found in Molnar and Kakihara [1](1990). 3.2. Orthogonally scattered dilations. Definitions 2.1 and 2.2 and (l)-(6) of Remark 2.3 are given by Bartle, Dunford and Schwartz [l](1955). [More about DSintegral, see Bartle [l](1956).] (7) of Remark 2.3 and Theorem 2.4 are found in Abreu and Salehi [l](1984). Proposition 2.5 is noted here. Orthogonally scattered dilation of a Hilbert space valued measure was first considered by Abreu [l](1970) in a special case (see also Abreu [2](1976)). Lemma 2.9 is due to Rosenberg [4](1981). Theorem 2.10 is proved by several authors (Niemi [4](1977) and Rosenberg [4]) and the idea of proving (3) => (5) is originally due to Pietsch [2] (1969). There are many papers dealing with Grothendieck inequality and Grothendieck constant KQ. We refer to Blei [1](1987), Gilbert [1](1977), Haagerup [1, 2](1985, 1987), Lindenstrauss and Peiczynski [1](1968), Krivine [1](1979), Pisier [1](1978) and Rietz [1](1974). Lemma 2.12 is due to Rosenberg [4] (see also Miamee and Salehi [2](1978)). Theorem 2.13 is from Niemi [4]. See also Miamee and Salehi [2] whose proof was completed by Houdre [1](1990). Chatterji [1](1982) gave some insight for o.s.d. of Hilbert space valued f.a. measures. The key idea of proving Lemma 2.12 and Theorem 2.13 is in Rogge [1](1969). 3.3. Gramian orthogonally scattered dilations. In Theorem 3.2, (1) ^ (2) is clearly stated in Kakihara [7] (1985), which is essentially due to Rosenberg [4] and (1) <=> (3) is proved by Kakihara [15]. Lemmas 3.3 and 3.5 are due to Makagon and Salehi [1]. Theorem 3.7 is from Rosenberg [4] and Makagon and Salehi [1]. In particular, (1) <=> (2) is originally due to Abreu [3] (1978). Theorem 3.8 is also in Makagon and Salehi [1]. Lemma 3.9 is by Kakihara [15]. Lemma 3.10 is in Makagon and Salehi [1]. [The key idea is comming from Pietsch [2] and Rogge [1].] Theorem 3.11 is proved by Pietsch [1](1967). Theorem 3.12 and Proposition 3.13 are due to Slowikowski [1](1969). Proposition 3.14 is proved by Makagon and Salehi [1]. Theorem 3.15 is stated and proved in Kakihara [20](1996). 3.4. The spaces Ll(F) and L2{F). Lemma 4.2 is well-known (cf. Hille-Phillips [1](1957)). Lemma 4.3 is from Mandrekar and Salehi [1](1970). The generalized inverse of a (not necessarily square) matrix was defined and studied by Penrose [1](1955) and of an operator on a Hilbert space was by Hestenes [1](1961). Lemma 4.5 is due to Hestenes [1]. Lemma 4.6 is proved by Fan [1](1949). Lemma 4.7 is

BIBLIOGRAPHICAL NOTES

147

proved by Mandrekar and Salehi [1]. Theorem 4.8 is new. (l)-(3) of Lemma 4.10 are from Mandrekar and Salehi [1] and (4) is noted here. Theorems 4.11 and 4.12 and Proposition 4.13 are also from Mandrekar and Salehi [1]. The finite dimensional case of Theorems 4.12 and 13 is due to Rosenberg [1](1964). Related topics can be seen in Robertson and Rosenberg [1](1968) and Welch [1](1973). 3.5. The spaces £*(£) and £ 2 ( M ) . The MT-integration theory is due to Morse and Transue [1, 2](1955) and [3, 4](1956), which is a functional approach. A measure theoretic approach is due to Chang and Rao [1](1983). In c) or c') of Definition 5.2, coincidence of two integrals ((5.5) and (5.6)) is superfluous (cf. Dobrakov [5](1990)). Even so, the original definition is still comprehensive. Example 5.3 is due to Chang and Rao [1]. Proposition 5.4, Theorem 5.5, Corollary 5.6 and Theorem 5.7 are due to Chang and Rao [2]. Definition 5.8 is newly given. Proposition 5.9 is noted here. More about bimeasure theory we refer to Dobrakov [2](1987), [3, 4](1988) and [5, 6](1990), Kluvanek [2](1981), Thomas [1](1970) and Ylinen [2]. When the measure spaces are LCA groups, bimeasure algebras are considered by Graham and Schreiber [1, 2, 3](1984, 87, 88), Gilbert, Ito and Schreiber [1](1985) and Ylinen [1](1975) (see also Varopoulos [1](1967)). 3.6. Riesz type theorems. Remark 6.2 is due to Dinculeanu and Kluvanek [1](1967) and Kluvanek [1](1967) for a locally convex space valued measure. Corol lary 6.4 is proved by Kluvanek [1]. Lemma 6.5 is noted here (cf. Kakihara [2]). Lemma 6.6 is essentially due to Dobrakov [1](1971). Theorem 6.7 is proved in Kak ihara [2]. Proposition 6.8 is noted here (see also Ylinen [2]). 3.7. Convergence. Theorem 7.1 is due to Kakihara [16](1992). Lemma 7.2 is due to Ressel [1](1974). Theorem 7.3 is proved in Kakihara [15, 16]. Theorem 7.6 is due to Ressel [1]. Theorem 7.7 is from Kakihara [16] and Proposition 7.8 from Kakihara [15]. Theorem 7.10, Corollary 7.11, Proposition 7.13 and Corollary 7.14 are due to Kakihara [16].

CHAPTER IV

MULTIDIMENSIONAL STOCHASTIC

PROCESSES

The preceding two chapters are devoted to necessary preparatory material to be used in multidimensional, mainly infinite dimensional, second order stochastic processes which is the topic of this chapter. This is done by considering Hilbert space valued stochastic processes, namely X = LQ(Q ; i?)-valued processes, where H is assumed to be separable throughout this chapter. After giving basic concepts on processes, we introduce the class of stationary processes, which is important in both theory and applications and with which many nonstationary classes are analyzed. Among nonstationary processes, harmonizable and V-bounded classes are of special interest and we investigate their properties in detail. As a generalization of harmonizable processes, Cramer and Karhunen classes are introduced and their properties studied. Series and moving average represenations are also important in applications, so they are considered in some detail. Further, approximation and convergence, and subordination properties of processes are discussed.

4.1. General c o n c e p t s In Section 1.1, we considered scalar second order stochastic processes, i.e., LQ(Q)valued processes on the real line R and, in Section 1.3, some motivation was given to treat infinite dimensional stochastic processes. In this section, we shall introduce fundamental definitions and results of X = L\{£l] #)-valued processes on an LCA group G. Definition 1. (1) A mapping x(-) : G -+ X is called an X-valued process on G or a Hilbert space valued second order centered stochastic process on G. We denote it by x or {x(t)} or {x(t,u>)}. (2) The operator covariance function F of a process x = {x(t)} is a T(H)-valued function o n G x G defined by T{s,t) = [x{s).x{t)}, 148

s,teG,

4.1. GENERAL CONCEPTS

149

and the scalar covariance function 7 is a C-valued function on G x G denned by y(s,t)=

(x(s),x(t))x,

s,teG.

Sometimes they are obtained as Tz and y$, respectively. (3) The vector time domain "Ho(x) of a process x = {x(t)} of X generated by the set {x(t) : t € G}, i.e., n0{x)

= e0{x(t)

is a closed subspace

:teG},

and the modular time domain ~H(x) is a closed submodule of X generated by the set {x(t) :t£G}, i.e.,

U{x) = e{x{t)

:teG}.

(4) A process {x(t)} is said to be (mean) continuous if the function x(-) is con tinuous in the norm of X, and to be weakly continuous if the function (x(-),y)„ is continuous for every y 6 V.o{x). {x(t)} is said to be bounded if there is a positive constant a > 0 such that ||a:(i)||x < a foi t e G. (5) Let x = {x(t)} be an X-valued process and y = {y(t)} be a Y = L\{Q.; H)valued process, where (fl,5, p) is a probability measure space. Then, x and y are said to be equivalent if there exists a gramian unitary operator U from H(x) onto H{y) such that Ux(t) = y(t) for t 6 G. x and y are said to be similar if there exists an invertible bounded module map T from ~H{x) onto H(y) such that Tx(t) = y{t) for t e G. When H = C, we treat the scalar or an JLg(fl)-valued process. For such a process the operator covariance function reduces to the scalar covariance function and the modular time domain to the vector time domain. We then simply call them the covariance function and the time domain of the process. With these definitions we collect some basic results as follows: P r o p o s i t i o n 2. (1) The operator covariance function of an X-valued process on G is a T(H)-valued p.d.k. on G x G. Conversely, for any T(H)-valued p.d.k. F on G xG there exist a normal Hilbert B{H)-module Y = Lg(0 ; H) for some probability measure space (fi,5> f1) and a Y-valued process on G whose operator covariance function is Y. (2) For an X-valued process x = {x(t)} on G with the operator and scalar covari ance functions T and 7, respectively, it holds that n(£) = Xr

and

H0{x) ~ £ 7 ,

where Xr is the r.k. normal Hilbert B(H)-module (cf. Section 2-4).

of T and f) 7 is the RKHS of 7

150

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

(3) An X-valued process {x(t)} on G is continuous iff its scalar covariance func tion 7 is continuous on G x G. If {x(t)} is weakly continuous, then 7 is separately continuous on G x G. If {x(t)} is bounded and 7 is separately continuous, then {x(t)} is weakly continuous. (4) Let x = {x(t)} be an X-valued process on G and y = {y{t)} be a Y-valued process on G where Y = LQ(£1;H). Then, x and y are equivalent iff their operator covariance functions are identical. Proof. (1) The first part is readily verified and the second follows from Theorem II.4.13 and its proof. (2) also follows from Theorem II.4.13 and its proof. More precisely, we can show that an operator V : ~H{x) —> Xr defined by Vx{t) = T{t,-),

teG

(1.1)

is a module map and preserves the gramian, so that it can be extended to a gramian unitary operator. (3) Suppose {x(t)} is continuous and let s,t 6 G be arbitrary. Then, for any h,k € G we get

\y{sh,tk) — 7(s, t)| = \(x(sh),x(tk))x

-

(x(s),x(t))x\

< \(x{sh)-x{s),x{tk))x\

+

< \\x(sh) - x(s)\\x\\x(tk)\\x —> 0

\(x{s),x{tk)-x(t))x\ + \\x(s)\\x\\x(tk)

-

x(t)\\x

as h, k —> e

since {x(t)} is continuous, where e is the identity of G. Thus 7 is continuous on G xG. Conversely, suppose 7 is continuous on G x G and let t € G be arbitrary. Then, for any k e G w e get \\x(th) - x{t)\\x

= j(th, th) - j(th, t) - j(t, th) + j(t,

t)-¥0

as h —> e since 7 is continuous on G x G. Thus {x(t)} is continuous. The first statement about weak continuity is obvious. As to the second statement, observe that, for each s 6 G, (x(-),x(s))x = 7(-,s) is continuous by assumption. n

Hence, for any y = J2 aix{si)i

{xi')^y)x

ls

continuous. For a general y e

choose a sequence {zn}%>=1 of the form zn = ]T a„jx(sn

j) such that \\zn -y\\

rl0{x) x

-4 0.

3=1

Then, (x{-),zn) is continuous for n > 1 and (x(-),y) = lim (x(-),zn)„, which is a uniform limit since {x(t)} is bounded. Therefore, {x(t)} is weakly continuous.

4.2. STATIONARY PROCESSES

151

(4) Suppose x and y are equivalent. Then there exists a gramian unitary operator U : H(x) —> %(?/) such that E/x(f) = j/(t) for t e G. Hence, for s,t 6 G we have

i y s , * ) = [»(»),»(*)] = [t/x(s),[/x(o] = [*(«),*(*)] =r fi (s,0Conversely, assume r 4 = Fg. Then we can define an operator V : H{x) —> H(y) by Vx(t) = y(t) for t 6 G and observe that V can be extended (by B(H)-linearity) to a gramian unitary operator from H(x) onto H[y) since Fz = Ty.

4.2.

Stationary processes

In Section 1.1, we considered an Lg(fi)-valued stationary process {x(t)} on R and derived an integral representation x(t) = f eUn£(du),

( e l

(2.1)

for a unique o.s. measure £ 6 caos(93,Z,g(fi)), where 25 is the Borel u-algebra of R. In this section, we introduce three stationarities for X-valued processes and obtain integral representations by suitable X-valued measures. Thus let G be the dual group of G and 23 g be its Borel a-algebra. As before the value of x £ G at t 6 G is denoted by (t,x)- Note that, if {x(t)} is an X-valued process, then {(x(t),)#} is an LQ(fi)-valued process for each e H. Definition 1. Let {x(i)} be an X-valued process on G. (1) {x(t)} is said to be stationary provided that its scalar covariance function y(s,t) depends only on the difference s i - 1 and that, if ^ ( s r - 1 ) = y(s,t), then 7 is continuous on G. (2) {x(t)} is said to be scalarly stationary if, for every 4> £ H, the Lo(f2)-valued process {(x(t), 4>)H} is stationary in the sense of (1) when H = C (3) {x(t)} is said to be operator stationary provided that its operator covariance function r(s,t) depends only on the difference s i - 1 and that, if ^ ( s i ^ 1 ) = F(s,t), then T is a T(H)-valued weakly continuous function on G, i.e., tr(aF(-)) is continuous for every a £ B{H). Now we obtain integral representations for stationary processes in the above sense. T h e o r e m 2. Let {x(t)}

be an, X-valued process on G.

(1) {x(t)} is stationary rcaos(^8Q, X) such that

iff there exists a unique regular o.s. measure

x(t)=

[ {t,x)S{dxl, JG

t^G,

£ 6

(2.2)

152

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

where rcaos{*&Q,X) = rca(Q5g,X) n eaos(23g,X). In this case, the scalar ance function 7 is written as

7 M ) = 7(s« _1 )= [ (st-\x)"{dx),

covan-

s,teG

JG

for a positive finite measure v 6 rca(25g,K + ) given by v{-) = ||£(-)ILY> 2-e-> v = vi(2) {x(t)} is operator stationary iff there exists a unique regular g.o.s. measure £, € rcagos(*BQ,X) such that (2.2) holds, where rcagos^&Q,X) = rca(*BQ,X) PI cagos{^&Q, X). In this case, the operator covariance function T is written as

r(s,t) = f(sr1) = f (st-\x)F(dx),

s,teG

JG

for a T+{H)-valued

measure

F

6 rca(f&Q,T+(H)),

where, in fact,

F(-)

=

[£(•),€(•)], i.e.,F = F4. Proof. (1) follows from (2) when we consider H = C. So we shall give a proof of (2), which is essentially same as the one given in Section 1.1 for an Lg(f2)-valued stationary process. Let {x(t)} be operator stationary and define, for each t £ G, an operator Uo(t) : H{x) -* H(x) by U0(t)x(s) = x(st), seG, or more generally by Uo{t)( ^ajxitj))

=

^ajxitjt),

where aj € B(H) and tj £ G for 1 < j < n. Since {x(t)} is operator stationary we have for s,u E G [U0{t)x{s),U0{t)x{u)}

= [x{st),x{ut)}

=t(su-1)

=

[x(s),x{u)].

Hence Uo{t) is well-defined and can be uniquely extended to a gramian unitary operator U(t) on ~H{x). It is easy to see that {{/(£)} t e G forms a continuous g.u.r. of G on H(x) (cf. Definition 11,5.1). Thus by Proposition II.5.4 there exists a regular gramian spectral measure P(-) on G such that

U(t)= fjt,X)P(dx),

teG.

JG

Putting £(■) = P(-)x(e),

we see that £ e rcagos(!8g, Jf) and

i ( t ) = U{t)x(e) = f(t,X)P(dx)x(e)= JG

[(t,x)Z(dx), JG

teG.

4.2.

153

STATIONARY PROCESSES

The other part is obvious. The unique measures £ obtained in the above theorem will be called representing measures of the processes. Also, v and F are called scalar and operator spectral measures of the process, respectively. Let x = {x(t)} be an X-valued process on G. It follows from the above proof that {x(t)} is stationary iff there exists a continuous unitary operator group {U(t)}teG acting on the Hilbert space Ho{x), the vector time domain, such that x(t) = U(t)x(e) for t e G. Hence ||a;(t)||x = ||a;(e)||jr for t g G. Thus every stationary process is bounded. Similarly, {x(t)} is operator stationary iff there exists a continuous gramian uni tary operator group {[/(i)}tgG acting on the normal Hilbert B(H)-mod\ile H{x), the modular time domain, such that x(t) = U(t)x(e) for t g G. Another characterization of operator stationarity is obtained from Proposition III.1.18, which implies that we need not distinguish scalar and operator stationarities. P r o p o s i t i o n 3. An X-valued process on G is operator stationary iff it is scalarly stationary. Proof. The "only if" part follows from Theorem 2 and Proposition III.1.18. As to the "if part, assume that {x(t)} is scalarly stationary. Then, for each 4> € H there exists £* g rcaos(93g,Lg(£3)) such that

{x(t),4>)H= f (t,x)S+(dx),

tea

JG

Now we have that

lie*ll(G)-||e*(G)||, = ||(i(t),0)H||8,

tec

by Theorem III.1.5 (4) and the stationarity of {{x(t),)H}, where ||-|| 2 is the norm in Lg(fi). If t>fc}£°=1 i s a CONS in H, we get OO

OO

°o>Mt)\\% = J2M**Mt))Jl = ,Z,\\Z*k{G)\\2r tea fc=l fc=l

Since the RHS of (2.3) is independent of t g G, we can define £ by OO

€(•) = £«**(•)**

(2.3)

154

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

and we see that £ e rcagos(Q3g,X), so that f is a representing measure of Therefore, {x(t)} is operator stationary.

{x(t)}.

Let {x(t)} be an X-valued operator stationary process on G with the representing measure £ e rcac/os(
teG,

1 being the identity operator on H as before. Proof. As is easily verified we have H{x) = S j ( G ) , where &^{G) = &{£(A) : A g L2(v^) is given by Vx{t) = (t,-),

teG.

