GAN SYATE UNIVERSITY LIBRARIES

will“mm n m mm

'nll

l‘l
l
3 1293 00599 5349

 

 

“ .

 

 

 

 

 

 

 

 

 

 

 

 

HERARY
Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

DILATION OF OPERATOR VALUED MEASURES
IN BANACH SPACES AND
HARMONIZABLE BANACH SPACE VALUED PROCESSES

presented by

Philip H. Richard

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Statistics

Glazijtgzu ([LC mm

M ajor professor

Date May 15, 1990

MS U i: an Afflrman'w Action/Equal Opportunity Institution 0-12771

 

o2§éaEQVK

3

l

 

PLACE IN RETURN BOX to roman this checkout from your rooord.
TO AVOID FINES return on or baton duo duo.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

L__4
fi—WLJ
1
TH

MSU Is An Affirmative Action/Eng Opponuniiy lnotltuion

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DILATION OF OPERATOR VALUED MEASURES
IN BANACH SPACES AND
HARMONIZABLE BANACH SPACE VALUED PROCESSES
by
Philip H. Richard

A DISSERTATION
Submitted to
Michi an State University
in partial ful illment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

1990

1

DZ;

’05

ABSTRACT

For Banach spaces .5 and y, we consider the problem of factoring a family
{T(A): A62} 9 LAX”, indexed by an algebra 2, through a single Hilbert Space H.
We obtain a sufficient condition for such a factorization through a reproducing kernel
Hilbert Space involving an indexed family of non—negative operators dand taking
values in the space of conjugate linear functionals on .3 In case 2 is trivial, this
condition is shown to be equivalent to known conditions guaranteeing the
factorization of a single Operator T: .34 }’ through a Hilbert Space. In addition,
under our condition, we get that the family {T(A), A52} 9 L(..%§y) has a spectral
dilation in the sense that T(A) = SE(A)R where R: .Z-o H and S: H -i y are
continuous linear Operators and {E(A): A52} is a finitely additive spectral measure
in H. As a consequence of this result we obtain necessary and sufficient conditions
for the existence of an orthogonally scattered dilation of a Banach space valued
measure {T(A): A62} of finite semivariation; T(A) = S ((A), 5 an orthogonally
scattered measure taking values in a Hilbert space H and S: H -. } continuous and
linear taking values in the Banach space 1. We study as an application a
representation of harmonizable stable processes in terms of stationary second order
processes. Additionally, we prOpose definitions for harmonizable and V-bounded
Banach space—valued stochastic processes indexed by a separable locally compact
Abelian group which are shown to be equivalent and consistent with the Hilbert space

valued case.

Copyright by
PHILIP H. RICHARD
1990

For Katie and Mama

iv

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation and admiration to Dr. V.
Mandrekar for guiding me through the work contained in this dissertation. Truly,
without his guidance, completion of my degree would have been impossible. Dr.
Mandrekar's technical assistance was exceeded only by his skillful handling of the
emotional highs and lows experienced by this student during the entire process. I
consider Atma a friend as well as a mentor. The three other members of my
committee also contributed significantly to the completion of my degree
requirements. Discussions with Dr. H. Salehi lead to suggestions eventually
incorporated in the thesis. Dr. Dennis Gilliland's administrative acumen and
personal council were invaluable to me. The classroom expertise of Dr. Sheldon Axler
was the basis for whatever analytic ability I now possess. I extend my deepest
gratitude for all their assistance. In addition to the members of my committee, the
word processing and secretarial abilities, along with the personal insights, of Ms.
Cathy Sparks are to be commended. Finally, I would be remiss in failing to thank all
members of the Statistics and Probability Department at Michigan State University

who have assisted my progress throughout my long tenure as a student.

CHAPTER
0
1
2

References

TABLE OF CONTENTS

Introduction
Notation and Preliminary Results -
Factorization and Spectral Dilation

Orthogonally Scattered Dilation and
2—Summing Operators

Harmonizable and 80:8 Processes

Operators on S(2,H)

vi

PAGE

13

24
33
48

51

CHAPTER, 9
W

Orthogonally scattered dilations of countably additive Hilbert space
valued measures were considered by Abreu [2] and generalized by Niemi [26].
Several subsequent papers ([1], [8], [21], [24]) give generalizations of these
results to finitely additive orthogonally scattered dilations and their
applications to harmonizable processes. It is by now well-known that all
these results are consequences of variations of Grothendieck's Theorem
([27],p54) that all bounded linear Operators defined on an L°°—space into a
Hilbert Space are 2—summing and, hence, 2—majorizable. A generalization of
this dilation problem to the spectral dilation of L(H,K)—valued measures of
finite semivariation was proposed by Rosenberg [28] who provided necessary
and sufficient conditions in terms of 2—majorizability.