R e m a r k 5. (1) Stationarity is invariant under equivalence but not under similarity in general. It follows from Proposition 1.2 (3) that every stationary process is (mean) continuous. Moreover, it is uniformly (mean) continuous as will be seen in the next section (cf. Proposition 3.7). (2) If G = R, then R = K and (2.2) reduces to (2.1) for some £ e caos(23,X). If G = Z, then Z = T, the unit circle in the complex plane, and we identify it with [0,27r]. In this case (2.2) reduces to x(n)=

[ einu

4.3. HARMONIZABLE PROCESSES

for a measure f 6 caos^j,

155

X), %j being the Borel a-algebra of T.

4.3. Harmonizable processes As was mentioned in Chapter I, harmonizability is a natural extension of stationarity since a harmonizable process {x(t)} on G is given by

*(*)= f {t,x)S(dx),

teG

JG

for a suitable vector measure £, not necessarily o.s. We introduce various harmonizabilitites, investigate some properties of them, and give examples of processes to distinguish each harmonizability. The stationary dilation is of special interest and is discussed in detail. Furthermore, vector and modular spectral domains are defined and their completeness is discussed. A bimeasure m G M = 0K(<8 g x Q3g ; C) (resp. M 6 97t = QJt( 0 there exist compact sets CijCa and open sets O i , 0 2 such that Cj C Aj C Oj for j = 1,2 and | | " i | | ( 0 1 \ C i , 0 2 \ C 2 ) < £ (resp. | | M | | ( 0 i \ C i , O 2 \ C ^ ) < e). M r and OH,, denote the sets of all regular measures in M and 3rt, respectively. Definition 1. Let {x(t)} be an X-valued process on G. (1) {x(t)} is said to be weakly harmonizable if its scalar covariance function 7 is expressible as (a strict integral) 7(»,t) = jL(s,x){t,x')m(dx,dX'),

s,teG

(3.1)

for some (necessarily p.d.) scalar bimeasure m 6 M r . Note that the RHS of (3.1) is well-defined because C(G) C £^(m) (cf. Section 3.5). {x(t}} is said to be strongly harmonizable if its scalar covariance function 7 is given by (3.1) for some bimeasure m e M r „ = M r n M„ of bounded variation (cf. Section 3.1). In either case, m is called the scalar spectral bimeasure of the process. (2) {x(t)} is said to be weakly (resp. strongly) operator harmonizable if its operator covariance function F is expressible as T(s,t)

= JJ^(s,x)J^X1)M(dx,dX'),

s,teG

(3.2)

for some operator bimeasure M € Wlrb = OTrnOT(, of bounded operator semivariation (resp. M E Wtrv = 9Hr n 9Jt„ of bounded variation). Note that the RHS of (3.2)

156

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

is well-defined since C(G) 0 1 C £,2t(M). In either case, M is called the operator spectral bimeasure of the process. (3) {x(t)} is said to be scalarly weakly (resp. scalarly strongly) harmonizable if, for each e H, the L§(fi)-valued process {{x(t),tf>)n} is weakly (resp. strongly) harmonizable in the sense of (1) when H = C. That is, the covariance function l{s,t) = ((X(S),4>)H, {x{t),4>)H)2 can be written as 70(s,t)= / /

(s,x)(t,X')m^{dx,dx'),

s,teG

for some scalar bimeasure m 0 E M r (resp. m 0 6 M r „ ) . As is clear from the definition, every scalarly stationary, stationary or operator stationary process is scalarly harmonizable, harmonizable or operator harmoniazable in both weak and strong senses, respectively. In Section 1.2, we derived an integral representation of an L§(fi)-valued harmonizable process on K. Similarly we can get integral representations for other harmonizabilities. T h e o r e m 2. Let {x(t)} be an X-valued process on G. (1) {x(t)} is weakly harmonizable iff there exists a regular measure rca(Q3g,X) such that z ( * ) = [ (t,x)t{dx),

*eG.

£

£

(3.3)

JG

(2) {x(t)} is weakly operator harmonizable iff there exists a regular measure £ € r6ca(25g, X) of bounded operator semivariation such that (3.3) holds. Proof. (1) The "if part is easily verified. For, let {x(t)} be given by (3.3) for some £ e r c a ( 3 3 g , X ) . Then, for s,teG 7(s,t)

= (Jjs,X)

ttdx),

JjtiX')

= JJg2(s,x)^Xt)(adx)A(dX'))x

Z(dx') =

JjS2(s,x)Kx7)rni(dX,dX'),

where the second equality holds by Proposition III.5.4 and m^ 6 M r is given by m(:(A,B) = {i{A)^(B))x for A, B € <8S. Thus {x(t)} is weakly harmonizable. The "only i f part is proved in exactly the same manner given in Section 1.2 for an Lg(f2)-valued weakly harmonizable process. (2) Assume that {x(t)} is expressed by (3.3) for some £ e rbcaCRg^). the operator covariance function T can be written as

r(s,t)

/>>xK(dx), l{t>y!)ZW) JG

JG

Then,

4.3. HARMONIZABLE PROCESSES

157

= JJ^(*,x)
»(*)= [ (t,x)v(dx),

teG.

JG

Then clearly {?/(£)} is weakly operator harmonizable, which follows from the first part of the proof above. Moreover, the operator covariance function of {y{t)} turned out to be that of {x(t)} since M = Mn. Thus x and y are equivalent by Proposition 1.2(4), so that there exists a gramian unitary operator U : H{y) —> H(x) such that Uy(t) = x(t) for t e G . Then it is easily seen that {x(t)} has an integral representation (3.3) with £ = Ur\ e r6ca(!8Q, X). The measures £ in (3.3) is called the representing measures of the processes {x(t)}. The next two corollaries are immediate consequences of Theorem 2, the integral representation of a harmonizable process, and we omit the proof. Corollary 3 . Every X-valued weakly harmonizable process {x(t)} on G is bounded. More fully, if £ G rca(93g,X) is the representing measure, then \\x(t)\\x < ||£||(G) for teG. Corollary 4. Any harmonizability is invariant under equivalence. More precisely, let x = {x(t)} be an X-valued process and y = {y(t)} be a Y-valued process on G, where Y is a normal Hilbert B(H)-module. Then: (1) / / x and y are equivalent and x is harmonizable harmonizable in the same sense as x.

in any sense, then y is

(2) / / x is weakly (resp. scalarly weakly) harmonizable and T : 'Ho(x) —> Holy) is a bounded linear operator such that y(t) = Tx(t) for teG, then y is also weakly (resp. scalarly weakly) harmonizable. (3) / / x is weakly operator harmonizable and T : ~H(x) —> H(y) is a bounded module map such that y(t) = Tx(t) for t e G , then y is also weakly operator harmonizable.

158

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

The integral representations of scalarly weakly harmonizable processes will be given in the next section (cf. Proposition 4.8), where 5(#,I/g(ft))-valued measures are involved. R e m a r k 5. The following set inclusion relations hold for classes of stationary and harmonizable processes: {scalarly stationary} = {operator stationary}

n

n

{stationary}

n

{strongly operator harmonizable}

n

n

{strongly harmonizable}

n

{weakly operator harmonizable}

n

{scalarly strongly harmonizable}

n

n

{weakly harmonizable}

n

{scalarly weakly harmonizable}

E x a m p l e 6. The above inclusions are mostly proper as seen by the following: We use essentially Example III.1.24. Let {
= f eltu(,{du),

teM.

(3.4)

JR

Then, since £ is o.s. but not g.o.s., {x(t)} is stationary but not operator stationary. (2) Stongly harmonizable but not stationary. Let / ( ^ 0) G LQ(Q) and define £ by

fc€AnN

Then, £ is not o.s. and satisfies that ( £ ( J 4 ) , f (-£?)) by Theorem III.1.22 (2) we see that

> 0 for every i , B e ! 8 . Hence

K | ( R , E ) = ||£(R)|&
defined by (3.4) is strongly harmonizable but not stationary.

(3) Scalarly weakly harmonizable but not scalarly strongly harmonizable. Let £ be in (2) of Example III.1.24. Then for every ifi e H, ||^II(R) 2 < a\(4>,iP)H\2 < oo,

4.3. HARMONIZABLE PROCESSES

159

but | m c J ( R , R ) = Y,\cjk\\(,ip)H\ = oo. Thus, {x(t)} defined by (3.4) is scalarly j,k

weakly harmonizable but not scalarly strongly harmonizable. (4) Strongly operator harmonizable hut not operator stationary. and define £ by

Let <j> ( / 0) € H

fceAnN Then, £ is not g.o.s. and satisfies that [f (A),f (5)] > 0 for every A, B 6 53. Hence, by Theorem III.1.22 (1) we see that |M£|(R]E)=||e(E)||| 0 be arbitrary. Since £ is regular, we can find a compact set C C G such that ||£||(C C ) < e. Since (t, ■) is uniformly continuous on C for f e G, we can choose an open neighbourhood Oe of e such that \(s,x) — {*,x)I < £ f° r all x € G if st'1 € Oe. Then we have for s.t € G with st~l £ Oe \x{s) - x(t)\\x

= II f ({s,X) II " G

~

(t,x))adx)

160

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

/ {(s, x) ~ (t, X))tidx)

Jc

<\\M-&-)\\JZ\\{C) <eU\\(G) + 2e. Consequently, {x(t)}

x

+ f {(s,x) - {*,X»zm Jc<=

+ \\(s, .)-(t, •>||0O||^||(cc)

is uniformly continuous.

We now consider stationary dilations of X-valued processes. D e f i n i t i o n 8. Let {x(t)} be an X-valued process on G. (1) {x(t)} is said to have a scalarly stationary dilation if, for each <j> e H, there exist a Hilbert space ig(O^) for some probability measure space (fi^, 5, P-) and an Lo(fi^)-valued stationary process {y(t)} on G such that L ^ f l ^ ) may be assumed to contatin Ll(£l) as a closed subspace and (x{t),4>)H = J+y^t),

teG,

where J^ : L ^ f i ^ ) —> LQ(Q) is the orthogonal projection. In this case, the family {{V(t)}} JkoOfy>)> ^V},4PH

W

^ ^ e a^so

ca

^e
a

scalarly stationary

dilation of {ar(t)}.

(2) {a:(£)} is said to have an operator stationary dilation if there exist a normal Hilbert B(H)-modu\e Y = LQ(Q,;H) for some probability measure space (fi,5, p) and a i^-valued operator stationary process {y(t)} such that Y contains X as a closed submodule and x(t) = Py(t), teG, where P : Y -4 X is the gramian orthogonal projection. The triple is also called an operator stationary dilation of {x(i)}.

{{y(t)},Y,P}

Let {x(t)} be an X-valued process on G and (/> £ if, and consider the Lg(f2)-valued proces {(x(£), )H}- If { M £ ) > < ^ ) H } is weakly harmonizable with the representing measure £$ S rca(58g,Lg(fi)), then it is easy to see that { ( i ( t ) , 4>)H} has a station ary dilation iff £,$ has a regular o.s.d., which is always the case by Theorem III.2.13 and Remark III.6.2 (3). Thus, we have: P r o p o s i t i o n 9. Every LQ(U)-valued weakly harmonizable process has a stationary dilation and hence every X-valued scalarly weakly harmonizable process has a scalarly stationary dilation. Similarly, we can say that every X-valued weakly harmonizable process has a stationary dilation in the sense that there exist a Hilbert space Y containing X as a closed subspace and a Y-valued stationary process {y(t)} such that x(t) = Jy(t) for teG, where J : Y —> X is the orthogonal projection.

161

4.3. HARMONIZABLE PROCESSES

Now the question is: When does an X-valued process have an operator stationary dilation? If {x{t}} is an X-valued weakly harmonizable process with the representing measure £ G rca(?Bg,X), then it has an operator stationary dilation iff £ satisfies one of the conditions of (1) - (9) in Theorem III.3.15, where all the measures in volved can be assumed regular in view of Remark III.6.2. We also give alternative characterizations of operator stationary dilation in terms of processes as follows, the proof of which is easily obtained by considering representing measures and Theorem III.3.15. T h e o r e m 10. Let {x(t)} ments are equivalent:

be an X-valued process on G. Then the following

(1) {x(t)}

has an operator stationary

(2) {x(t)}

is weakly operator

state

dilation.

harmonizable.

(3) For some CONS {4>k}
{{j/fc(«)},io(Ofc),«4} fc6N of {{x{t),
such that £

dilations

2

\\yk{t)\\2

k

< ao far t €

k—l

G, where ||-||2,jt is the norm in Lg(fifc) for k > 1. (4) For every CONS {4>k}^Li of H the conclusion of (3) holds. We now give a characterization of weak operator harmonizability in terms of a family of contractive operators of positive type. Recall that A(X) denotes the set of all bounded linear operators on X with gramian adjoints (cf. Definition II.2.3). An operator S 6 A(X) is said to be gramian contractive if [Sx,Sx] < [x,x] for x 6 X. A family of operators {T(s) : s 6 G} C A(X) is said to be of positive type if Tis'1) = T(s)* for s 6 G and for each finite set { s i , . . . , s n } C G and each {xi,. . . , xn} C X it holds that n

n

j = l k=\

T h e o r e m 11. Let {x(t)} be an X-valued weakly operator harmonizable process on G. Then there exist a normal Hilbert B(H) module Y = LQ(Q,;H) containing X as a closed submodule, j/o G Y, and a weakly continuous gramian contractive family of operators {T(s) : s € G) C A(Y,X) with T(e) being the identity on X such that, when it is restricted to X, it is of positive type, and x(t) = T(t)yo for t 6 G. Conversely, suppose that {T(s) : s € G} C A(Y,X) is a weakly continuous family of gramian contractive operators such that, when restricted to X, it is of positive type and T(e) is the identity on X, where Y is a normal Hilbert B(H)-module containing X as a closed submodule. Then, {T(t)x 0 } defines an X-valued weakly operator harmonizable process for every x0 g X.

162

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

Proof. Assume that {x(t)} is weakly operator harmonizable. By Theorem 10 it has an operator stationary dilation {{y(t)},Y,P}. By a remark after Theorem 2.2 there is a weakly continuous gramian unitary operator group {U(t) : t £ G} such that y(t) = U{t)y{e) for t 6 G. Putting T(s) = PU{s) for s g G, we claim that {T(s) : s g G} is the desired family. In fact, {T(s) : s 6 G} C A(Y,X) is clear, T(e) = P is the identity on X and ||T(s)|| < ||P||||*7(s)|| < 1. Weak continuity of {T(s) : s £ G} follows from that of {U{s) : s £ G). To see that {T(s) : s g G} is of positive type, let Ti(-) = T ( - ) | x , the restriction of T ( ) to X . Observe that for s g G and x,y E X

= [PU{S-l)x,y]

[T^s-^y]

=

=

[U(S)'x,Py}y

[x,U(s)y}Y=[x,PU(s)y}

= [x,r 1 ( S ) 2 / ] = [T1(S)*a:,2/], implying that ^ ( s - 1 ) = Ti(*)*- Moreover, for any finite sets { s 1 ; . . . , s n } C G and { x i , . . . , xn} C X , one has n

n

n

j=lk=l

n j=lk=l

n

n

^^2[u(skyu(S])x3,xk} j=lk=l n

n

> 0. L

j=i

k=i

Conversely, suppose that there is a family {T(s) : s G G} C i4(F, X) with the properties mentioned above. Write X = H ® K and Y = H ® K\. Since Y is assumed to contain X as a closed submodule, K is a closed subspace of K±. Since i4(y, X) can be identified with 1 ® S(ii"i, X) (cf. the remark after Corollary II.3.4), {T(s) : s g G} can be written as T(s) = l®a(s),

sgG,

where {o(s) : s g G} C B(Ki,K) is a family of contractive operators. Let a(s) be the restriction of a(s) to K for s g G. Then {a(s) : s g G} is of positive type in the sense that n

n

j=ifc=i

for every n > 1, s 1 ; . . . ,s„ G G and A , . . . , / „ g K, where ( - , - ) K is the inner product in if. By the Principal Theorem of Riesz and Sz.-Nagy [1, pp. 475-476] we

163

4.3. HARMONIZABLE PROCESSES

have a Hilbert space K2 containing K as a closed subspace and a weakly continuous group {u(s) : s £ G} of unitary operators on K2 such that d(s) = Ju(s) on K for s £ G, where J : K% —\ K is the orthogonal projection. Here we may assume that K2 = 6o{u(s)f : s £ GJ £ K}. If x0 £ X = H ® K, then the Y2 = H ® K2valued process {y{t)} defined by y(t) = (l®u(t))xo for t £ G is operator stationary. Moreover, the X-valued process {T(t)x0} is weakly operator harmonizable since T(t)x0

= (1 d{t))x0

= (1 ® Ju{t))x0

= (1 ® J ) ( l 0 w(t))i 0 ,

which is a gramian projection of an operator stationary process. To end this section we discuss vector and modular spectral domains of weakly harmonizable processes in connection with Kolmogorov isomorphism theorem. Definition 12. Let {x(t)} be an X-valued weakly harmonizable process on G with the representing measure £ € rca(23g, X ) . The vector spectral domaoin of {x(t)} is the space S?t(m^) of all Borel functions / on G for which ( / , / ) is strictly m^-integrable (cf. Section 3.5). If {x(t)} is weakly operator harmonizable with £ € r&ca(25g, X), then the space -C 1 ^) is called the modular spectral domain of {x{t)} (cf. Section 3.5). It follows from the definition that, for a weakly operator harmonizable process {x(t)}, the modular spectral domain £ 1 (f) is a closed submodule of X and is isomorphic to the modular time domain 'H(x), i.e., £ x (£) = T-i{i) = ©<:(G), so that the Kolmogorov isomorphism theorem holds. Now we examine the vector spectral domain £^(m^). Recall that this is a pre-Hilbert space with inner product and norm given by

(f,9)m( = f [ (f,g)dm€,

||/||mf = (f,f)l(

JGJG

for / , g 6 £,(771^), respectively. Denote by P(G) the set of all trigonometric polyno mials on G, i.e., the set of all functions (p of the form n

(p(-) = ^ajitj,-),

aj eC,tj£G,l<j

1.

>=i

Then we have the following theorem. T h e o r e m 13. / / {x(t)} is an X-valued weakly harmonizable process on G with the representing measure £ £ rca(5Bg,X), then the vector spectral domain £^(mc) is a Hilbert space. Moreover, the vector time domain and vector spectral domain are isomorphic, i.e., "Ho(i) — £^(m^)-

164

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

Proof. Let {i],Y, J} be a regular o.s.d. of £ and suppose that this is minimal, i.e., e0,v = &o{v(A) ■■ A £ %o(l/) is given by

f e L2K

fdri, JG

Putting 6 0 ,£ = 6 0 {£(A) : A 6 33^} = H0(x),

we have the following diagram:

Y = e0„ = H0(y) <-" "

X D ©0,4 = H0(x)

L2(vv)

->
-

where J i is the orthogonal projection on L2(yn) corresponding to J , which is clearly given by J\ = U~1JU. We also define £^ = J i L 2 ( ^ ) , which is a closed subspace of L2(u,j). We shall show that £ 2 (m^) is isomorphically and isometrically embedded onto £^ and finish our proof. We have for t e G that, by JU = UJi,

I (t,x)adx)

= x(t) = Jy(t) = JU{(t,-))

JG

UJx «*,-)) = /

{Ji(t,-))(x')r,(dX').

(3.5)

JG

Putting ft{x')

= {Ji(t, -)){x'),x'

e G, we see that

ft 6 £ 2 ( m ? ) = L\0,

t e G

(3.6)

(cf. Theorem III.5.7). In fact, let t e G be fixed and choose a sequence {gn}%Li C L°(G) of CSg-simple functions such that # n -> / t i/^-a.e. and hence f-a.e., and llfln - /*]|a,^„ -> 0 as n -> 00, where | H | 2 , ^ is the norm in L2{vn). Then for every A € 25Q it holds that

/ 9ndiJA

\

gtdi

I {9n

-9t)lAd£

JA

9i)^-Adr} JG

JG

/ Ji[{9n - gt)lA] dri JG

165

4.3. HARMONIZABLE PROCESSES = \\Ji[(9n-ge)lA}\\2il/Ti

<

< \\9n - 9ih,u., -* 0 e

\\{gn-9e)lAh,vv

( « , / - > oo),

(3.6)

a

*- > { //i 3 " ^ } „ = 1 is Cauchy sequence in X. Thus / t 6 £*(£) = £j(m^). If 7 is the scalar covariance function of {x(t)}, then we have L [ (s,x)'tt^f)rni(dx,dx')

= j{a,t)

= f

JGJG

fM~Mxl>
JG

for s,t e. G. Define a mapping V : Wo{x) —> £*("i^) by Vx(t) = (t,-),

teG.

Then V is an isometry since for s,t £ G (x{s),x(t))x

= j{s,t)

=

ff(s,x)(t,x')m^dx,dX')

= «'>-),(t>-))mf

=

(Vz(*)JVx{t))mt

and clearly V can be extended to the whole space Ho(x) by linearity. Since £ and rri£ are regular, it is not hard to see that Goo(G) is dense in £»(G). Goo(G) is clearly dense in L2((/T)) since r) is regular. Then we see that P(G) is also dense in these spaces since each function in Coo(G) can be uniformly approximated on a compact set by a sequence of functions in P(G) (cf. Naimark [1, p. 413, Corollary 4]). Now consider £^ = JiL2{vn). Note that {/t : t £ G} C £^ by definition. For / € Coo(G) C L2{vn) we have that Jif £ £5 and the mapping V\ : f t-» Jif from Goo(G) C £^(m^) into L2(yv) is an isometry since Iro^

= Jr£j(x)f(x')mi(dX,dx')

= ( j f / d i , J f d^j

= ( I Jifdr,, f Jifdr,) \JG

JG

)Y

= f \Jif\2dvn = \\Jif\\l JG

Since Coo(G) is dense in £»(m^), V\ can be extended to an isometry on Z2(m^)

into

Finally we claim that Vj is from £,l(mv) "onto" £^. Thus we take any / £ £<; C L2(uv). Then we only need to show that / £ £^(m^) = L 1 ^ ) because Jif = f. Choose a sequence {g„}^=i C L°(G) for which gn —> / v^-a.e. and \\gn — f^2,v„ —> 0 00 as n —> 00. Then in a similar manner as in (3.6) we can verify that { J. gn d(,} n=l 1 forms a Cauchy sequence in X. Consequently we see that / £ L ^) = £j(m^ Therefore the proof is complete. The above theorem is also refered to as the Kolmogorov isomorphism

theorem.

166

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

4.4. V - b o u n d e d processes V-boundedness of Ljj(O) -valued processes was defined by the use of Fourier anal ysis. In this section, we introduce three different V-boundednesses for X-valued processes and derive some relations with harmonizabilities. Let L X (G) be the L x -group algebra of an LCA group G, i.e., the Banach *-algebra of all C-valued Borel measurable functions on G which are integrable w.r.t. the Haar measure g of G. For \p 6 i x ( G ) its Fourier transform Tip is defined by

^v(x)

f(t,x)
xeG,

JG

which is in Co(G). Definition 1. (1) An Lg(f2)-valued process {x(t)} on G is said to be V-bounded if it is norm continuous and bounded, and if there exists a constant C > 0 such that

L

< c||^||c

ei'(G),

where the integral is in the sense of Bochner. [Note that ip(t)x(t) grable w.r.t. g.}

(4.1)

is Bochner inte

(2) An X-valued process {x(t)} on G is said to be scalarly V-bounded if, for each 4> € H, the Lo(fi)-valued process {(x(t),4>)H) is V-bounded in the sense of (1). (3) An X-valued process {x(t)} on G is said to be V-bounded if it is norm con tinuous and bounded, and if there exists a constant C > 0 such that

L

< CMc

veL\G).

In order to formulate a stronger notion of V-boundedness, we need to work on the Fourier transforms of operator valued functions. Denote by Ll(G;B{H)) the Banach *-algebra of all B(.ff)-valued Bochner integrable (w.r.t. g) functions on G whose product (convolution), involution and norm are respectively given by

($ * $)(t) = f $( s )^( s -4) g{ds), JG

H i = I \m)\\Q{dt) JG

$*(t) = ^(r 1 )*,

167

4.4. V-BOUNDED PROCESSES

f o r * , * eL1(G;B{H)) and t € G. Moreover, L1 (G;B{H)) is a left J?(#)-module. The Fourier transform J"* of * e L X (G; £?(#)) is defined by

F*(x) = [ (t,xMt)g(dt), JG

xeG.

As is simply verified, the RHS is a well-defined Bochner integral. Observe that Ll{G;B(H)) = LX{G) ® 7 B{H), the tensor product of Ll(G) and B{H) with the greatest crossnorm 7, and the algebraic tensor product L 1 ( G ) 0 B(H) is dense in it. Let CQ(G ; B(H)) be the C*-algebra of all £?(i/)-valued norm continuous func tions on G vanishing at infinity with the sup norm ||-||oo, the product of pointwise multiplication and the involution of taking pointwise adjoint. We have that C0(G;B(H)) = C0{G) ®A B(H). As was observed in Section 3.6 (Lemma III.6.5), we have C0{G\B{H)) C L°°(£;B{H)) for every £ € bca{
then T$> 6 C0(G,B{H))

= {jF* : * e L 1 (G; B{H)))

and ||.F*||oo <

is dense

inC0(G;B(H)).

-

(3) J ^ * * * ) = ( J * ) ^ * ) and J"($*) = (J"$)* /or * , * e L^G;

B{H)).

Proof. (1) and (2) follow from the relations

II^WH < / \\(t,xMt)\\ e(dt) = 11*11!,

x

e G,

./G

^ ( ^ ( G ) 0 £ ( # ) ) = ^ ( L ^ G ) ) 0 B(ff) C C0(G) 0 5 ( H ) and from the fact that T{L\G)) = {Tf : f £ L ^ G ) } is dense in G 0 (G). (3) can be verified in a manner similar to that for Fourier transform theory. Definition 3. An X-valued process {x(t)} on G is said to be operator V-bounded if it is continuous and bounded, and if there is a constant C > 0 such that
(4.2)

JG

We need a Fubuni type theorem for harmonizable processes: L e m m a 4. (1) If {x(t)} is an Ll(Q)-valued (resp. X-valued) weakly harmonizable process with the representing measure £ £ rcai^Bg, LQ(Q)) (resp. rca(*BQ,X)), then [
JG

V £ Ll{G).

(4.3)

168

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

(2) If {x(t)} is an X-valued weakly operator harmonizable process with the repre senting measure £ e rbca(^BQ,X), then [ *{s)x(s)g{ds)=

$eLl(G;B(H)).

[ F*(x)Z(dx),

JG

(4.4)

JG

Proof. (1) Let / e I>l(£l) be arbitrary and observe that for

= j

v(s)(x(s),

f)2

g(ds)

= J rt'){Jg<s>x)t(dx),f) e(ds) = J

j^(s)(s,x){adx)j)2Q(ds), since ( £ ( - ) , / ) 2 e r c a ( « 8 e , C ) ,

=

[f(s,x)
JG

by Fubini's Theorem,

= I FH>{x)Wx)J)2 JG

Since / 6 LQ(Q.) is arbitrary, we get (4.3). (2) I f $ = £ fc=i

lAtafcei°(G;B(/f))nL1(G;B(if)),thenby

/ $(s)x(s)

g{ds) = Y"a f c / lAk(s)x(s)

JG

(1) we have that

g(ds)

JG

fc=i

= X>* /. ^ A „ (x) adx) = ! r*(X) adx)fc=i

J

G

JG

l

For a general $ 6 L (G ; B(H)) we can find a sequence {$ n }£° = 1 C L°(G ; B ( H ) ) n Ll{G;B{H)) such that ||$„ - $||i ->• 0. Hence it holds that

/ *„(«)*(*) e(ds) - / *(*)*(*) e(ds)

JG

JG

X

< / ||*„(«) JG

*(B)||||X(*)||X

o,

g(ds)

169

4.4. V-BOUNDED PROCESSES

fr*n{x)ttdX)-

JG

[ F*(x)ttdx)

JG

<\\?*a-T*U\\ZUG)

as n —> oo. Therefore we have / $ ( s ) x ( s ) g(ds) = lim / $ n ( s ) x ( s ) g(ds) " ^ " J G

JG

= lim f F$n(x)ttdx)= " ^ i g

f

F*(x)Z(dx),

JG

i.e., (4.4) holds. Now we can prove the following theorem. T h e o r e m 5. An X-valued process on G is operator V-bounded iff it is weakly oper ator harmonizable. Proof. Suppose that {x(t)} is an X-valued weakly operator harmonizable process with the representing measure £ 6 rbca^Q, X). Then it follows from Lemma 4 that for $ &Ll{G;B{H)) i $(s)x{s)

g(ds)

JG

< \\m*>u\UG),

JG

so that {x(t)} is operator V-bounded with the constant C = ||£||o(G). Conversely, assume that {x(t)} is operator K-bounded with a constant C > 0 in (4.2). Define an operator T0 : T{Ll{G \B{H))) -> X by T 0 ( ^ ) = f t>(s)x(s)g(ds),

SeL^G-BiH)).

JG

Then To is

a

module map and is bounded since ||T0(^$)||x
^>eLL{G-B{H))

by assumption. Since T{Ll(G;B{H))) is dense in C0(G;B(H)) by Lemma 2 (2), T0 can be extended uniquely to a bounded module map T : C0(G;B(H)) —> X. Then by Theorem III.6.7 there exists a unique £ € r6ca(93g,X) such that T ( $ ) = / $ ( x ) f (dx)

$ e C 0 (G ; B(£T))

170

IV. MULTIDIMENSIONAL STOCHASTIC PROCESSES

and ||T|| = ||£|| 0 (G). Hence we have for $ e T{F*)

Ll(G;B{H))

= f <S>(s)x(s) g(ds) = [ F*{x) £(<**)■ JG

(4-5)

JG

Now the R.HS of (4.5) can be rewritten as [ F*{x)S(dX)=

I

JG

[

(8,X)t>(s)Q(ds)Z(dx)

JG JG

(G;B(H)) = jG {JMx)Z(dX))m g(ds), $ € Ll by Lemma 4 (2). Thus for every $ € Ll(G; B(H)) we get j

*(») (x(s) - J(s,

x) Z(dx)) Qids) = 0,

which implies that

*(*) = [ (t,x)ttdx),

teG.

JG

Therefore {x(t)}

is weakly operator harmonizable by Theorem 3.2.

Corollary 6. Let {x(t)} be an X-valued process on G. (1) {x(t)} is scalarly V-bounded iff it is scalarly weakly (2) {x(t)} is V-bounded iff it is weakly harmonizable.

harmonizable.

Proof. When H = C, Theorem 5 reduces to (1). (2) is verified in the same manner as in the proof of Theorem 5. There is another characterization of stationary dilation in terms of scalar Vboundedness. P r o p o s i t i o n 7. An X-valued process {x(t)} on G has an operator stationary di lation iff it is scalarly V-bounded and, for every ONS {4>k}^=i in H, the constants oo

Cj>k's can be chosen so that ^

C\

< oo, where, for each k > 1, C$h > 0 is the

fc=i

constant in (4.1) for the LQ(Q)-valued

process {(x(t), k)ii}-

Proof. We first prove the "only if' part. Suppose that {x(t)} has an operator stationary dilation {{y(t)},Y, P}. We may assume that Y = Lg{Sl;H) for some probability space (fi, 5,//)■ Let n e rcagos(%Q,Y) be the representing measure of {y{t)}- Then {x(t)} is weakly operator harmonizable with the representing measure

4.4. V-BOUNDED PROCESSES

171

f = Pr} € r&ca(93g, X). Moreover, it is scalarly weakly harmonizable, where £^(-) = (?(•)> )H} for each E H. Thus by Corollary 6 (1) {x(t)} is scalarly V-bounded. Let {fc}j£=i be a CONS in # containing { > * } £ r Then we get for each A: > 1

sup

/

JG

&/<* ( d x )

sup ■

: ^ g L ^ G ) , H^Hoo < 1 ^

by Lemma 4 (1), sup

:feC0(G),

ff(x)Z*k(dx)

< 1

JG

HIUJ(G),

(4.6)

by Corollary III.6.4,

= II^JI(G)<||^J|(G) = ||^(G)||2l where J : Ll{tt)

-*■ Lg(f2) is the orthogonal projection and rj^(-) = (??(•),)H G

rcaos(*Bg,L;;(fi)) for e H. Putting G^fc = | | ^ t ( G ) | | 2 for k > 1, we see that

E

C

fc=i

|

t

= E

lfo*»(G)Hs < E

fc=i

11^* (G)111 = HG)\\2Y

< oo

fc=i

since 7;(-) = Yl (v(')t V,)JjV'fci where |[-||y is the norm in F . fc=i

Next, we shall prove the "if part. Let {ipk}^=i be a CONS in H. Since {x(t)} is scalarly V-bounded, it is scalarly harmonizable. For each k > 1 let £*fc g rca(iBg,LQ(fi)) be the representing measure of {%$k(t)}, where x^k{t) = (x{t),rpk)H

for £ g G. Then we have by (4.6) that ||£** 11(G) < C^k, the latter conoo

stants being square summable. Define £(•) = J2 £^fc {')*Pk- Then £ is unambiguously defined and is in rca(25g,X) since | | e | | ( G ) 2 < 1 6 sup ||e(A)|& = 16 sup £ 1I^)H A£
<16^||^||(G)2<16^^t
172

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

by Theorem 111.1.5(1) and assumption. Now we see that {x(t)} nizable with the representing measure £ since oo

oo

~

x(t) = £ (x(t),i>k)Hi>k = ^

Jt,X)

= f (t,x)f^C"'(dx)^k= JG

an

Ck(dxWk

k=lJ°

fc=l

Let {4>k}k*=i t"e

is weakly harmo

f(ttX)S(dx),

JG

k=1

teG.

y ON sequence in H. Then we have oo

oo

£ll^II(G)2<XX
where £^(-) = (£(•)> )H € rca(93g,.Lo(£2)) for € H■ Consequently, by Theorem III.3.15, £ has a g.o.s.d., i.e., {x(f)} has an operator stationary dilation. Finally, we derive an integral representation for a scalarly weakly harmonizable process using Fourier transforms. P r o p o s i t i o n 8. Let {x(t)} be an X-valued scalarly weakly harmonizable process on G. Then there exists a regular rj € wca(93g, B(H, Ljj(Q))) such that

(0, x(t))H = f (t, x) v(dx)4>,

teG^eH.

(4.7)

J G

Proof. By assumption, for <j> G H there exists a (^ e rca(93g, L Q ( O ) ) such that

{4>,x(t))H= f(t,x)Z+(dx),

teG.

JG

Since, for each <j> e H, sup ||(^,ar(t)) K || 2 < ||f*||(G) < oo, we have by the Uniform t£G

Boundedness Principle that sup \\x(t)\\ < oo, where x(t) is regarded as an operator teG

in B(H,Ll{Q)) and ||x(t)[| is the norm in it. For each

,x{s)) HQ{ds),

4>eH.

By a similar computation as in (4.6) we have for 4> e H s u p { | | 2 > | | 2 : v € L\G),

U ^ l ^ < 1} = M+\\(G) < oo.

4.5. CRAMER AND KARHUNEN CLASSES

173

Hence, again by the Uniform Boundedness Principle, Bnp{||£*||(G):0eff,MH
Halloo < 1} = C < co.