In [16], Lindenstrauss and Pelczyi’iski gave a generalization of 2—Summing
Operators to the class of Operators in L(xy) which admit a factorization
through a Hilbert space, i.e. Hilbertian Operators. Our first result (Theorem
2.15) gives a relationship between the spectral dilation of Operator-valued
measures and the factorization of a family of Operators using an apprOpriate
concept of 2—majorizability. In the case of a single Operator our condition is
equivalent to that of Lindenstrauss and Pelczyr’iski (Theorem 2.18). This is
done in Chapter 2.

In Chapter 3 we consider using factorization of a particular Operator
through a Hilbert Space to obtain an orthogonally scattered dilation of a
y-valued measure for a general Banach space y. It is observed that an
orthogonally scattered dilation of a (fl—valued measure is equivalent to

2—summability of an associated Operator defined on a class of bounded

scalar—valued functions into I. In this case, the factorability of this
Operator, its 2—summability and the existence of an orthogonally scattered
dilation of its associated measure, (Theorem 3.9), are equivalent. AS a
particular consequence, we consider an interesting condition given in a recent
paper by Makagon and Salehi [19]. We Show in Theorem 3.10 that under
their condition an L(H,K)—valued measure has an orthogonally scattered
dilation. Here H,K are Hilbert Spaces.

Using a result of Maurey concerning the 2-summability of operators from
an L°°—Space to a Banach space ] of cotype 2 we consider, in Chapter 4,
y—valued harmonizable processes. These processes are shown to have a
stationary dilation in an apprOpriate Hilbert space (Theorem 4.8). As the
main consequence of this, using results of Maurey [22] and Krivine (see
[27],p100), we give a more detailed spectral representation in the case of
stable processes than that obtained by Makagon and Mandrekar [18],
(Theorem 4.12). The cotype 2 case motivates definitions of harmonizability
and V—boundedness of a general Banach space—valued process. We prove
equivalence of these two concepts using techniques of our earlier results
(Theorem 4.16).

We want to emphasize that, except for Chapter 2, our study is mainly
directed towards orthogonally scattered dilations. The last chapter is devoted
to a brief discussion of an interesting Open problem in Makagon and Salehi
[19] concerning the prOperties of a particular K-valued Operator defined on
the class of .3—valued simple functions S(2,..3) when it is tOpologized using

two different norms and [I . ”00. Here we consider a very Special

in“,
case of a result in [19], namely when K = C and Show that when their

operator is -continuous, it is in fact not only 2—Summing but

Him

sup" 00’1“”th of their Operator implies its

in“,
integral. Also, in this case, ||°|I

|| -||m 2—summability (Theorem 5.3). We have concentrated our effort
towards orthogonally scattered dilations for two reasons. First, such dilations
have direct applications in the prediction of stochastic processes. The second
reason is that the only interesting sufficient conditions for the existence of a
spectral dilation of an L(H,K)—valued measure appear in [19], i.e. that of
[Lump—continuity and ll-Ilm 2—summability of an Operator defined on
S(E,H). Unfortunately, as pointed out in [19], in general a spectral measure
has neither of these prOperties. We should also mention a recent paper of
Miamee [23], which extends known results on spectral dilation to

L(..£H)—valued measures. This work, however, simply uses the original

techniques deveIOped in [19].

QEAEIIEB, l
NTAIN N P LIMIN Y

Let .5 I be Banach spaces over the complex field C with
norms [I'llg , ||~|| I , respectively. We will denote by L(5fl) the
space of all bounded linear Operators from .5 into I equipped with the
usual operator norm, ”T" = sup{||Tx||y: xc5 ||x||$$ 1}, TcL(.5}).

In case .5 = y we will write L(.5) instead of L(.5.5). In case
I = C, L(5C), the dual Space of .5 will be written as .5“ and
< x*,x > will stand for the evaluation of a functional x*e.5 * at a

point xe.5 The collection of all conjugate linear functionals on .5 will

be denoted by 25*. Thus 3* = {<;,_->: x*c.5 *}.