Thus ||e 0 ||(G) < C\\(J)\\H for 4> € H. Now define, for each A € B g , r?(A)0 = £*(A) ((p £ H). Then it is easy to see that T] € wco(5Sg, B ( # , L | ( 0 ) ) ) and is regular since £* is uniquely determined and regular, and ||T;(^)|| 2 = U f * ^ ) ^ < ||£*||(G) < C|| e H

(4>,x(t))H=

[ (t,x)Z+(dx)=

JG

[

(t,x)v(dx),

JG

i.e., (4.7) holds. [Note that, by Lemma III.1.6, the (total) semivariation of r\ is given

by 117,11(5) = C]

4.5. Cramer and Karhunen classes In this section, we define several classes of Cramer and Karhunen processes on an LCA group G and obtain their integral representations as well as some other properties. Definition 1. (1) An L§(fi)-valued process on G is said to be of weak (resp. strong) Cramer class if its covariance function 7 is representable as ■y(s,t)=

(I

s,teG

(5.1)

for some measurable space (0,21), some p.d. scalar bimeasure m G M (resp. M„) and some family {tp{t,•) : t € G} C £»(m) (cf. Section 3.5), and where the integral is a strict m- (resp. Lebesgue-Stieltjes) integral (cf. Definition III.5.2). (2) An X-valued process {x(t)} on G is said to be of scalarly weak (resp. scalarly strong) Cramer class if, for each e H, the Z/Q(fi)-valued process {{x(t), 4>)H} is of weak (resp. strong) Cramer class in the sense of (1). (3) An X-valued process is said to be of weak (resp. strong) Cramer class if its scalar covariance function 7 is representable as (5.1). (4) An X-valued process is said to be of weak (resp. strong) operator Cramer class if its operator covariance function T has a representation of the form r ( s , t ) = If

$(s,0)M4(d0,dA)*(t,A)*,

s,t e G

(5.2)

174

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

for some measurable space (0,21), some £ e 6ca(2t,X) with M c e 9% (resp. M^ £ mv) and some family {$(*, •) : t <E G} C £ : ( 0 (cf- Section 3.5). We note that if (9,21) = ( G , % ) and or $(*,•) = ( V ) l f° r * 6 G, then the above definition reduces to the harmonizable case. We also note that the (weak or strong) Cramer class reduces to the (weak or strong) Cramer class when H = C by Theorem 3.13 and its proof. D e f i n i t i o n 2. (1) An L§(0;)-valued process on G is said to be of Karhunen its covariance function 7 is expressed as y{s,t)=

[ tp{s,9jv(i~¥ji/{d0),

s,tEG

class if

(5.3)

Je for some finite positive measure space (9,2l,i/) and some family { e H, the Lo(fi)-valued process {(x(t),
s,teG

for some measurable space (9,21), some operator measure F e c a ( 2 l , T + ( / f ) ) and some family {$(*, •) : t e G} C L2(F) (cf. Section 3.4). When (9,21) = (G,Q3 g ) and
f ${t,0)£{d9), Je

teG,

(5.4)

which follows from the definition and (III.5.14). Hence, these two classes are invari ant under equivalence and, moreover, under transformations by bounded module maps. For other classes we have:

4.5. CRAMER AND KARHUNEN CLASSES

175

P r o p o s i t i o n 3. Let {x(t)} be an X-valued process on G of weak Cramer class relative to a p.d. scalar bimeasure m 6 M and a family {(p(t, •) : t e G} C £»(m) on some measurable space (0,21). Then there exists an X-valued measure £ G ca(2t,X) such that m = m^ and

x{t)=

I ip{t,0)Z(dO), Je

teG.

(5.5)

Proof. Let {x(t)} be an .X-valued process of weak Cramer class on G, so that its scalar covariance function 7 is of the form (5.1). Since m £ M is p.d., we get a RKHS f j m of m and an f) m -valued measure 77 6 ca(2t, fj m ) denned by 77(A) = m(j4, ■) for A E 21. Define an f5 m -valued process {?/(£)} by 2/(£)= /
teG,

which is well-defined since £»(m) = Z,1(T7) by Theorem III.5.7. Then {?/(£)} is of weak Cramer class since its scalar covariance function j y is identical to 7, which is easily seen by m = mv. This implies that there exists a unitary operator U : %o(y) —> "Ho{i) such that Uy(t) — x(t) for teG, where HQ{X) and Ko(y) are vector time domains of a; and y, respectively (cf. Proposition 1.2 (2)). Then, putting £ = Urj e ca(2t, X ) , we have the representation (5.5). P r o p o s i t i o n 4. Let {x(t)} be an X-valued process on G of operator Karhunen class w.r.t. an operator measure F e ca{%,T+{H)) and a family {$(£,■) : t 6 G} C L2(F) on some measurable space (0,21). Then there exists an X-valued measure £ E cagos{%,X) such that F = F^ and x{t)=

f $(*,(?)$(dfl), Je

teG.

(5.6)

Proof. The proof is quite similar to that of Proposition 3 above, so it is left to the reader. Note that for a process of Karhunen class we also have its integral representation. The measures appearing in (5.4), (5.5) and (5.6) are called the representing measures of processes {x(t)} of Cramer and Karhunen classes, respectively. We also note that, if a process {x(t)} has an integral representation given by (5.4), (5.5) or (5.6), then it is of (operator) Cramer or (operator) Karhunen class. T h e o r e m 5. Let x = {x(t)} be an X-valued process on G such that H(x) particular, T-LQ{X) is separable. Then it is of operator Karhunen class.

or, in

176

IV. MULTIDIMENSIONAL STOCHASTIC PROCESSES

Proof. We may assume that DimW(i) = N0 since U{x) is separable. Let (9,21) be a measurable space and F e ca(»,T+(.ff)) be such that Dim L2{F) = H0. Choose gramian bases {x n }£° =1 of H(x) and { $ „ } ~ = 1 of L2(F) and let U : L2{F) -> "H(i) be the gramian unitary operator such that £/<2>n = xn for n > 1. Define £(A) = t / ( i A i ) ,

A e 21.

Then, £ is c.a. and g.o.s. since [t(A),S(B)]

= [U(1A1),U(1B1)}

= [lAl,lBl]F

for .4, i? e 21, where [■, -]F is the gramian in L2{F). isomorphism L2{F) £ © ? ( 0 ) = H(z) such that [/$=

=

F(AnB)

Thus F = F^ and (/ gives the

$ ei2(F)

/" $ d £ , ./e

(cf. Proposition III.4.13). Then we have that OO

/

x{t) = J2ak{t)xk

OO

= u ( j 2

fc=l

v.

a

^

t

t e G

^ ^

^k=l

>

'

where ajt(£) = [ar(t),a?fc] for k > 1. If we define OO

*(*,-) = X>*(*)*fc(0.

(£G»

fc=i

then {$(*, ■) -teG}

C L2(F)

since

i.fc OO

= Y^ak(t)lxk,xk]ak(t)*

=T(t,t)

for teG, where T is the operator covariance function of {z(£)}. Consequently, we have a representation x{t) = U$(t,-)= Therefore {x(t)}

[ ${t,9)Z(d9),

teG.

is of operator Karhunen class by Proposition 4.

4.5. CRAMER AND KARHUNEN CLASSES

177

As harmonizable processes have stationary dilations, processes of Cramer class have "Karhunen dilations," which is stated in the following. T h e o r e m 6. (1) An X-valued process {x(t)} on G is of weak operator Cramer class iff it has an operator Karhunen dilation, i.e., there exist a normal Hilbert B(H)-module Y containing X as a closed submodule and a Y-valued process {y(t)} of operator Karhunen class such that x{t) = Py(t) for t e G, where P : Y —» X is the gramian orthogonal projection. (2) An X-valued process {x(t}} on G is of weak Cramer class iff it has a Karhunen dilation, i.e., there exist a Hilbert space Y containing X as a closed subspace and a Y-valued process {y(t)} of Karhunen class such that x(t) = Jy{t) for t € G, where J : Y —> X is the orthogonal projection. Proof. (1) Suppose that {x(t)} is of weak operator Cramer class with the operator covariance function T satisfying (5.2). Then it has an integral representation given by (5.4). Let {n,Y, P} be a g.o.s.d. of f, which exists by Theorem III.3.15, and define a F-valued process {y(t)} by

y(t)=

f $(t,0)r,{d9), Je

tea

Note that, since {$(*,•) ■ t e G} C £*(£) C L 2 (F„), the above integral is welldefined. Since {y(t)} is of operator Karhunen class and x(t) = Py(t) for t € G, {x(t)} has an operator Karhunen dilation. The converse implication is rather obvious. (2) Let £ 6 ca(2t, X) be the representing measure of {x(t)} given by Proposition 3 and consider its minimal o.s.d. {n, Y, J}. Since {
[
tea

Je Then {y{t}} is a Y-valued process of Karhunen class and x(t) = Jy(t) i.e., {x{t)} has a Karhunen dilation.

for t e G,

It follows from (3.5) and (3.6) that a weakly harmonizable process is of Karhunen class. Moreover, we note that every weakly operator harmonizable process is of operator Karhunen class, which is seen from a similar relation as (3.5).

178

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

4.6. Series representations Three types of series representations are considered for X -valued processes on G, which are called ordinary, modular and tensor series representations. Relations among these representations are given in connection with the separability of the vector and modular time domains of a process. Definition 1. (1) An X-valued process {x{t,ui)} series representation (o.s.r.) if

on G is said to have an ordinary

oo

x{t,u)

= ^2ak(t)xk(u>), k=i

(6.1)

where the series converges in the norm ||-||x for each t € G and, for k > 1, ak(-) is a C-valued function on G and xk € X (= LQ(£1 ; H)\). (2) An X-valued process {x(t,u>)} on G is said to have a modular series repre sentation (rn.s.r.) if oo

x(t, u)) ~ ^2 ak{t)xk(u)),

(6.2)

fc=i

where the series converges in the norm |||].Y for each t € G and, for k > 1, ak(-) is a B(H)-va\\ied function on G and xk 6 X. (3) An X-valued process {x(t,ui)} on G is said to have a tensor series represen tation (t.s.r.) if oo

x{t,u) = ^2d»k{t)fk(u),

(6.3)

fc=i

where the series converges in the norm \\-\\x for each t E G and, for k > 1, ^fc(-) is an H-valued function on G and fk € L\ (fl). If {x(i, u>)} has an o.s.r., then it has a m.s.r. since we can take ak(-) = ak(-)l k > 1 in (6.1). Moreover, we have: P r o p o s i t i o n 2. Let x = {x(t,uj)}

for

be an X-valued process on G.

(1) x has an o.s.r. iff its vector time domain "Ho(i) is separable. (2) x has a m.s.r. iff its modular time domain H(x) is separable, or equivalently, iff x has a t.s.r. Proof. (1) follows from first part of (2) when H = C. (2) Suppose that x has a m.s.r. given by (6.2). Then, clearly H{x) = &{xk : k > 1} and it is separable.

4.6. SERIES REPRESENTATIONS

179

Conversely, if H(x) is separable, then we can find a countable set {xk ■ k > 1} forming a gramian basis for 'W(i), so that, for each t e G, x(t, ■) can be written as oo

x

(t,-) = Yl

[x{t),xk]xk{-).

fc=l

Putting ak(t) = [x(t),xk] for k > 1, we obtain a m.s.r. (6.2). Suppose %{x) = H ® K is separable, i.e., K is separable. Take a CONS {fk}kLi for K and a <j> e H of norm 1. Then, by Proposition II.3.12, {4> ® /k}fcLj forms a gramian basis for %(£). Hence we have that oo

x(t,w) = '$2[x(t),fk4>]fk{w)1

teG,

fc=i

where ($®/fc)(w) is identified with fk(u)4> for w 6 fi. Putting k\4> for fc > 1, we obtain a t.s.r. (6.3). Finally assume that {x(t,ui)} has a t.s.r. (6.3). Then we see that for each t € G x(t) e e0{ 1} ® 6 0 { / f c : fc > 1}, where &o{' • • } is the closed subspace of i / or L Q ( ^ ) generated by the set {••■}. Let K = ©o{/fc '■ k > 1}. Then H{x) = H ® K and, hence, W.(x) is separable. The following corollary is easily verified. Corollary 3. Let x = {x(t,uj)} the operator covariance functions

be an X-valued process on G with the scalar and 7 and F, respectively.

(1) / / x has an o.s.r. of the form (6.1) such that {xk}kxL1

is ON in X, then

00

7 ( s , t) = Yl ak(s)ak(t),

s,teG.

fc=i

(2) / / x has a m.s.r. of the form (6.2) such that {xk}kxL1 then

is gramian ON in X,

00

T{s,t) = YJhk{s)bk{t)*,

s,teG,

fc=i

where bk(t) = ak(t)[xk,xk]^ for k > 1 and the series converges absolutely in the trace norm. (3) / / x has a t.s.r. of the form (6.3) such that {fkJkLi is ON in £g(Q), then 00

r(s,o = 5Z<Ms)®<Mt), fc=i

s,teG,

180

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

where the series converges absolutely in the trace norm and ® is in the sense of Schatten. In the representations (6.1) and (6.2) of an X-valued process {x{t,ui)} the time dependent deterministic functions ak(t)'s and afc(£)'s are uniquely determined pro vided that {xk}kxL1 is ON or gramian ON, respectively. Similarly, in the repre sentation (6.3) we can prove the uniqueness of the time functions as follows: Sup pose that {x{t,w)} has a t.s.r. of the form (6.3) with {fk}^Li ON in Lg(fi) and oo

x(t,ui) = J2 V'fcW/fc^) be another t.s.r. Take any 4> 6 H and observe that fc=i

fjd>, £(^fc(*)-^fc(*))/*

(^■(0-<M*)) = o,

teG,j>i.

fc=i

Since 4> e H is arbitrary, we conclude that 4>j(t) = i>j{t) for each t e G and j > 1. P r o p o s i t i o n 4. Lei £ = {x(£)} 6e an X-valued process on G. Then the vector time domain Ho(x) is separable if one of the following conditions holds: (1) G is separable and {x(t)} is weakly continuous or, in particular, continuous. (2) The underlying probability space (£7, 5, M) *5 separable or, in particular, 5 has a countable generator. (3) G is countable. In this case {x(t)} has all kinds of s.r. 's. Proof. (1) Suppose that G is separable and {x(t)} is weakly continuous. Let Go be a countable dense subset of G. We claim that 'HQ(X) = Go{x(t) : t g Go}- For, suppose x e U0{£) 6 &o{x{t) : t e G 0 } . Then (x,x{t))x = 0 for t € G 0 . This implies that ( x , x ( t ) ) x = 0 for t 6 G since {a;(i)} is weakly continuous. But it then follows that x = 0. Therefore Ho{x) is separable. (2) is obvious since L$(Q) is separable in either case, and (3) is also clear. Now we consider s.r.'s in r.k. spaces associated with (scalar or operator) covariance functions of X-valued processes. T h e o r e m 5. Let x = {x(t,w)} be an X-valued process on G with the scalar and the operator covariance functions 7 and T, respectively. Let f>7 be the RKHS of 7 and X? be the r.k. normal Hilbert B(H)-module of T. (1) / / {x\}\eA is a CONS in the vector time domain 'Ho(x) and ct\(-) (xx,x(-))x for A 6 A, then the family { a A ( - ) } A e A forms a CONS in f j 7 .

=

(2) / / {x\}\eA is a gramian basis for the modular time domain Ti{x) and a A ( ) = [XA,2:(-)] for A G A, then the family {a\{-)}\£A forms a gramian basis for Xr.

181

4.6. SERIES REPRESENTATIONS

Proof. (1) follows from (2) when H = C. So we prove (2). Let X0 be the set of all n

T(H)-valued functions on G of the form J2 ajF{sj, ■), where a,j € B(H) and Sj G G 3=1

for 1 < j < n , n > 1. Define an operator (70 : -X0 -> W(x) by

v

i=i

'

j=i

Then, f/o commutes with the module action of B(H) and preserves the gramian as we can easily see. Since Xo is dense in X? and the range of UQ is dense in T-L(x), UQ can be uniquely extended to a gramian unitary operatror U ■ Xr -> ~H(x). For each A e A, x\ can be written as oo x

for some a\j

6 B(H) ax(-) =

and s\j

\ =

/,aA,jz(SAJ)

E G, j > 1. Then we have that

[x\,x(-

= Ylax>iF(Sx'i>')

^a\jx{sx,j),x{-) = U X

( Ylax
3

= U

lxx

'

3

Therefore {a,v(-)}A€A forms a gramian basis for X?Corollary 6. Let x = {x(t,w)} be an X-valued process on G with the operator covariance function F. If it has an m.s.r. of the form (6.2) such that {xk}'^L1 is a gramian basis of W.{x), then the family {a/t(-)*}fcLi °f T{H)-valued functions on G forms a gramian basis of the r.k. normal Hilbert B(H)-module Xr of F. T h e o r e m 7. Let x = {x(t,u>)} be an X-valued process on G with the operator covariance function T. If it has a t.s.r. of the form (6.3) such that {/jb}E=i is a CONS in K C Ll(tt), where U(x) =H®K, then {<j>k{)}f=l forms a CONS in the RKHS S,T of T. Proof. If we are given an ON set {fk}kLi in L>o(ty, t n e n t n e t i m e functions tj>^s are uniquely determined as was remarked just before Proposition 4. Hence we may assume that, for each j > 1, 4>j{t)=

x(t),/>
teG

182

IV. MULTIDIMENSIONAL STOCHASTIC PROCESSES

for some 6 H of norm 1. Since fj{-)4> e H(x) for each j > 1, we have that for some { a i > m } ~ = 1 C B ( H ) and { t , - , m } ~ = 1 C G

fj(-)=

^a.hrnx{t j,mj 7 771 =

i

3>l-

1

Then it holds that for each j > 1

/(•) = [i(-)»/^ Y,

^V'J'

[x{-),x(tj,m)]a'jim4>

y.&j,rn.3>vjjim)

= ^r(-,i;,m)a*m0.

Consequently, ^ 6 Sjr for each j > 1. Moreover, it holds that for j , k > 1

= ^

(r(tktn,tjym)alm4>ta*kn
see Definition 114.22,

7Ti, n

= ^ Z ( a fc,n[ X ( t ':,«)' a: ( t J,™)] a i,m < ? ;, ^) H >

771,77

^ L

771

77

J

/

H

= ((/fc,/i)2(0®?)'/'»H = {fk,fjhThus, { 0 * } ^ ! is ON in £ r . By Corollary 3 we get I?(s,t) = £ <£fc(s) ® (/>fc(t) for s,teG.

Then by the proof

fc=i

of Theorem II.4.24 (see also Proposition II.4.8) we see that {<j)k}f=l is complete in f)r. Therefore, it is a CONS in i } r . In the above theorem it holds that K ~ S)i by Theorem II.2.24.

4.7. MOVING AVERAGE REPRESENTATIONS

183

4.7. M o v i n g average representations In this section, moving average representations for AT-valued processes on R will be considered. These were usually denned with o.s. measures and we extend them to any vector measure which is not necessarily o.s. D e f i n i t i o n 1. An X-valued process {x(t)} on R is said to have a moving average representation (m.a.r.) if there exist an X-valued measure £ g bca{^&,X) and a function $ g -C1 (£) such that x(t)=

I ${t-u)£{du),

teR,

(7.1)

JR

where {$(* - -) : t € R} C -C 1 ^) (cf. Section 3.5). R e m a r k 2. If an X-valued process {x(t)} has an m.a.r., then there exists an n € capos(iB,y) and a function *P g ^(.F,,) such that x(t)

/ *(t-u)77(du),

t el,

where {77, Y, P} is a g.o.s.d. of £. This is the usual m.a.r. defined so far. Moreover, {x(t)} is of operator Karhunen class. T h e o r e m 3 . Let {x(t)} be an X-valued process on R. / / it has an m.a.r. of the form (7.1) with £ g bca(*&,X) and $ = JT*, where * g L 1 ( R ; B ( f f ) ) , then {x{t)} is strongly operator harmonizable with a representing measure of bounded variation. Proof. Note that $ = T^ g C 0 ( ! ; £ ( # ) ) C £}(Cj. Then we have that t(t) = / # ( t - u) £(du) = I T{t - u) £(du) JR

J«

J R \ J&

e *(*-«)^$(A)rfAje(du)

= f eltxf{X) JR

( f e~iXu £(d«) ) dA, \ JR )

by Lemma 4.4

= f eux^{\)y{-X)d\,

(7.2)

JR

where dA is the Lebesgue measure, the last integral is in the sense of Bochner and {y{t)} is a weakly operator harmonizable process with the representing measure £. Define n : <8 -S- X by T?(A)= /

9{t)y(-t)dt,

Ae
(7.3)

184

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

Then we see that 77 6 ca(
= f p(t)y(-t)\\xdt< JR

/||y(-t)||x||*(OII*
Now by (7.2) and (7.3) it holds that x{t) = JRetturi(du) for t 6 M, so that {x(t)} is strongly operator harmonizable with the representing measure 77 of bounded varia tion. It follows from (7.3) that the representing measure of {x{t)} is absolutely contin uous w.r.t. the Lebesgue measure with the RN-derivative *&(-)y(—)• E x a m p l e 4. Let us consider a special case of the above theorem. Suppose that an X-valued process {x(t)} has an m.a.r. of the form (7.1) with £ 6 cagos(
Tv{u)=

JR

=

Tf{u)a.

JR

For 77 e ?;ca(Q3,X) defined by (7.3) in this case, we have MV{A,B)=

I

■H{u)TJ{-u + v)a^{vydudv,

A.BeiB.

JJAXB

Finally, the operator covariance function Tf of {x(t)} is written as r*(s,i)= //(A)$(s-A)a$(i-A)*
s,teR

4.8. A p p r o x i m a t i o n and convergence In this section, we shall consider ^-valued processes on R. So we deal with the measurable space (R, <8). At first we study approximation of a process by a sequence of strongly (operator) harmomzable processes with the representing measures of bounded variation. This class of processes has very nice properties. Secondly, some convergence properties of sequences of processes are obtained.

4.8. APPROXIMATION AND CONVERGENCE

185

Let us denote by {woh} and {soh} the classes of all weakly operator harmonizable and strongly operator harmonizable processes on E. In particular, {soh}v denotes the subclass of {soh} consisting of strongly operator harmonizable processes whose representing measures are of bounded variation. That is, {x(t)} g {soh}v iff there exists f g uca(Q3,X) such that x{t) = I eitu£(du),

t e t

JR.

We first note: L e m m a 1. If x & {woh} and Dim'H(x) < oo, then x g

{soh}v.

Proof. Let £ g bca(?&,X) be the representing measure of x. Since Dim'H(x) = D i m 6 4 ( R ) < oo, it follows from Theorem III.7.1 (1) that £ £ vca( 0 for every t g R. Moreover, the convergence is uniform on each compact subset of R. Proof. Since x = {x(t)} is continuous by Proposition 3.7 and hence the modular time domain 7i(x) is separable by Proposition 6.4, we can find a gramian basis {x/c}^L1 of H(x). Thus x(t) is written as CO

x{t) = ^2[x(t),xk]xk,

teR

fc=l

(cf. Theorem II.3.6). Let £ g 6ca(25,X) be the representing measure of x and define for n > 1 n

x„(t) = 22 [x(t),xk]xk,

teR,

fc=i n

/t=i

Then it is easy to check that {xn(t)} £ {woh} with the representing measure £ n for n > 1. Moreover, since D i m 6 j n ( R ) = n < oo, xn g {so/i}„ for n > 1 by Lemma 1. The convergence ||a;„(t) — asfOltx —>■ 0 for £ g R is obvious. The uniform convergence follows from the metric approximation property of the Hilbert space H(x) (cf. Diestel and Uhl [1, VTII.3]). More generally we can prove the following, which tells us that a fairly wide class of processes can be approximated by sequences from {soh}v.

186

IV. MULTIDIMENSIONAL STOCHASTIC PROCESSES

T h e o r e m 3. / / {x(t)} is an X-valued bounded and continuous process on R, then there exists a sequence {xn(t)}'^'_1 C {soh}v such that \\xn(t) — i ( t ) | | x ~> 0 for every t 6 R. Moreover, the convergence is uniform on each compact subset of R. Proof. Let a > 0 be given and put

/27TCT

and, for t e R, x
j,

s e t

Then, xCT,t(-) : R —> X is norm continuous and Bochner integrable w.r.t. the Lebesgue measure ds. Define a ya(t) = -^=/
V2nJ

t e R.

JR

It can be seen that ya{-) : R —» X is norm continuous and Bochner integrable w.r.t. ds. Moreover, define an X-valued measure (,„ on 33 by

ZM)= [ y*(t)dt,

Ae

JA

and an X-valued process {xa(t)}

by j eUuUdu),

z„(t)=

te

JU

Then we see that £CT 6 i/ca(Q3,X) since |^|(R)= /

||y
< CO,

JR

where ||-||i is the norm in L 1 ( R ; X ) , the Banach space of all X-valued Bochner integrable functions on R w.r.t. the Lebesgue measure dt. Now we have for t 6 R xa{t) = [ eu" Udu)

= [ eUuy„(u)

J&

= f eltu^=fa{u)( JR

V27T

= -== / / e

V2^

du

JR

JJR

f \JR

ltu

x^u(s)ds]du J

■— exp ( - JL_]a5( s )exp F (- ius - —)

V2^V

V la V

V

a

)

dsdu

4.8. APPROXIMATION AND CONVERGENCE

187

,2

by F u b i n i ' s t h e o r e m for Bochner integrals (cf. Hille a n d Phillips [1]), V2n JR

\

a

2

I

Finally, we check t h e convergence ||a;CT(t) —x(t)||jf —> 0 as a —> oo. R e w r i t e xa(t) u

1 f t \ I \ot-Y til u\ , x„(t) = -== I x t + - ) e x p ( - - ! -—- - — )du, v V27r 7 K V °' °" 2 / Then we have that for t € R V^F{x CT (t) - x(t)} = |

as

2

m

teR.

{*(t + £ ) - * ( t ) } exp ( - l ^ t ^ i - £ ) d« +

/

R

X

W

e x p ( - ^ ) { e x p ( - ^ ) - l }

= / 2/o-,t(w)dM+ / z0tt{u)du, 7R

d u

(say).

,/R

Let a > 0 be arbitrarily large and e > 0 be arbitrarily small. Choose (5 > a such that /

exp I

1 du < e.

Since x(-) is uniformly c o n t i n u o u s o n [—/3, /?], there exists <5 > 0 such t h a t 5 < (3 — a a n d \\x(u) — x(v)\\x

< s for every u , « e E w i t h \u — v\ < 6. T h u s for a >

T

o ya,t(u)du

sup |t|
VR

X

 t (w)||, Y du+ s u p / ||y CT ,t(w)ILYdw \t\0 C

2

ex /3 2 / < (v/27+2C)e, (8.1)

where C = sup ||x(£)||x < oo by assumption. On the other hand, there is 5' > 0 such that \eu —1| < e if |u| < <$'. Hence we have for a > 0 such that sup \t\
zait(u)du JR

X

~-

r2

0 ' < C{\ + v^F)e. (8-2) From (8.1) and (8.2) we conclude that \\xa(t) - x(t)\\x -> 0 as a -> oo uniformly on [—a, of]. This shows the uniform convergence on compact subsets of R. We now consider convergence of sequences of processes. We are interested in the following assertion (A): (A) If a sequence of X-valued processes with the property (P) converges in a certain sense to an X-valued process, then the limit also has property (P). When (P) is the stationarity, we can prove the following. T h e o r e m 4 (Continuity T h e o r e m ) . Let {xn{t)}n~l ^e a seQuence of X-valued stationary processes on R such that \\x„(t) — x(t)\\x —> 0 for every t € R and {x(t)} is norm continuous at t — 0. Then, {x(t)} is also stationary. Moreover, if £ n (n > 1) and £ are the representing measures of {xn{t)} and {x(t)}, respectively, then £n => £, i.e., {£n}«Li converges weakly to £. Furthermore, the convergence of \\xn(t) — x(t)\\x —> 0 is uniform on each compact subset of R. Proof. Let */„(-) = ||f»(-)|H and yn{t) = JRe'tuun(du) = (xn{t),xn{0))x for t £ R. By assumption 7(t) = (x(t),x(0)) v = lim 7 n (£) exists for t G R and continuous at t = 0. Thus by Levy's continuity theorem (cf. Rao [6, p. 233]) there exists a positive finite measure v 6 ca(Q3,R + ) such that y(t) = I eltuv{du),

(El

JR

and vTl => v. Now (xn(t),xn(s))x = jn(t - s) implies y(t - s) = (x(t),x(s))x every ( , s e E and {x(t)} is continuous since \\x(t + h)-

x{t)\\x

= 2 7 (0) - 7(ft) - 7 ( - f t ) ^ 0

for

as ft ->■ 0

for every t 6 R. Thus, by definition, {x(t)} is stationary and hence there exists £ e caos(93,X) such that x{t) = JReltu £(du) (t e R), so that u(-) = U(-)\\2X by Theorem 2.2. Next we show that £ „ = > £ . Let S be the set of all continuity points of £, i.e., S = {t E R : £({*}) = 0 } . Note that 5 is dense in R since Sc is at most countable because £ is o.s. and ||f(R)||x < oo. Let a,b e S with a 0 be arbitrary. Choose ei > 0 such that e1{5 + 2a) + 2e2, < e, (8.3)

189

4.8. APPROXIMATION AND CONVERGENCE

where a = sup {||£„(K)||x, ||£(R)||.Y : n > 1} and choose c > max{|o|, \b\} + 1 such that sup {Un(Cc)\\x, U(Cc)\\x : n > 1} < ei, (8.4) where C = [—c, c]. We can find, for each p > 1, a continuous function fp : R —> [0,1] such that fp = 1 on [a,b] and / p = 0 on ( — oo,a

) u ( 6 - | — , oo). Let p > 1 be

fixed and choose a trigonometric polynomial g of period 2c such that sup \fp(s) -g{s)\

< ex.

|»|
We then have | | / p — g\\oo < 2 + e\ by the periodicity of p and the choice of fp. Observe that

[ fpdtnJR

[ fp<% JR

<\[{fP-g)dZn x

I JR

+ X

j gd{in

0

rfS

[ifp-9)d£ Since x n ( i ) = JReltwd£n n 0 > 1 such that

—» JRe'tud£

= x(t) by assumption, we can choose an

0

< Ei,

n > n 0.

Moreover, by (8.4) and the choice of fp and g, we get

f(fp-g)

< £ i i / „ ( C ) ' + (2 + ei)£i < ei(2 + a) + e\.

d£r.

JR

We then have that for n > n0

J fVdZnJR

I

fpdZ

<e,

(8.5)

JR

which implies that, for every p > 1, || JRfpd(,n — fRfpd£\\x ~^ 0. For each S > 0 let Bs = [a — S,b + fi\. For every e > 0 there is some 5 > 0 such that ^(Ba) < e2 and some (^ e (0,5) with vldBsJ = 0, i.e., z/({a - <5i, b + 6±}) = 0. Since vn => v, there is an nx > 1 such that i/„(5^J < e 2 for n > m . Then choose q > 1 such that / , = l(o,6] on BSl and 2

= f \fq~ l ( a , 6 ]| 2 ^„ < Vn{BSl) < E2

/ /,«*fn " &,(M]) II JR

X

JR

190

IV. MULTIDIMENSIONAL STOCHASTIC PROCESSES

for n > ny. And the above inequality holds if £ n is replaced by £. By the previous argument we can choose n 2 > 1 such that (8.5) holds for p = q and n > n2Consequently for n > max{ni,n2} it holds that \\Znt(a,b})-Z{(a,b])\\x<

U(a,b})-

f fqd^n

+

[fgd£n-[fqdZ

[ fqdZ-a((a,b}) JR

<3e. 0 for every a,b E S with a < b, and this This shows that ||£ n ((a,6]) — £((a,6] holds if a — —oo or b = oo. Then it is easy to see that

\UB)-t{B)\

as n

for every B e 33s, the algebra generated by the set {(a, b] : a,b 6 5 } . Note that <j(23s) = 33, i.e., 33 s generates 33. It follows that, for any e > 0 and any A e 33 there exists a B e 33 s such that i/{AAB) < e2. Moreover, AAB € 33s implies that there is some n 3 > 1 such that vn(AAB) < e2 and ||£„(B) -£(B)\\x < e for n > n 3 . Thus we have for n > max{n 1 ; n2,ri3}

\\UA)-aA)\\x<\\UA)-uB)\\x
+ \\uB)-aB)\\x + \\aB)-aA)\\x \\UB)

- Z(B)\\X

+

v{AAB)*

< 3e. Therefore £ n => f. Finally, we prove the uniform convergence. It is easy to see that the sequence {a;„(i)}^L 1 is uniformly equicontinuous. If C C K is compact and e > 0, then we obtain the uniform convergence on G by choosing a suitable finite <5-net for C. The above theorem is regarded as a generalization of Levy's continuity theorem. Also it is valid for Lg(f2)-valued processes. Corollary 5. Let {xn(t)}^=1 be a sequence of X-valued operator stationary processes on R such that \\xn(t) —x(t)\\x —> 0 for every t 6 R and {x(t)} is norm continuous at t = 0. Then, {x(t)} is also operator stationary. Moreover, if £ n (n > 1) and £ are the representing measures of ( x n ( t ) } and {x(t)}, respectively, then £ n => £. Furthermore, the convergence \\x„(t) - x(t)\\x —> 0 is uniform on each compact subset of R. Proo/. By Theorem 4 we see that {x(t}} is stationary with the representing measure £ e caos(33,X) such that £ n => £ and that the convergence ||x„(£) - a;(t)||x -> 0

4.8. APPROXIMATION AND CONVERGENCE

191

is uniform on every compact subset of R. We only have to verify that {x(t)} is operator stationary. For each <j> E H, {(xn(t),<j>)H}^LI is a sequence of Lg(fi)valued stationary processes on M, \\(xn(t),)H - (x(t),) \\ —► 0 for all f e l , and ( x ( - ) , 0 ) H is norm continuous at t = 0. Hence, again by Theorem 4 {(x(£),)//} is stationary. Since this is true for every E H, {x(t)} is scalarly stationary. Now Proposition 2.3 implies that {x(t)} is operator stationary. If (P) in (A) is the harmonizability, then we have: T h e o r e m 6. Let {xn(t)}^L1 be a sequence of weakly operator harmonizable pro cesses on R with the representing measures {£n}^Li Q 6ca(53, J \ r ). / / ||x n (i) — x(t)\\\ —> 0 for every ( 6 R, {x{t)} is continuous on R, and {\\(in\\0(Wj}^'=1 is bounded, then {x(t)} is also weakly operator harmonizable. Proof. Since {||^ n || 0 (R)}^L 1 is bounded, there is some a > 0 such that ||£ n || 0 (R) < a for n > 1. Hence (eltuUdu)

B»(*)HJC

< ll^||o(R) < a

JR

for every t € R and n > 1. Thus | | x ( i ) | | x < a for t E R. Now by Lemma 4.4 it holds that for every $ E L1(R;

[ ${t)xn(t) dt = [ F$(u)(a(du), JR

B(H))

n > 1.

JR

Moreover, for n > 1 we have that ||$(«)x n (i) - $(t)x(t)\\x

-> 0 a.e.(£),

||*(*)in(t)||x
«w-(t).

Hence, by the Bounded Convergence Theorem for Bochner integrable functions we get I $(t)x(t)dt= JR

n

lim $(t)xn(t)dt= ^ ° ° 7M

lim "-* 00 JR

T®{u)£n(du).

Consequently dt

lim n—i-oo

/ .F$ («)&,(**; ./R

IV.

192

MULTIDIMENSIONAL STOCHASTIC PROCESSES

Thus the set |

f m)x(t)dt:$eL1{R;B{H)),

||^*|U < l |

is bounded in X. Since {x{t)} is assumed to be continuous, it is operator ^-bounded, or equivalently, weakly operator harmonizable by Theorem 4.5. The above theorem holds if weak operator harmonizability is replaced by weak harmonizability and the boundedness of {||f„|| 0 (R)}£Li by that of {||£n||(R)}£Li-

4.9. Subordination In this section we consider subordinations of processes on an LCA group G. We begin with general results and discuss the operator stationary case in detail. Definition 1. Let x = {x(t)} and y = {y{t)} be X-valued processes on G. Then, x is said to be subordinate to y if H(x) C %{y). The operator cross covariance function F^y of x and y is defined by r ^ ( s , t ) = [x(s),y(t)]t

s,teG.

As in Section 2.4 we denote by F(G) the set of all T(H)-valued p.d.k.'s on G x G. Recall that, for r i , r 2 e T(G),Ti > T2 means that r x - T 2 > 0 , i . e . , r x - r 2 e T{G). For T e r ( G ) , X(T) denotes the r.k. normal Hilbert B{H)-modu\e of T with the gramian [■, ]rIf Y is a normal Hilbert B(H)-module with the gramian [•, ]y, then the Cartesian product Y x Y becomes a left B(H)-module with T(H)2x2-valued gramian [•, -}YXY, where

„a . xv _—f „(a. TXi,a„.Tx\*2 j

,

fxv J »1„ - Ir^i'yily j " x r„ — r „ i

•

fci^M ,

for x = (»!, x 2 ) \ y = (2/1,3/2)* e V x Y, T(H)2x2 is the set of all 2 x 2 matrices with entries from T(H), and "'" stands for the transpose. We have the following technical result. T h e o r e m 2. Let Ti,T2 € T{G) and T : G x G ->■ T ( H ) 6e a function. following statements are equivalent:

Then, the

(1) There exist a normal Hilbert B(H)-module Y and a Y x Y-valued x(t) = (x(t),y(t)) whose operator covariance function T is written as r(.,t) = [ x

W l

x

W

]

y x y

= (£ j£j]

^

^

,

M

EG.

process

(9.1)

193

4.9. SUBORDINATION

(2) $«(■) = T(;ty e X ( r i ) /or every t £ G and, i/ r # ( « , t ) = [ * „ * i ] r , /or s,t e G, then F$ < T 2 . (3) * t ( . ) = r ( i , ) e X ( r 2 ) /or even/ i e G and, t/ F * M ) = [ * s , * t ] r a for s,t e G, then Tq, < F i . Proof. (1) => (3). Note that Fj = F 5 , T 2 = Ty and F = F ^ . Write x{t) = Xi(t) + x2(t) where x^t) <= H{y) and x 2 (i) 6 ft(j/)# = Ye H{y) for i e G. Since ft(y) £* X ( r 2 ) by Proposition 1.2(2), where the isomorphism V : 'H(y) -> X(T2) is given by (Vz)(-) = [z,j/(-)] for z e H(y), for xi(i) there corresponds a unique
**(•) = [*i(t),»(0]y = [«W.»(i = r(v)e^(r,),

teG.