In particular, we will make use of the following Banach spaces. For
a locally compact Hausdorff space S, 00(8) is the space of continuous
(scalar) functions defined on S which vanish at infinity equipped with the
tOpology of uniform convergence. In case S is compact, we denote
CO(S) by C(S). Additionally, given a measure Space (A,.9;A) and
1 S p < oo, Lp(A,.9;A) is the Space of functions f (up to A—equivalence)

on A satisfying []A|f(x)|p A(dx)]1/p

topological space and .7 is the Borel sets of A we write .7 = 5A).

= ||f||p < 00. In case A is a

1.1 Definition ([27], p9). Let U: .5 -. y be an Operator between
Banach Spaces. We say that U is 2—Summing if there is a constant C

such that, for all finite subsets (xi) in .5 we have
EllUxill; s c sup {EMX,)|2= 9063*.Ilcpll s 1}.

The following is not the most general form of the Pietsch

Factorization Theorem. It, however, suffices for our purposes.

1.2 Pietsch Factorization Theorem ([27], p11—13). Let S be a locally
compact Hausdorff space and I be a Banach Space. Then

U: CO(S) -» I is a 2—summing Operator if and only if there is a
probability measure A (a regular Borel measure with A(S) = 1) and a

constant C such that for every chO(S),
"Uri; .<. c [ |f(s)l2 Mas).

1.3 Qgrgllory ([27],p12). Let U: CO(S) -. I be a 2—summing
Operator. Then U = AB factors through the Hilbert space
L2(S.~%(S).A);

00(3) 3. L2(s,.aS),i) 5.. y
where B is the natural inclusion map and A is linear and bounded.

Conversely, any Operator UcL(Co(S), I) which admits such a
factorization is 2—Summing.

In fact, ([27],p12) any 2—summing Operator U: .5 -+ I has a
factorization through the Hilbert space L2(K,5K),A) where K = B i,
the unit ball in .5" equipped with the weak*— topology and A is a
probability measure on B ‘2!

In dilation theory ([26],[8]) of bounded vector measures one uses the
fact that every bounded Operator on. CO(S) into a Hilbert space H is
2—summing. The analogue of this result, when a Banach space I takes
the role of H, is known provided I has some additional structure. To

describe this we need some definitions.

Let D = {—1, +1}°° and p be the infinite product of
1/2[6_1 + 5+1] on D where 6t is the Dirac measure. We denote by

th

en: D -+ {—1, +1} the n coordinate of D.

1.4 Definition ([27],p32). A Banach Space I is of cotype 2 if for any

(1)
sequence (yi) in I, E ciyi converges p—almost everywhere implies
i=1

°° 2
:31 "yin, < m.
We note that ([27],p34) if (A,.7,/\) is any measure space and
1 S p S 2 then Lp(A,.9;A) is of cotype 2. Thus, by definition, any
subspace of Lp(A,.9;,\), 1 g p 5 2, is of cotype 2. Also, any Hilbert

space is of cotype 2.

1.5 Thmrem ([27],p62). Let K be a compact Hausdorff Space and I
be a Banach space of cotype 2. Every bounded linear Operator
U: C(K) -i I is 2—summing.

We have noted that each 2—summing Operator factors continuously
through an L2(K,5K),/\)—space. We now state a characterization of
Operators which factor continuously through a Hilbert space due to

Lindenstrauss and Pelczyr’iski.

1.6 Definition ([27],p21). Let .5 I be Banach spaces. We say that an
Operator U: .5 -+ I factors through a Hilbert space if there is a Hilbert
Space H and Operators BcL(.5H) and AcL(H, I) such that U = AB.
Any Operator which factors through a Hilbert Space is called Hilbertian.

1.7 Thmrem ([27],p25). An operator U: .5 -i I is Hilbertian if and
only if there is a constant C with the following property: For every n

and every nxn unitary matrix (aij),2 we have, for all x1,x2,...,xn in .5

i2 || 2 3i ijlel < C 1'21 ||xj ||2
=1 j=l I ‘3
It is widely known that certain classes Jlof Operators, e.g. Hilbertian

Operators ([27],p26) are identifiable as members of the dual space of an
appropriate tensor product space. We first define the injective tensor
product of two Banach spaces ([27],p2).

Let .5 I be Banach Spaces and .5 O I be their algebraic tensor

product. Each element 11 in .5 O I has a representation

u = 2 x G yi for (x.) in .5 and (y.) in I. Moreover, we
i-l l i 1

can regard u as a bilinear form on 5x I" as mapping an element

11
(x*,y*) in .5"xI“ into 2 x*(x.)y*(y.).
i=1 i 1

We can also associate to u a finite rank Operator ii: .5" -+ I

n
which is weak*— continuous defined by H(x*) = 2 x*(xi)yi,x*c.5"‘. Using
i=1

this identification we place the injective tensor product norm ||-||v on

.50 I by

n
inn, = "11"de = sup {iglx"‘(xi)y*(yi)= var. yams". |ly*|| s 1}.
The completion of .5 O I equipped with the injective tensor product

v
norm is denoted by .5 O I.

1.8 thhendigk ThQrem ([10],p231). A continuous bilinear functional i!)

v
on .5 x I defines a member of (.5 O I)* if and only if there exists a

regular Borel measure p on B j“ x B i such that, for each xc5 ye I,

whey) = *I é *X*(X)Y*(Y) dit(X*.y*)-
B 5 I

v
In this case, the norm of it as a member of (.5 O I)* is
precisely In] (B “3:" x B i), the variation of it. Note that here, as
at
before, B 3", B i denote the unit balls of .5, I" respectively.

This allows the following definitions.

1.9 Definition ([10],p232). A continuous bilinear form it on .5 x I is

v
integral whenever i/i determines a member of (.5 9 IO“. The integral

norm of i1), denoted by Ilibllint, is the functional norm of i/2 as a

member of (.5 g M“. A continuous linear Operator T: .5 a I is an
integral operator whenever the bilinear form r on .5 x I“ defined by
r(x,y*) = y*(Tx) is integral. The integral norm of T, denoted by
IITllim. is ”rim.
We will sometimes have occasion to apply the above Operator theory
to spaces generated by the collection of .5—valued simple functions S(2,.5)
on a set (I where .5 is a Banach Space and E is an algebra (or

a—algebra) of subsets of Q. We let 1A denote the indicator function of
the set A62.

D
For f= E 1 X4: 8(2),.5) let
i=1 AiJ

"rim, = sup{llf(W)l|5= wen} = sup {iijgllAjiwixjii$= wen}.
Then (S(2,.5),||-||8up) is a normed linear Space. In completing this space
we obtain the Banach Space B(2,.5) of bounded totally Bochner
Z—measurable .5—valued functions on fl([ll],p83). We note that in case

.5 = C and 2 is a a—algebra, B(2,C) is the Banach space of

bounded Z—measurable functions on Q.

1.19 Theorem. Let Q be any set, 2 be an algebra (or a—algebra) of
subsets of (I, and .5 be a Banach space. The space B(E,.5) is

v
isometrically isomorphic to B(2,C) O .5

v
m. Define J: B(2,C) 0 .5 -. B(E,.5) by
n n
J( 2 fi 0 xi)(w) = 2 fi(w)xi.
i=1 i=1
n n
Then ||J(.ZI1 fi 8 xi)"B(2.5) = sup {Ilig1 fi(w)xi||$: wrfl}
i= ’
n
= sup {IX*( 2 fi(w)xi)[ wcn’ X*£'3*r")(*” S 1}
i=1
n
= sup {Iligl x*(x,)f,i,,pz we. |lx*|| s 1}
n
= sup {MEI x*(xi)f,)|= xw, newer, llx*||,||ull s 1}
l:
n
= sup {L21 x*(x,)u<f,)i= cw, p6[B(2.$)]*. ||x*l|. Hull 5 1}
1:
n
= [Ii-El fi 9 xillv-
v
Thus B(E,C) 0 .5 is isometrically isomorphic to the closed linear

n

subspace of B(2,.5) generated by functions of the form 2 fi(')xi where
i=1

f1,f2,...,fn eB(E,C) and x1,x2,...,xnc.5 But this subspace contains S(E,.5),

the .5-valued simple functions on (0,2), which is dense in B(E,.5),
completing the proof.

Alternatively, following [19], we could place the Mackey tOpology on

10

the dual pair 8(2),.5) and the class of all 5—valued finitely additive set

functions on ((2,2).