Putting r j ( s , t ) = [ x J ( s ) , x J ( t ) ] y fors,« 6 G a n d j = 1, 2, we see that T\, Tj £ T(G),

Ti = r i + rf and r«(s,t) = [*.,*t]r, = [*i(s),*i(0]y = r}(s,t) for M £ G. Thus we have T^ < Fx(3) => (1). Suppose that * t ( - ) = T(i, •) 6 X(T2) for r € G and T* < T 2 where r * ( s , i ) = [* 3 ,* t ]r 2 for s, t G G. Put y(f) = F 2 (i,-) for t e G. Then {«/(£)} is an ,X(r 2 )-valued process with the operator covariance function F 2 . Let = fyt for t € G. Similarly for r 0 = T 2 - T$ £ T(G) we have an A:(r 0 )-valued Xl(t) process {x2{t)} with the operator covariance function r 0 . Consider the direct sum r = X ( r 0 ) f f i X ( r 2 ) and identify X(T0) =X{To)®{0} and X(T2) = {0} © X ( r 2 ) . Then the K x Y~-valued process {x(i) = (x(i), y(i))} has the operator covariance function F ( s , t ) given by (9.1), where x(t) — Xi(t) + x2{t) for t e G. (2) «=> (3) is easily seen. Based on Theorem 2, we can derive some characterization of subordination as follows. T h e o r e m 3. Let {x(t)} and {y(t}} be X-valued processes on G with the operator covariance functions F j and Ty, respectively and the operator cross covariance func tion Tjy. Then, {x(t)} is subordinate to {y{t)} iff *£((■) = r ^ t , ■) 6 -^(r^) for t € G and [¥.,¥*]r, = r e ( M ) , s,teG. Proof. Observe that, in the proof of Theorem 2, Tt{x) C H(y) iff xi{t) = x(t) for

zeGiffr^ = r i = r1. Corollary 4. Lei F i , r 2 (1) {x(i)} and {y{t)} Ti and F2, respectively P : %(j/) —> %(£) is t^e

6 T(G). TTien the following conditions are equivalent: are X-valued processes with operator covariance functions and x(t) = Py(t) for t g G, w/zere 'W(i) C "H(y) and gramian orthogonal projection.

194

IV. MULTIDIMENSIONAL STOCHASTIC PROCESSES

(2) / / * t ( . ) = r x ( t , - ) for t e G, then * t e X{Y2) s,teG. (3) X ( r i ) is a closed submodule of

and [*„ * t ] r a = Ti(»,*)

f°r

X(r2).

Proof. (1) => (2). By (1) we have * t ( ) = T^t, •) for t € G, and hence Theorem 3 applies. (2) => (3). It follows from (2) that | | * f | | r i = | | * t | | r , f ° r * e G. Thus X ( r x ) is a closed submodule of X ( r a ) . (3) => (1). Let Q : X(T2) -► ^ ( F i ) be the gramian orthogonal projection. Since 7f (x) = X(T X ) and H(y) = X(T2) by isomorphisms Vi and V2 respectively, H ( i ) can be regarded as a closed submodule of H{y) and the gramian orthogonal projection P : "H(y) -» %(£) corresponds to Q. Thus for £ G G Py(t) = V1-1QV2y(t)

= V^QT2{t,

•) = Vflrt{t,

•) = x(t).

Corollary 5. Suppose that an X-valued process {x(t)} has a decomposition x(t) = y(t) + z(t) for t € G and 7l{y) -L %{z). Then, either y and z are subordinate to x or both y and z are not subordinate to x. Moreover, y and z are subordinate to x iff one of the following conditions holds: (1) # t ( . ) s r f ( t , 0 e X ( r « ) and [ * . , * t ] r . =Ty(s,t)fors,teG. (2) Fy and Tf are orthogonal, i.e., if T 6 T(G) is such that T < Ty and T < T j , then T = 0. When y is weakly operator harmonizable, we have: P r o p o s i t i o n 6. Let x and y be X-valued processes, where y is weakly operator har monizable with the reprersenting measure £ £ rfcca(2$g,X). Then, x is subordinate to y iff x is of operator Karhunen class with a representing measure £. Proof. Suppose that x is subordinate to y. Then, for each t £ G we have that x(t) 6 U{x) C H(y) = G^G) S £*(£). Hence, there is some $ ( t , - ) e fi1^) such that i(t) = / $(t,u)£(du), teC, ./G

i.e., x is of operator Karhunen class with a representing measure £. The converse is almost obvious. Now we consider the case where x and y are operator stationary. Definition 7. (1) Two X-valued processes x and y are said to be operator stationarily correlated if their operator cross covariance function riy(s,t) depends only on the difference s £ _ 1 .

4.9. SUBORDINATION

195

(2) For a pair of measures £,77 € rca(93g,X) we say that £ and n are gramian biorthogonal if [£(^4), 77(B)] = 0 for every disjoint A,B € 53g. In this case, we put FE,(inB)=[f(l),,(B)],

A,BeS

g

and see that Fiv 6 rca(
s,teG.

(9.2)

JG

Proof. The "if part is rather clear, so we prove the "only i f part. Suppose that x and y are operator stationarily correlated. For each s E G define an operator Us on H(x) + ri{y) = {x + y : x € U{x),y 6 rl{y)} into itself by U„(x(u) + y(v)) = x(us) +y(vs), Then we see that for s,U\,u2,

u,v € G.

V\, v2 e G

[U„{x{iii) + 2 / K ) ) , U3(x(u2) + y(v2))] = [x(uis) + y{vis),x(u2s) = [x(uis),x(u2s)]

+

+ [y{vis),x(u2s)] = [x(ui),x{u2)]

+ y(v2s)] [x(uis),y(v2s)]

+ [y(vis),y(v 2 s)]

+ [x(ui),y{v2)]

+ [y(vi),x{u2)]

+

[y(vi),y(v2)],

because of operator stationary correlatedness, = [x(«i) + y(vi), x(u2) + y{v2)}. This implies that, for each s 6 G, Us is a well-defined operator and can be extended to a gramian unitary operator on the closed submodule Xi generated by 'H(x)+'H(y). Moreover, it can be seen that {Us : s € G} forms a weakly continuous g.u.r. of G on Xi- Thus by Proposition II.5.4 there exists a regular gramian spectral measure P(-) on G such that Us = fg(s,x) P{dx) f° r s G G. Consequently we have that £(•) = P(-)x(e) and TJ(-) = P(-)y(e), so that they are gramian biothogonal. The equality (9.2) is readily verified. D e f i n i t i o n 9. Let £,77 € rca{*&Q,X). £ is said to be subordinate to 77 if ©^(G) C &V(G). £ is said to be fully subordinate to 77 if 6^(^4) C &ri(A) for every A e 25g.

196

IV. MULTIDIMENSIONAL STOCHASTIC PROCESSES

In order to characterize full subordination we need some preparation. L e m m a 10. (1) If £ e ca(Q3g,X), then it belongs to ca
6i(G) = 6i(A)®&i(Ac),

Ae^Q.

(2) If Ze ca(<2>Q,X), x 6 X and A e 5 3 g , then x € Ge(A)# where £ o £ ( - ) = \x,£(•)]• and $ e L2(F{),

(3) If £ £ cagos(*BQ,X)

iff \x o £\(A) = 0,

then it holds that

('
Aeed,

where, as before, F{(-) = [£(•)>£(•)]■ (4) 7/ £ 6 cagosCBQ,X) and x 6 X, then there exists a &x € L2(F^) (xot)(A)=

( <S>xdFc,

such that

Ae
JA

Proof. (1) is obvious. (2) follows from the following two-sided implications: x e &i(A)*

<=> [x,£(AnB)] ^^

= 0 for every B e 3 3 e

\\(x ° 0(&)\\T

Yl

= 0 for every TT 6 11(A)

A6TT

(3) is easy to check and (4) follows from (4.13) in Proposition III.4.13 and (3). R e m a r k 11. (1) Let F 6 ca(
/ $dF,

and E € vca(
If there exists

A e %

- /J .A then $ is called the Radon-Nikodym

derivative (RN-derivative) 2

denoted by $ = — . $ is unique as a point in L (F). are identified if $ F ^ = # F ^ we can write "P^ = — — — . arc

of E w.r t F and is

[Recall that $ , * e £ 2 ( F )

J/-o.e., where i/(-) = |F|(-).l In Lemma 10 (4) above,

197

4.9. SUBORDINATION

(2) Let f E cagos(<&d,X), $ 6 L 2 ( ^ ) and i 6 l Since f o x E vca(<&d,T{H)) by Theorem III. 1.5 we can define the integral of $ w.r.t. £ o a; and show that |

$d((oi)=

/#d£>x,

Ae!8g

Lemma |£ o x|(-) (2) If (3) If

12. (1) Lei £,7/ € rca(*&Q,X). Then, £ is /«//?/ subordinate to n iff
A e B G>

so that the conclusion follows from Theorem III.1.5. (3) is seen from (2), Lemma 10 (4) and Remark 11 (2). Now we can characterize full subordination for c.a.g.o.s. measures. T h e o r e m 13. For £,77 E rcagos(?&Q,X) the following conditions are equivalent: (1) £ is fully subordinate to rj. (2) £ is subordinate to 77, and £ and 77 are gramian biorthogonal. (3) There exists a $ E L2{FV)

such that

i{A) = f §dT], JA A

AE<&;

G-

(4) |£ o x|(-)
M-HJ&MTZ)'-

A

^

Proof. (1) => (2). We only have to prove gramian biorthogonality of £ and TJ. Let A,B E 55 g be disjoint. Then

i(A) e ©€(ii) c 6^(BC) c e„(Bc) c &V(B)*

198

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

by (1) and Lemma 10 (1). Hence [f(A),7?(B)] = 0 since 77(B) e <Sn(B). Thus £ and r] are gramian biorthogonal. (2) => (3). By Proposition III.4.13 and Lemma 10 (4) we have that d(i{A)or,) dF„ JG

drj,

Ae
Now (£(A) o ri)(B) = Fin{A n B) for A , B e 2 3 g , which, together with Lemma 12, implies that dF,,

G

dF„ '

Thus we get

dFc Taking $ = —|3L g L 2 (F,,), (3) follows. (3) => (4). (4) follows from the implications: \r]ox\(A)

= 0 = > ( ? 7 o i ) ( J 4 n S ) = 0 for every B e B g

(^oi)(AnB)

/

$dr?,x

= /

•/Ana

$d(?7ox)=0,

J/AnB ARB

for every B G 23 g by (3) and Remark 11 (2) | € o i | ( X ) = 0. (4) =>■ (1). Let A € 55g. Then the following implications hold: x e &V(A)*

= > 17/ o at I (A) = 0, =*\Sox\{A)

= Q, #

==>at€ 6 C ( A ) , Hence 6((A)

by Lemma 10 (2) by (4) by Lemma 10(2).

C 6 „ ( i l ) . Thus (1) holds.

(2), (3) => (5). By (2) and Lemma 12 $ in (3) can be taken to be ^ K

dF„

holds that F 4 (A) = [ £ ( A ) ,£(>!)] =

/ *dr,} [

J A

JA

<&dr?

=

[1A*,1A*]

im^m* ****

Then it

4.9. SUBORDINATION

199

(5) =* (1). Let B E KQ be fixed and PV(B) : X -> &V{B) be the gramian orthogonal projection. Then as in the proof of (2) => (3) we have that

= f lA$dT)= JB

where $ =

f

JlAnB A<

Qdq,

Ae
dFt

* • Hence we get by (5) that for J4 6 23 g

[Pn{B)£{A),Pv{B)t{A)]

=

[lAnB*AAnB*]Fn

= I

$dFv$*

=

Fi{Af]B).

JAnB

Thus, if AC (1) holds.

B, then £(A) € &V{B).

This shows that 6 ? ( S ) C 6V(B).

In the above theorem, the function

dFe1 Z E ^(F^) dFv

Therefore

can be regarded as a "sym-

dt,

bolic" RN-derivative — of £ w.r.t. n. From the above theorem we can characterize art gramian orthogonal scatteredness as follows. Corollary 14. For £ E 6ca(Q3g,X) the following conditions are equivalent: (1) £ is g.o.s. (2) There exists an r\ E cagos{?&Q,X) to which £ is fully subordinate. (3) There exists an r\ E cagos(23g,X) £0 which £ is subordinate and such that they are gramian biorthogonal. (4) There exist an n E cagos{*&Q,X) and a <3? 6 L2(FV) such that $ d-q,

AE 93;

JA

(5) There exists an -q E ca(/os(23g,X) such that |£°x[(-) <S |»joa;|(-) for every xEX. Proof. Clearly (1) implies (2) - (5) by taking 77 = £. Equivalences among (2) - (5) are easily seen from the proof of Theorem 13. (2) => (1). Let A,B E^&Q be disjoint. Then by (2) £(A) 6 GV{A) and £{£) e 6 „ ( B ) C © ^ ( ^ c ) C ev(A)*. Hence [£(4) £(#)] = 0. Thus £ is g.o.s.

IV. MULTIDIMENSIONAL STOCHASTIC PROCESSES

200

Now we can state subordination for operator s t a t i o n a r y correlated processes. Corollary 15. Let x and y be X-valued operator stationary processes on G and assume that they are operator stationarily correlated, so that their representing mea sures are P(-)x(e) and P(-)y(e) for some regular gramian spectral measure P(-). Then the following conditions are equivalent: (1) (2) (3) (4)

x is subordinate to y. P{-)x(e) is fully subordinate to P(-)y{e). P(-)x(e) is subordinate to P(-)y(e). There exists a $ 6 L2(F$) such that P(A)x{e)=

I $dPy(e), JA

A e
(5) \Px{e) o z\(-) -C \Py{e) o z|(-) for every z e X. (6) F± can be written as

(7) x is of operator Karhunen class with the representing measure Here, Fz, Fy and Fig are defined respectively by F i ( - ) = [P()x(e),x(e)},

Fy(-) = [P()y(e),y(e)},

FXy(-) =

P(-)y(e).

[P(-)x(e),y(e)}.

Proof. This follows from Propositions 6 and 8 and Theorem 13.

Bibliographical n o t e s Most of the definitions and results on infinite dimensional stochastic processes are based on and motivated by one dimenstional and finite dimensional cases. 4-1- General concepts. Proposirion 1.2 is fundamental. (1), (2) and (4) are from Kakihara [7](1985), and (3) is well-known (see e.g. Parzen [1](1959)). 4-2. Stationary processes. Theorem 2.2 gives integral representations for station ary or operator stationry processes, which is essentially due to Khintchine [1](1934) and Payen [l](1967), respectively. Proposition 2.3 is in Kakihara [15](1992) and the Kolmogorov isomorphism theorem (Theorem 2.4) is due to Kolmogorov [1](1941) for the one dimensional discrete case, Wiener and Masani [l](1957) for the finite dimensional case and Mandrekar and Salehi [ll(1970) for the present case.

BIBLIOGRAPHICAL NOTES

201

4-3. Harmonizable processes. Harmonizable processes are introduced by Loeve [2](1948)(strong case) and byRozanov [l]( 1959)(weak case). (1) and (2) of Theorem 3.2 is due to Rozanov [lj and Truong-van [1](1981), respectively (see also Kakihara [7](1985)). Corollaries 3.3 and 3.4 are well-known. Example 3.6 is mostly from Kakihara [15]. Propositions 3.7 and 3.9 are due to Niemi [1, 2](1975). Theorem 3.10 is from Kakihara [15]. Theorem 3.11 is due to Rao [4](1982) for X = Lg(fl) and noted here for a general X. Theorem 3.13 is also due to Rao [5](1984) for a finite dimensional case and the present case is given here. Mehlman [1] (1991) proved that £ 2 ( M ) is a Hilbert space for M G 9 ^ ( 2 3 x 03 ; J5(C*)). Related topics may be found in Cambanis and Liu [1](1970). 4-4- V-bounded processes. V-boundedness was first introduced by Bochner [1] (1951) for one-dimensional case and for infinite-dimensional case by Kakihara [7] (see also [15]). Lemma 4.2 is in Kakihara [7]. (1) and (2) of Lemma 4.4 are Miamee and Salehi [2](1978) and Kakihara [7], respectively. Theorem 4.5 is proved by Kakihara [7]. Corollary 4.6 is due to Niemi [1] (see also Miamee and Salehi [2]). Propositions 4.7 and 4.8 are proved in Kakihara [20](1996) and [15], respectively. Various equivalence conditions among weak harmonizability, V-boundedness and Fourier transforms of vector measures are summarized in Rao [4]. 4-5. Cramer and Karhunen classes. The Cramer class (in the strong sense) was introduced by Cramer [2](1951). Rao [4] used the terminologies "of class (C)" and "weakly of class (C)" for strong and weak Cramer classes in our terminology. Weak and strong opertor Cramer classes are defined in a similar manner here. It is desirable to define operator Cramer class using functions in £?t(M) for a p.d. operator bimeasure M rather than those in £*(£) for some £ e bca(0,21). Some further investigation is needed. The Karhunen class was defined by Karhunen [l](1947) and the operator Karhunen class in Kakihara [11](1988). Proposition 5.3 is due to Rao [4] and Proposition 5.4 is new. The one dimensional case of Theorem 5.5 is due to Cambanis [1](1975) and the present style by Kakihara [11]. (1) of Theorem 5.6 is noted here and (2) is essentially due to Chang and Rao [2] (1986) (see also Rao [3](1982)). Related topics can be seen in Rao [1](1978). 4-6. Series representations. A usual series representation for a process and its relation to separability of the time domain were considered many years ago. Propo sition 6.2 and Corollary 6.3 are stated in Kakihara [11]. Proposition 6.4 is noted by Cambanis [1]. (2) of Theorem 6.5 and Corollary 6.6 are in Kakihara [11]. Theorem 6.7 is proved by Kakihara [11] and [17] (1994). 4- 7. Moving average representations. The one dimensional case of Theorem 7.3 is due to Chang and Rao [3](1988) and the present case by Kakihara [15]. Mehlman [1] and [2] (1992) considered various types of m.a.r.'s for finite dimaensional processes. 4-8. Approximation and convergence. Lemma 8.1 is noted by Kakihara [9](1986) and [15]. Proposition 8.2 is due to Niemi [1] for the one dimensional case and Kakihara [9] for the present case. Theorem 8.3 is due to Dehay and Moche [1](1986)

202

IV.