1.11 Definition ([10],p1). Let Q be any set, 2 be an algebra of
subsets of Q, and Z be a normed linear space. A set function
T:2-»Zissaidtobe

(i) Finitely additive (f.a.) provided
T(A1U A2) = T(A1) + T(A2) for any two disjoint A1,A2c}l

(ii) Countably additive (c.a.) provided 2 is a a—algebra and

co 00
T( U Ak) = E T(Ak) in the norm topology of Z for all sequences
=1 =1

R
(Ak) of pairwise disjoint members of 3.

(iii) In case Z = L(5 I) where .5 I are Banach Spaces T is
00 (D
countably additive (c.a.) if for each M5 T(kU Ak) x = k2 T(Ak) x

in the norm to pology of I. We note this coincides with (ii) in case

.5=C.

1.12 Definitien ([10],p4). Let (I be any set, 2 be an algebra of
subsets of (2,5 be a normed linear space and T: 2 -i .5 be a finitely
additive set function. The number ||T||8vc[0,ao] defined by

n
[lTllgv = sup IIkEItkT(Ak)“~5 where the supremum is taken over all

finite E—partition (Ak) of Q and scalars (tk) with ltkl 51 is
called the semivariation of T. Any T for which "T”sv < 00 is called

a measure of finite semivariation.

11

1.13 flfheerem ([10],p9). If 2 is a a—algebra and T: 2 -i .5 is

countably additive then T is necessarily of finite semivariation.

The space of all finitely additive measures on (0,2) of finite
semivariation is a normed linear space and will be denoted by (M(B,.5),
||-||sv). In fact, when .5 is a Banach space, so is (M(2,.5), ll-IISV),

[19].
Returning to S(2,.5), the set of all E—Simple .5—valued functions of

n .
the form f = 1'21 lijj, A1,A2,...,An62, x1,x2,...,xnc.5 let
n
Ilflla, = sup {Ijgl < Twas,- >|= remote), IITIISV s 1}.

1.14 Lemma [19]. Let .5 be a normed linear Space. Then
(a) (8(5), ""00) is a normed linear Space.
(b) the mapping er: (Mew). II-IISV) -» (sets, II-llw)* defined

11
by <I>T(f) = J < f,dT > 2 < T(A.)x. >, f =
i=1 1 J i

is an antilinear onto isometry.

|| Mt:

1x.
iAjJ

We will denote the closure of S(2,.5) with reSpect to the norm

 

II-Ila, simply by (36%). II-Ilml
In Chapter 2 we discuss the spectral dilation of Operator valued

measures. Before proceeding we recall the following definition.

1.15 Definitien ([30],p301). Let 0 be any set, E be an algebra
(er—algebra) of subsets of (I and H be a Hilbert Space. A finitely
additive (c.a.) set function E: 2 -» L(H) is said to be a finitely additive

(c.a.) spectral measure in H if

12

(i) E(A) is an orthogonal projection Operator in H for each Ad?
(ii) E(Al) E(A2) = 0 for any two disjoint APA2 in 2 and
(iii) Em) = I.

HAPTE
F ZAINADP LILTN

Let 2 be a a—algebra of subsets of a set 0 and H be a separable
Hilbert Space. Let {E(A): A52} be a countably additive spectral measure
in H. In ([20],p508—510), Masani showed that there exists an
L+(W)—valued c.a. measure F such that the map

v: p5L2(fl,2,F: W) -. ] <p(A) Eo(dA)5H
is a unitary Operator. Here W is a subspace of H and E0 = Rester.

Moreover E(A) = VoM oV-l. It is easy to see that L2(Q,2,F: W) is

1
A
isometric to H(F), the reproducing kernel Hilbert space generated by the
kernel K((A,x),(A',x')) = <F(AnA')x,x'>H. In addition, multiplication by

l A’ MIA, goes into the spectral measure in H(F) defined by

E(A)f(A',x) = f(AnA')x, the so—called canonical Spectral measure in H(F).
When one considers a spectral dilation of a family
{T(A): A52} g L(.5I) of the form T(A) = SE(A)R with S: H -+ I
and R: .5 -i H continuous and linear and H some Hilbert Space, we can
consider constructing on apprOpriate H(F) and a canonical Spectral measure
E in H(F). Thus, the question of obtaining a spectral dilation of a family
{T(A): A52} of the above form reduces to factoring the family through a
single reproducing kernel Hilbert space H(F). The general problem of
factoring an operator T5L(.5 I) through a Hilbert space was studied by
Lindenstrauss and Pelczyi’iski [16]. Our object here is to generalize this,
relate it to the spectral dilation problem, and Show that for a single Operator
our condition is equivalent to the one given by Lindenstrauss and Pelczyr'iski

(see ([27] ,p25).

13

14

2.1 Definitien. Let 5 I be Banach Spaces and {T(A): A52} 9 L(5 I)
We say that {T(A): A52} factors through a single Hilbert space H if
there exists a Hilbert space H, constants 01,02 < co and bounded linear
Operators AA: .5-» H, BA: H -+ I with "AA" 5 Cl’ "BA" 5 C2 such
that for each A52, T(A) = B AA A‘

Since our construction will depend upon a family of Operators
' {F(A): A52} defined on .5 and taking values in the conjugate dual space
of 5 we begin by considering conjugate linear functionals and positive
Operators taking values in the conjugate dual space of .5 (see [25]).

For a Banach space 5 let 3" be the collection of all bounded
conjugate linear functionals on .5 Then, for T5L(5:5*),

< T(alxl), (123:2 > = 01 52 < Tx1,x2 >, x1,x255 011,025C.

2.2 Definitien. Let 5 I be Banach Spaces and T5L(5 I) We define
the conjugate adjoint T*5L(I“,3*) of T by
T*(y*) = y*oT, We}.

We note that in case I = H is a Hilbert space, the conjugate adjoint
of T5L(5H) satisfies < T*h,x > = < h,Tx >H, X55 h5H. In particular,
in taking .5 = H also, we get the usual definition of the operator
T*5L(H). Here, of course, we have identified H and H“, which are

isometrically isomorphic under the identification h H < -,li > , h5H.

2.3 Dfiinitien. Let .5 be a Banach Space. An Operator F5L(5Z5 *) is
said to be positive provided

15

(i) < F xl,x2 > = <F' x2,x1 > for all x1,x2 in .5 and
(ii) <Fx,x>20forallxin.5

The collection of all positive Operators in L(53 *) will be denoted by
mm).

The following is a direct extension of the definition of Rosenberg [28] for
measures taking values in L(H,K) where H,K are Hilbert spaces (see

Remark 2.6).

2.4 Dgijnition. Let 2 be an algebra (a—algebra) of subsets of a set D
and 5 I be Banach spaces. A set function T: 2 -o L(5 I) is said to
have a finitely additive (c.a.) 2—majorant F if F: 2 -+ L+(5T5 *) is

finitely additive (c.a.) and there exists a constant C < as such that for

each positive integer n, A1,A2,...,An 52 and xl,x2,...,xn55
n

n
[[21T(Aj)xll S Cj212 < F(A nA.)x ,x. >
k=1 k J k J

j=1 I
Mpg. Let K be a compact Hausdorff space, 2 be the Borel sets
in K, and it be a positive, finite, regular Borel measure on K.

Define T: 23-» L(C(K), L1(K,2,p)) by
T(A) f = lAf, f5C(K)

 

and F: 2 —. L(C(K), C(K)*) by
< F(A)f. g > = ]( 1A(X) f(X) STE #(dx), fee-C(K)-

Then for each A52, F(A)5L+(C(K), C(K)*) and for each

16

f1,f2,...fn 5C(K), A, A.2,..,An 52“.

n

"1.3 Tf(Ak) k||1= || 2 lAfkll1=(](_l § 1Ak(X)fk(X)l #(dx))

11k
< MK) ]( I 1.5-1 1Ak(X) fk(X)l pox)

 

II II

= H(K) 1.51 .31 [ 1Ak(x) Ike) rAjix) rjixi u(dx)
II II

= H(K) 1,311 1A “AA (x) fk(x) in max)

= ”(K) :2: :21 < F(AknAj) fkf >

so F is a 2—majorant of T.

2.5 Remark. Suppose H,K are Hilbert spaces and G is a finitely
additive (c.a.) 2—majorant of the L(H,K)-valued set function T. Let
F: 2 -» L(H,H*) = L(H) be defined by
< F(A)xl,x2 > = < G(A)x1,x2 >H, A52, x1,x25H. Since each G(A) is
positive, each F(A)5L+(H,H*) = L+(H). Thus F is a 2—majorant of T
in the sense of Rosenberg [28].

We are now ready to construct a single reproducing kernel Hilbert Space

through which a 2-majorized family {T(A): A52} g L(5 I) factors.

.7 ' i 11. Let T be any set and K be a complex—valued function
on TxT. The function K is a positive definite kernel if

n n

igl jgl aj ai K(tj ,it) 2 0 for all al,a.2,... .,an 5C, tl,t2,.. .t, n5T.