MULTIDIMENSIONAL STOCHASTIC PROCESSES

for the one dimensional case and Kakihara [12](1988) for the present case. Theorem 8.4 is due to Ressel [1](1974). Corollary 8.5 is in Kakihara [15]. One dimensional case of Theorem 8.6 is proved by Matsuyama and Kakihara [1](1989) and the infinite dimensional case by Kakihara [15]. Related topics can be seen in Cambanis and Masry [1](1971). 4-9. Subordination. The main idea of Theorem 9.2 through Corollary 9.5 is due to Siraya [l](1977) who treated the one dimensional case and they are newly formulated in the present style. Proposition 9.6 is noted here. Proposition 9.8 is due to Payen [1]. Lemma 9.10 is by Masani [4](1970), Ressel [2](1976) and Kakihara [14](1991). Lemma 9.12 is due to Ressel [2], Niemi (7](1984) and Kakihara [14]. Theorem 9.13 and Corollaries 9.14 and 9.15 are due to Rosenberg [2](1969), Mandrekar and Salehi [2](1970), Ressel [2], Niemi [8] and Kakihara [14].

CHAPTER V

SPECIAL

TOPICS

Fundamental definitions and results on X-valued processes on an LCA group G are developed in the previous chapter. In this chapter, some further topics are considered for processes mainly on R or Z. The Wold decomposition is a decom position of a process into deterministic and purely nondeterministic parts and gives some insights for estimation problems which will be discussed in the next chapter. Cramer's decomposition is defined for a harmonizable process and decomposes its representing measure into absolutely continuous and singular (w.r.t. the Haar mea sure of G) parts. Relations between these two decompositions are noted. A process of KF-class is defined to have stationarity in an asymptotic sense. Properties of such a process are discussed in connection with harmonizability. Uniformly bounded lin early stationary processes are defined and characterized. Existence of general shift operators are also considered. Periodically correlated processes are of interest. A process in this class is not stationary but has a period and it associates a stationary process. Some properties of this class are investigated. In the final section, isotropic processes, processes on hypergroups, and processes on locally compact groups are briefly mentioned, where definitions are precisely given but results are stated without proofs since these topics need much preparation and may be explored elsewhere.

5.1. Wold decompositions Let X = L2(fl;H). We confine our attention to X-valued processes on the real line R, although the results obtained in this section are also true for processes on Z. Wold decompositions are defined for such processes and a detailed analysis is given for weakly (operator) harmonizable processes in connection with their (operator) stationary dilations. Definition 1. Let x = {x(t)} (1) The observation

be an X-valued process on R

submodule W.(x,t) up to iime t 6 R and the remote past on 1 }

204

V. SPECIAL TOPICS

Ti(x, — oo) are denned respectively by M{x,t) = &{x{s) :s
H{x,-oo)

=

f)H(i,t). teR

[Note that %(x,oo) = H{x), the modular time domain.] (2) {x(t)} is said to be deterministic if U(x, - o o ) = ~H{x) and to be nondetermin istic if 7i{x, — oo) C T-L(x), a proper set inclusion. An extreme case of nondeterminism is as follows. {x(t)} is said to be purely nondeterministic if H(x, —oo) = {0}. Note that determinism and pure nondeterminism are invariant under similarity and hence under equivalence (cf. Definition IV. 1.1). The following theorem, known as "Wold decomposition," states that every process can be decomposed uniquely into its deterministic and purely nondeterministic parts. T h e o r e m 2 (Wold D e c o m p o s i t i o n ) . Let {x(t)} there exists a unique decomposition

x(t) = xd{t) + xp(t),

be an X-valued process.

Then

(el

(l.l)

of {x(t)} such that (1) {xd{t)} is deterministic and {xp(t)} is purely nondeterministic. (2) {xd(t)} and {xp(t)} are subordinate to {x(t)}. (3) {xd(t)} and {xp(t)} are gramtan orthogonal, i.e., [xd{t),xp(u)] = 0 for every t,ueR. Moreover, H(x) = U{xd) ®H{xp). Proof. Let P : X —> H{x, —oo) be the gramian orthogonal projection and define xd{t) = Px(t),

xp(t) = (I -P)x(t),

teR.

Then (2) and (3) are easy to see. To prove (1) observe that, for each t 6 R, H{xd,t) C ~H(x,-oo) and H{xp,t) C H(x,— oo)*, the gamian orthogonal comple ment oiU{x,-oo), so that H(x,t) = H{xd, t) © 7i(xp,t). Hence we have U(xp, - o o ) C P | [H(x, - O O ) # n H(x, t)} = H(x, - o o ) # n U(x, - o o ) = {0},

which implies that {xp(t)}

is purely nondeterministic, and

H{x, - o o ) = p | [H(xd, t) ®U{xp,

t)} = n(xd,

= 1t{xd, - o o ) C H(i, - o o ) ,

- o o ) © H(x„ - o o )

205

5.1. WOLD DECOMPOSITIONS

which implies that {xd{t)} is deterministic. As to the uniqueness of the decomposition (1.1), let x(t) = x%{t) + x2(t), t e R be another decomposition such that (1), (2) and (3) hold. Then we have that x(t) = xd(t)+xp{t) = xi{t) + x2{t) fort e K , H(ii) =H(ii,-oo) = H{x,-oo) and H{x2,-oo) = {0}. Thus for t € R we get xd{t) -x±{t) = x2(t)-xp(t) € H(x,-oo)f\ H(x, —oo)#, from which we conclude that Xd{t) = xi(t) and xp(t) = x2(t) for t 6 R R e m a r k 3. Suppose that a process is defined on an LCA group G. Let X be a family of subsets of G which is translation invariant in the sense that A e X and s £ G imply sA e X. We can define determinism and pure nondeterminism as follows. An X-valued process {x(t)} on G is said to be deterministic w.r.t. X if H{x, A) = &{x(t) : t € A} =%{x) for every A eX and to be purely nondeterministic w.r.t. X if n TL(x,A) = {0}. Then, we can prove Theorem 2 in this setting. Aei Moreover, the following theorem is also true in this setting. T h e o r e m 4. Let {x(t)} be an X-valued process, and {xa(t)} and {xp(t)} be its Wold decomposition components given by (1.1). Then, (1) {x(t)} is operator stationary iff {xd(t)} and {xp(t)} are such. (2) {x(t)} is weakly operator harmonizable iff {x^t)} and {xp(t)} are such. (3) {x(t)} is strongly operator harmonizable with the representing measure of bounded variation, i.e., {x(t)} 6 {soh}v, iff {xd(t)}, {xp(t)} 6 {soh}v. Proof. (1) The "if part. Let £d,£ p £ capos(Q5,X) be the representing measures of {xd(t)} and {xp(t)}, respectively. Then, since £j and £p are gramian biorthogonal, f = £d + £,p € ca<70s(Q3,X), which is the representing measure of {x(t)}. Thus {x(t)} is operator stationary. The "only i f part. Let {f/(s)} se R be the gramian unitary group associated with {x(t)}, so that x(t) = U{t)x(0) = U{t)xd{0) + U{t)xp{0) for t € R Note that for i€R U(t)U{x,

- c o ) = U{t) p | Uix, s) = P | U(t)H(x,

= f)U{x,

s)

S + t) = n{i, -OO)

s£R

and hence U(t)H{x, - o o ) # = H{x,-oo)*. Since U{t)xd(0) G H(x,-oo) and U{t)xp(0) e 7i(x, - o o ) # we get xd{t) = U(t)xd{0) and xp{t) = U(t)xp(0) for t € R This shows that { x ^ t ) } and {x p (t)} are operator stationary. (2) and (3) are easy. If {x(t)} has strongly operator harmonizable components in (1.1), then {x(t)} is also strongly operator harmonizable. However, the converse is not true in general.

206

V.

SPECIAL TOPICS

A simple necessary and sufficient condition for a strongly operator harmonizable process to have strongly operator harmonizable components in (1.1) is that one (hence both) of its components is strongly operator harmonizable. Let £ G bca(*B,X) and {77, Y,P} be its g.o.s.d. Y need not contain the whole space X but 6 5 ( E ) . Recall that {T),Y, P} is a minimal g.o.s.d. of £ provided that Y = 6„(R) D 6 € ( R ) and £ = Prj, and if {r/,Y',P'} is such, then Y is regarded as a closed submodule of Y'. If {x(t)} is weakly operator harmonizable with the representing measure £ e ftcaf^B,^) and {y{t)} is an operator stationary dilation of {x{t)} with the representing measure 77 e cagos( ~H{y, —00) be the gramian orthogonal projection. Note that PH{y,t)

= H(x,t),

t€H

Hence we have P-H{y,-00)

= H(x,-oo),

PH{y) = H{x).

Consequently it holds that for t e E Pyd{t) = PPiy(t) Pyp(t)

€ PH(y, - 0 0 ) = H{x, - 0 0 ) ,

= P(I - P 0 y ( t ) e P{H(y)

9 H{y, - 0 0 ) } = H(x) Q U{x, - 0 0 ) .

Since xd{t) E %{x, - 0 0 ) , xp(t) € %{x) 0 H{x, - 0 0 ) and ar(t) = Py[t) for i G R, we conclude that xd{t) = Pyd(t), xp{t) = Pyp(t), teR. Therefore the first assertion follows. As to the second assertion the "if part follows from the first assertion of the theorem. To see the "only if1 part assume that {x(£)} is deterministic and y{t) = Vd{t) + yv{t) is the Wold decomposition. Let rj,rjd,r]p e cagos(
5.1. WOLD DECOMPOSITIONS

{y(t)} is deterministic. If {x(t)} that {y(t)} is so.

207

is purely nondeterministic, we can prove similarly

Let {x(t)} be an X-valued process on E and <> / e H. Consider the Lg(fi)-valued process { ^ ( t ) } = {(x(t),(t>)H} and define for t e R ~Ho(x^,t) =
Ho(x^, - 0 0 ) = P | H0{X^, t), teR which are called the time domain and the remote past of {x^(i)}, respectively. Since H is assumed to be separable, we can choose a CONS {4>k}kLi of H. Then it holds that 00

00

fc=i

*:=i

H ( i , i ) =00/ / ® 6 o ( ' Q ' H o (v i 0 f c

( v

/

00

y v U Wo(i*», -00) I, H{x) « F ® 6 0 (fc=i | J «o(

fc=i

fc=i

'

^fc=i

For each n > 1 define n ,i)),

teR,

fc=l Then, for each n > 1, {xn(t)} is regarded as an [Lfttt)] -valued process and it holds that x n ( t ) —> x(f) in the norm of X for every i e R . That is, {x„(t)},n > 1 can be considered as a finite dimensional approximation of {x(t)}. Hence we immediately have P r o p o s i t i o n 6. Let {x(t)} be an X-valued process on R and {xn(t)},n > defined by (1.2). Let x(t) — Xd(t) + xp(t) and xn(t) = xn,d(t) + xn the Wold decompositions of {x(t)} and {xn(t)}, respectively. Then it holds Xn,d(t) —> X xp(t) as n —> 00 in the norm of X for every

1 be 1 be that ieR.

Purely nondeterministic and operator stationary processes will be considered in Section 6.1 along with prediction problems.

208

V. SPECIAL TOPICS

5.2. Cramer d e c o m p o s i t i o n s Let G be an LCA group with the dual group G. If F <= rca(
Z„Zaeca(%,X)

(2.1)

of £ for which it holds that (1) £ s is singular w.r.t. v, denoted £ _L v, (2) £ a is absolutely continuous w.r.t. v, denoted £a
UA) = aAnAct),

UA) = Z(An A.),

there

Ae%.

In the above theorem, (2.1) is called the Lebesgue decomposition of £ w.r.t. v by A,. If 0 is locally compact, 21 is its Borel u-algebra and £ is regular, then £ s and £ a are also regular. Next we consider Cramer decompositions for operator stationary processes. T h e o r e m 2 (Cramer D e c o m p o s i t i o n ) . Let {x(t)} be an X-valued operator sta tionary process on G with the representing measure £ E rcagos^g, X) and the operator spectral measure F = F^ € rca( < 8g, T+(H)). If £ = £ s + £ a is the Lebesgue decomposition of £ w.r.t. the Haar measure g of G by a set A, e 'Bg, then F

= ^

+ Ft,

(2.2)

is the Lebesgue decomposition of F w.r.t. g by the same At. Hence, x(t) = xa{t) + xa{t),

teG

(2.3)

5.2. CRAMER DECOMPOSITIONS

209

gives a decomposition of {x(t)} into two operator stationary processes with singular representing and operator spectral measures, and absolutely continuous representing and operator spectral measures, respectively. Proof. This immediately follows from Theorem 1. The decompositions (2.2) and (2.3) are called the Cramer decompositions of the operator spectral measure and the process, respectively. If, in the above theorem, we are first given the decomposition (2.2) by a set B„ £ 2$g, then we also have the Lebesgue decomposition £ = £3 + £Q by the same set B,. More generally, we shall define the Cramer decomposition of a weakly operator harmonizable process as follows. D e f i n i t i o n 3 . Let {x(t)} be an X-valued weakly operator harmonizable process on G with the representing measure £ 6 rbca(^&Q,X) and £ = £ s + £ a be the Lebesgue decomposition of £ w.r.t. g by the set A, E Q3g. Let {xs(t)} and {xa{t}} be weakly operator harmonizable processes with the representing measures £s and £ a , respectively. Then the decomposition x(t) =xs{t) is called the Cramer decomposition of

+ xa(t),

teG

{x(t)}.

The following theorem is a key to obtain relations between the Cramer decompo sitions of weakly operator harmonizable processes and those of operator stationary dilation processes. T h e o r e m 4. Let E, £ r6ca(23g,X) and ( = £ s + £ a be its Lebesgue decomposition w.r.t. g by A, £ 93g. Then there exists a regular g.o.s.d. {n, Y,P} of £ for which its Lebesgue decomposition n = ns + r\a w.r.t. g is by the same At, and rj8 and na are the g.o.s.d. 's of £s and £ a , respectively. Moreover, if {n,Y,P} is minimal, then £ s = 0 (resp. £ a = 0) iff r?s = 0 (resp. na = 0). Proof. Let {£, Z, P) be a regular g.o.s.d. of £ and ( = (s + Ca be the Lebesgue decomposition w.r.t. g. Define TJ by

n(A) = (s(A^At)

+ Ca(A),

Ae<3d.

Clearly 7/ € rcagos(%$Q, Z). If 7j = 77,, + na is the Lebesgue decomposition w.r.t. g, then ns{-) = Cs(- l~l A»), na = Co and r]s,7]a 6 rcagos(?&Q,Z). Observe that for

Ae<3d V(A

n A,) = (S{A n A.) + ca(A n A„) = Q(A n At) = n,{A),

210

V. SPECIAL TOPICS

r,(A n A%) = C,(A n Al n At) + (a(A n ACJ = c«(>l n AJ) + (a(A n A.) = Ca(A) = IJ„(A), aQ

since Co < Q d g(A n A») = 0. Hence n = n3 + na is by A,. We show that r/„ is a g.o.s.d. of £„. It suffices to prove that

/ $ dZs, I $ df,l < f $ dF„3 **, _JG

<J G

J

$ G L°(G; B(£T))

»/ G

(cf. Theorem III.3.2). Observe that for every $ = £ o , - l ^ G

L°(G;B{H))

3=1

= ^ai[^inA»),((4ni,)]oJ i.fc

< ^ Q j [C(Aj n A,),C(Aj n A,)]aJ, since f is a g.o.s.d. of £, = 5^aj[C.(-Aj n A,)Xs(Aj

n A,)]a*,

since Co < Q and £(A,- n A,) = 0, 1 < j < n,

= J2^h(Aj),Vs(Aj)}a'3

= [*dF„,

$*.

Hence ns is a g.o.s.d. of £a by Theorem III.3.2. Similarly we can show that na is a g.o.s.d. of f0. The last assertion is easy to verify. The following corollary corresponds to Theorem 1.4 and is a direct consequence of Theorem 4. Corollary 5. Let {x(t)} be an X-valued weakly operator harmonizable process on G with the Cramer decomposition x(t) = xs(t) +xa(t). Then there exists an operator stationary dilation {y{t)} of {x(t)} with the Cramer decomposition y(t) = y3(t) + ya{t) for which {ys(t)} and {ya(t)} are operator stationary dilations of {x„(t)} and {xa(t)}, respectively. Moreover, if {y{t)} is minimal, then xs(t) = 0 (resp. xa(t) = 0) iff y„{t) = 0 (resp. ya(t) = Q). Let {^fc}^! be a CONS in H and £ e r6ca(Q3 g ,X). Then it holds that OO

211

5.2. CRAMER DECOMPOSITIONS

where as before ^ ( - ) = (£(•), tf>)H e rca(33g,Lg(fi)) for <£ 6 H. If £ = &, + £a and £<£ = ?0,s + £<£,a are the Lebesgue decompositions w.r.t. g, then we have oo

oo

Jfc=l

fc=l

Let {x(t)} be an X-valued weakly operator harmonizable process with the represent ing measure £. Note that for each n > 1 the process {xn(t)} denned by (1.2) is also n

weakly operator harmonizable with the representing measure £ n (-) = J^ £,4>k(-)4>kfc=i

Thus we obtain P r o p o s i t i o n 6. Let {x(t)} be an X-valued weakly operator harmonizable process on G and x(t) = xs(t) + xa(i) be the Cramer decomposition. If, for each (j> e H, x${t) — X0 jS (t) + a^0,a(*) is the Cramer decomposition of {x^t)}, then it holds that oo

*»(*) = 2ja ; **,a(*)0fc, fc=l fc=l

oo

Xa(t) = y ^ J ^ , t t ( t ) ^ f c ,

teG.

Hence, if xn(t) = xniS(t)+x„ia(t) is the Cramer decomposition of {xn(t)} for n > 1 defined by (1.2), i/ien x „ s ( ( ) —> x s (£) and s n ,a(*) -► xa(t) as n —> oo in the norm of X for every t e G. In the rest of this section we consider the case where G = R or Z. In this case we defined the Wold decomposition for every X-valued process. So the question is: If the process is weakly operator harmonizable, do Wold and Cramer decompositions coincide? We begin with a definition. Definition 7. For an X-valued weakly operator harmonizable process {x(t)} on G = R or Z let x{i) = Xd(t) +xp(t) and x{t) = x3(t)+xa(t) be its Wold and Cramer decompositions, respectively. If Xd{t) = xs(t),

xp(t) = xa(t),

hold, we say that {x(t)} satisfies the Wold-Cramer

teG concordance.

Some necessary and sufficient conditions for the Wold-Cramer concordance are given in the following which is easily proved from Theorem 1.4, Proposition 1.5, Corollary 5 and Proposition 6. P r o p o s i t i o n 8. Let {x(t)} be an X-valued weakly operator harmonizable process on R or Z and {xn(t)} be defined by (1.2) for n > 1. Then, the following conditions are mutually equivalent:

212

(1) The (2) The {»(*)} of (3) The (4) The {Vn{t)} of

V. SPECIAL TOPICS

Wold-Cramer concordance Wold-Cramer concordance {x(t)}. Wold-Cramer concordance Wold-Cramer concordance {xn(t)} for every n > 1.

holds for {x(t)}. holds for some operator stationary

dilation

holds for {xn(t)} for every n > 1. holds for some operator stationary

dilation

It follows from the above proposition that the concordance problem is reduced to the operator stationary case, where some necessary and sufficient conditions for the concordance are known. Here we don't discuss further.

5.3. T h e KF-class In this section, we consider a class of X-valued processes on R which are operator stationary in an asymptotic sense, called the KF-class. We shall show that this class contains {soh} but not {woh}. Also we consider weakly operator harmonizable processes with cr-finite operator spectral bimeasures. Definition 1. Let {x(t)} be an X-valued process on R with the operator covariance function T. Then it is said to be of KF-class if T is Bochner integrable w.r.t. the Lebesgue measure ds x dt on every bounded Borel subset of R2 and if there exists a function T : R —>■ T(H), continuous in ||-|| T , for which the following equality holds: f ( f e ) = lim - f T(s + h, s)ds, t >0 - ° t J0

heR.

(3.1)

F is called the associated operator covariance of {x(t)}. It can be seen that f is p.d. (see Lemma 2 below), so that there exists an F 6 ca(25, T + ( # ) ) such that f(h)=

f ethuF(du), J«

heR.

F is called the associated operator spectral measure of the process. L e m m a 2. Let T be the operator covariance function of an X-valued process for which T is defined by (3.1) and is continuous. Then, F is p.d. Proof. Let n > 1, hi, ■ ■ ■ , h„ 6 R and a 1 ? . . . , a„ 6 B(H) be arbitrary. Then we have that 1 Y^ Ojf {ht - hk)a'k = hm jYjan 3,k

3,k

/"' J

°

r

( s + hJ ~hk,s)

ds a'k

5.3. THE KF-CLASS

213

t-hk rt-ni,

1 .

= lim - > a, /

T(u + hi,u + hk) dual hk

= lim - (

/

+ / + /

^°°t\J_hk i

Jo

Jt

)y^

ajT(u + hj,u +

hk)aldu

Jj ^

/■' r

Therefore T is p.d. We can define a process of KF-class on Z as follows. An X-valued process {x(n)} on Z is said to be of KF-class if there exists a function f : Z —> T(H) such that f ( f c ) = lim — — - V

T(n + k,n),

fc

6 Z,

t->CG It + 1 * ' n= — t

where F is the operator covariance function of {x(n)}. F e c a ( 2 3 T , T + ( # ) ) such that

f(k)= f eikuF(du),

In this case, there exists an

ke

Jj

where T = [0,27r] (or the unit circle in the complex plane) and *Bj is its Borel a-algebra. For a bimeasure M on 23 x 23 its value at A x B is denoted indifferently by M{A,B)OT

M{AxB).

P r o p o s i t i o n 3. / / {x(t)} is an X-valued R, then it is of KF-class. More fully, if of {x(t)}, then the associated operator spectral measure F are respectively given

strongly operator harmonizahle process on £ <E bca(23,X) is the representing measure covariance T and the associated operator by

£{h) = I eihuMs(du,dn),

he

JR

F(A) = Mi({AxA)r\A), where A = {(t,t)

A €23,

: t £ R} is the diagonal.

Proof. Since M^ is of bounded variation by assumption, M^ can be uniquely extended to a measure on 23 ® 23, the Borel a-algebra of R 2 . Then we have that - [ r{s + h,s)ds=t Jo

[

[

* JO J-oo

[ J-oo

el^s+^u-3V>Mi{du,dv)ds

214

V. SPECIAL TOPICS OO

/

/»CO

-I

ft

elhu-j

I

ei{u-v)adsM^{du,dv), t J0

-co J-oo

by Pubini's theorem, J

/•CO

—(

eihu

Mf(dut du)

as t -> oo,

by Bounded Convergence Theorem, where T is the operator covariance function of {x(t)}. This completes the proof. L e m m a 4. Let {xn(t)}^=1 be a bounded sequence of X-valued processes of KF-class on R sac/i t/mt \\xn{i) -x{t)\\x -> 0 (n -> oo) uniformly on E. T/ien, {x(t)} W a/so o/ KF-class. Proof. Let F n and T be the operator covariance functions of {x n (i)} and (x(t)}, respectively. It follows from the assumptions that | | r n ( s , t) — r ( s , i ) | | T -4 0 uniformly on R 2 . Thus we get that

Hm i rT(s +

/i, s)ds = lim

Tn(h)

n—yoo

exists for every h € R and is continuous on R, where r n the associated operator covariance of {x„{t)}. Therefore {x(t)} is of KF-class. We can generalize Proposition 3 by considering weakly operator harmonizable propcesses with cr-finite operator spectral bimeasures. Definition 5. A T(H)-valued bimeasure M € M = 9Jt(95 x 1. P r o p o s i t i o n 6. / / {x(t)} is an X-valued weakly operator harmonizable process on R with a a-finite operator spectral bimeasure, then there exists a sequence {xn(t)}^L1 of X-valued strongly operator harmonizable processes on R such that \\xn(t) —x(t)\\x —> 0 uniformly on R. Proof. Let £ € 6ca(05,X) be the representing measure of {x(t)}. By assumption M € is a-finite with an increasing sequence {An}'^'=1 C 05. For each n > 1 define U-)=£(-n.4n), Zn(*)= I eltuZn(du),

Mn = Min,

(3.2)

teR.

(3.3)

215

5.3. THE KF-CLASS

Since M„ € Ttv, {xn(t)} \xn(t) -

is strongly operator harmonizable. Moreover, x(t)\\x

[

eUuZ{du)

< 11^11(^)^0,

since An I 0. Hence, this convergence is uniform on E. T h e o r e m 7. If {x(t)} is an X-valued weakly operator harmonizable process on K whose operator spectral bimeasure M is a-finite with an increasing sequence {An}n°_1 C 33, then it is of KF-class with the associated operator spectral measure F given by F{A)=

lim M( (An An) x( An An))P\ A),

A6
(3.4)

Proof. Let £ E bca(^&,X) be the representing measure of {x(t)} and define £ n and {xn(t)} by (3.2) and (3.3), respectively for n > 1. Let T and T n be operator covariance functions of {x(t)} and {xn(t)}, respectively. Note that ||x„(£)||x 5: IICnll(R) < ||C||(K) < oo for every n > 1 and t 6 R and hence {x„(£)}£°=1 is a bounded sequence of strongly operator harmonizable processes of KF-class. By Proposition 6 we have ||ar„(t) — a?(t)||x ~> 0 uniformly on M, so that {x(t)} is also of KF-class by Lemma 4. To obtain the formula (3.4) we need to work more. The uniform convergence of {xn(t)}n°=1 to {x(t)} implies that | | r „ ( s , t) - F(s,t)\\T -> 0 uniformly on K 2 . Hence, for each e > 0 there exists an N(e) > 1 such that - / T(s+h,s)ds~t Jo

t Jo

Tn(s + h,s)ds

n > N(e), t>0,heR.

<e,

By Proposition 2, for each n > 1, there exists a T(n,e) - f rn{s + h,s)dst Jo

[ elhu J&

Mn(du,du)

<e,

(3.5)

> 0 such that t>T(n,e),h£R,

(3.6)

where Mn is defined by (3.2). It then follows from (3.5) and (3.6) that - f F(s + h, s)ds- f elhu t Jo Jv.

Mn(du, du)

< 2e,

n> N(e),t>T(n,e),

he (3.7)

For each n > 1 define Fn by

Fn(A)=Mn((AxA)nA))

AeS.

(3.8)

216

V. SPECIAL TOPICS

Then, we shall show that { F n } ^ is a bounded and increasing sequence in c a ( 2 5 , r + ( F ) ) , where Fn < F n + 1 means that the operator inequality Fn(A) < Fn+i(A)

holds for every A € «8. For q G N and r 6 Z define Fq = [ ^ , ^ - ) -

For

each q G N the set {Fq : r e Z} forms a partition of 1 , and U^Ig x I£) | A as rSZ

5 —)• oo. Hence it holds that for every n > 1 and .A G 33

F„(A) = M ( ( ( A n An) x(An An)) n = lim M({(An

A)

An) x [An An)) n | J ( / , r x JJ) J

= lim M( (J ((Ani„nij) x f i n ^ n 7 J ) ) = lim y"M(An A„n/I,,4nA„n JI) > o. q—*oo

rgZ

The finite additivity of F„ is easily seen and the countable additivity follows from the fact that, if Bk | 0, then ||jF„(5fc)||T = ||Af„((B fc x Bk) n A ) | | r -* 0. Thus, {F n }£° =1 C ca( 1 and A € 33 we see that Fn(A) = Fn+1(A

n An ) <

Fn+1(A),

so that {Fn}^=1 is increasing. To see that {FnJ^Lj is bounded, let rj G cagos{?&, Y) be a g.o.s.d. of £ and observe that

0<M(A,A)
A €38,

where F,,(-) = [??(■),?7(-)]y Hence, we have for n > 1 and A G 33 that \\Fn(A)\\T =

lim VM(A n An n JI, A n An n /I) lirn^llMtAn^n^An^n/;)^

r6^

S Jim E q—YOD

1 1 ^ ( A n A » n / , r ) L = Wpv(A n A n ) l | r < | | F n ( K ) | | T < o o .

rgZ

Consequently, { F n } ^ - ! is bounded. Since, for each A G 33, { F ^ A ) } ^ is a bounded and increasing sequence in T+(H), it is also a Cauchy sequence. Hence, there exists an element F(A) G T+(H) such that \\Fn(A) - F(A)\\T -+ 0. By Vitali-Hahn-Saks-Nikodym Theorem (cf. Sec tion 3.1) we have F G ca(33, T+{H)).

217

5.3. THE KF-CLASS

Now, for any bounded Borel function / on R with a bound a > 0 we have that [ f(u)Fn(du)JR

[ f(u)F(du) JR

=1 / T

f{u)F(di

\ JArn

Hence, for any e > 0 there exists an N'{e) > 1 such that j elhu Mn(du, du) - / eihu F{du) JR

< e,

n> N'(e),

h g

(3.9)

JR.

(cf. (3.8)). From (3.7) and (3.9) we deduce that there exists a T(e) > 0 such that j eihu F{du)

- / T{s + h,s)ds~ t Jo

< 3e,

t > T(e), h g

JK

Therefore, we conclude that f(h) = j elhuF{du),

fiet

Jm

This completes the proof. E x a m p l e 8. (1) Weakly operator harmonizable with a o-fmite operator spectral bimeasure but not strongly operator harmonizable. Let f £ 6ca(Q3, X) be denned as in Example III.1.24 (2). Then, M^ is cr-finite with an increasing sequence {[—ra,«]} and |Mj|(R,R) = oo. Thus {x(t)} with the representing measure £ satisfies the assertion. (2) Of KF-class but not weakly operator harmonizable. Let E H and g g L\(£l) be of norm 1 and define a function / on R by f(t) = 1 if t e [2n, 2n + 1) and = 0 if t E [2n + 1, 2n + 2) for n e Z. Define an X-valued process {a;(t)} by x(i) = f(t)g for £ € R. Then we can show that {x(t)} is of KF-class but not a weakly operator harmonizable process since it is not continuous. (3) Weakly operator harmonizable but not of class KF. Assume that LQ(Q,) is separable and let {fk}^Li be a CONS in Lg(fi) and $ e H be of norm 1. Then, putting x{n) = f„4> for n g Z, {x(n) = / n ^ } is a gramian ON basis of X by Proposition II.3.11 and hence an operator stationary process on Z. Define a sequence {afcjfcez C C by

«fc

2 2 n < |fc| < 2 2 n + 1 , 2

2n+1

< Ifcl < 2

2n+2

fcgZ,

n>0

k E Z, n > 0

V. SPECIAL TOPICS

218

and an X-valued process {y(n)} by y(n) = anx(n),

n E Z.

P u t A = 1 0 a e A(n{x)) = A(X), where a £ B(L§(fi)) is denned by afk = akfk for k € Z. It is easy to see that {y{n)} is weakly operator harmonizable by Corollary IV.3.4 since y(n) = Ax{n) for n € Z but

,lim ^TTT E I M *

t->oo 2t + 1 fc=-t ■*—'

does not exist since 2!"+ - 1

t

" 5 ^ a T l E 11^)11^ = ^ , 2(2^1- 1 ) + 1 v

fc=-t

y

E 2

fc=-2 "

IW*)ll2r=3. l

+ +l

2 2 ™+ 2 —1

t

^ a T T E llv(*)l&-^s >8(aaw+ ,. 1) + 1

E

2

Ily(*)ll3r-|.

Therefore {y(n)} is not of KF-class.

5.4. Uniformly b o u n d e d linearly stationary processes In this section, we consider uniformly bounded linearly stationary processes on an LCA group G, which form a generalization of stationary processes. The shift operator group for a general process is defined and when it is uniformly bounded, we get a uniformly bounded linearly stationary process. We shall characterize this class of processes. Throughout this section, we assume that every process {x(t)} is nontrivial, i.e., x(t) ^ 0. Definition 1. Let {x(t)} be an X-valued process on G. P u t <5i(£) = I Y^ajxfa)

:au...

,an<EB{H),tx,...

•> j=i

, i n e G, n>

ll, >

which is dense in H{x) = 6 ( 6 i ( i ) ) . {x(t)} is said to admit a shift operator group if, for each ft E G, an operator T^ on ©i(£) given by Th f Y^ o.jx{tj) j =

Y^ajx{tjh)

5.4. UNIFORMLY BOUNDED LINEARLY STATIONARY PROCESSES

219

is well-defined. In this case, every Th is densely defined and {Th}heG forms a group of B(H)-\mea.T operators (i.e., module maps) on &i(x). When every Th is bounded, we still denote by Th the unique extension to ri(x), so that Th € A(H(x)) for h € G. Moreover, if {ThjheG is uniformly bounded, i.e., if there exists a constant a > 0 (a > 1 may certainly be taken) such that ||T),|| < a for h 6 G, then the process {x(t)} is said to be uniformly bounded linearly stationary (u.b.l.s.). Thus {x(t)} satisfies: akx(tkh) < a 2

2 J a,jx(tjh), ^ j=i

(4.1) L

fc=i

j'=i

fc=i

for every n > 1, t l t . . . ,tn 6 G, oi, ■.. , a„ € B(H) and h R e m a r k 2. Let {x(t)} function V.

eG.

be an X-valued process on G with the operator covariance n

(1) If {x(i)} satisfies the condition that £ ]

a

j^{tjjtk)al

= 0 for some aj e -B(-ff)

>,fc=i n

and tj £ G (1 < j < n) implies ^

o,jY(tjS, tks)a*k = 0 for every s 6 G, then {x(£)}

j,k=l

admits a shift operator group. (2) If {x(t)} is weakly operator harmonizable with the representing measure £ € 6ca(23g, X), then it admits a shift operator group iff for any tp g P(G)

/a

V = 0 £-a.e.

where P ( G ) is the set of all trigonometric polynomials on G. In fact, the "if part is obvious. To see the "only i f part, assume {x(t)} has a shift operator group {Th}heG and

0} is the same as the one with respect to nz defined in (2.31) since nz is in bca(rS+,Y). Note that (x{t),(j>)H € Lg(fi) for 0} to the filtering problem is called an optimal filter. In this case the estimate x(t) = / 0 ' K.(t, s) dz(s) is also called a solution to the filtering problem. Let T(t,s) = [x(t),x(s)]x, t,s > 0 be the operator covariance function of the process {x(t),t > 0}. Then, by using the uncorrelatedness of x0 and £, it can be represented as r{t,s)=U(t)[xo,xo]xU(ay + | I W(t)W _ 1 («)$(u)^(d«), f

U(s)U-\v){v)Z{dv X (2.32)

for t,s > 0. If, in particular, £ is g.o.s., then T{t,s) =U{t)[x0,xQ}xU{s)'

+U{t)l

{U-x§)dFK{U-x$)*\u{s)*

J

for t, s > 0, where t A s = min{£, s}. L e m m a 7. For eac/i t > 0, x(t) = J0 IC(t, s) nz(ds) is a solution to the filtering problem iff the cross gramian [x(t),z(v) — z(u)] = 0, 0 e H and t > 0 define i W =

{(i,fl

H

:ie4

where £^#(17*, *) = {Bl[o,t] : B E Sijc^iVi)}- N o t e t h a t these two spaces are closed subspaces of Ljj(fi) a n ( ^ t n e s e c ° n d is a closed subspace of the first. Also note that for x, y e X [x,y}x = 0<=^ ((x,(/))H,{y,4>)H)2

= 0 for every 6 H.

By the above fact and the usual orthogonal projection lemma in a Hilbert space we obtain the following two-sided implications: {x(t),4>)H-{i{t),)H\\2=

min

{x{t)A)H-( ' Material

K

f Bdnz,4>) Jo >n

271

6.2. KALMAN FILTER

for every 4> 6 H ( ( x ( i ) , 0 ) H , ( y ( t ) , 0 ) f l ) 2 = O for every (j/(£), )// 6 L(4>,t) and

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