2,8 flfhmrem ([3],[9]). Let T be any set and K be a positive definite
kernel on TxT. Then there exists a unique Hilbert space H(K) consisting

of a linear space of functions f: T -. C such that

17

(i) K(t,-)5H(K) for each t5T and

(ii) f(t) = < f,K(t,-) > H(K) for each t5T, f5H(K).

Moreover, H(K) coincides with the closed linear subspace in H(K)
Spanned by the set {K(t,-): t5T} under “'“H(K) = < -,- >III{12()' The
space H(K) is called the reproducing kernel Hilbert space spanned by K.

m n
2.2 Bgmark. For f(o) = 2 aiK(si,-),g(-) = 2 ij(tj,°)5H(K),
i=1 j=1

f ‘2’? ii a

In particular,
< K(S,°), K(t,°) > H(K) = K(S,t), 8,tCT
and

“Huang“, = K(t,t)-

2,19 Emma. If g1,g2,...,gn5H(K) and gnag in H(K) then
gn(t) 4 g(t) for each t5T.

M. For t5T, we have, by the Schwartz Inequality,
lgn(t) " 8(0' = l < 8n " g, K(t,‘) >H(K)| 5 "3n ‘ gl|H(K)“K(t,')llH(K)
= llg,l - gllmx) flit—715,1; -» o as n -» 00 if gn -» g in H(K).

2,11 jlfhggrem [21]. Let .2” be a Banach space and F: 2 -+ L(fl *) be
a finitely additive set function on ((2,2) such that F(A)5L+(.£3' *) for
each A52. Let T = 2x3 Then the kernel K: TxT 4 C defined by
KF((A1,xl),(A2,x2)) = < F(AlnA2)X1’x2 >, A1,A252, x1,x.25$ is positive
definite.

18

We call KF the kernel generated by F and denote its reproducing
kernel Hilbert space by H(F). Thus if
H = Span {fz 2x34 C: f() = < F(An-)x,* >, A62, xc.3}

m n
w1th < f,g >H = 1‘21 jgl akbj < F(AjnAk) xk’xj > where

1:1
H(F) is the closure of H with respect to the norm ||-|| = < -,- >136?“
Also, for ch(F), f(A,x) = < f,< F(An-)x,* >>H(F)’ Ad}, er Thus by

the pr0perties of F, we get that for each fixed A62, f(A,.)s < u(A),-> 63*.

m n
f() = 1‘21 ak < F(Akn-)xk,* >, g() = E bj < F(Ajn-)xj,* > (H then

Moreover, u: 2 -+ 3* is a finitely additive measure.

2.12 flemark. In case .3 = {Axo: MC} is one dimensional, u(-)x0 is a
complex-valued f.a. measure, absolutely continuous with respect to the
nonnegative f.a. measure < F(-)x0,x0 > (since

u(A)x0 = < u, < F(An-)xo,* >>H(F)) and whose density with respect to
< F(-)x0,xo >, all]; (L2(fl,2,F). This provides a unitary transformation
between L2(fl,2,F) and H(F) by um) = A f(w) F(dw), ch2(Q,2,F).

The following pr0position was essentially proved by Miamee and Salehi
[25, Theorem 2.8].

2.13 Promsition. Let E be an algebra of subsets of a set 9,3 be a
Banach space, and F: 2 -» UPC/£3") be finitely additive.
Define
E: 2 —. L(H(F)) by E(A)f = fA, A52, ch(F)
where

fA(A1,x1) = f(AnA1,x1), A162, x161

19

Then E is a finitely additive spectral measure in H(F). If 2 is a
a—algebra and F is countably additive the E is countably additive.

2.14 Qefinition. The L(H(F))-valued set function E of Proposition 2.13
is called the canonical finitely additive spectral measure in H(F).

2.15 fIfheQrem. Let 2 be an algebra (a—algebra) of subsets of a set 9
and a} be Banach spaces. Suppose {T(A): A62} g L(xy) has a
finitely additive (c.a.) 2-majorant F. Then

(a) T(A) = SE(A)R where E is the canonical finitely additive (c.a.)
Spectral measure in H(F) and R: .34 H(F), S: H(F) —. I are bounded
and linear.

(b) {T(A): Adi} factors through the Hilbert space H(F).

M. (a) Let H = span {fz 2x34 C: f() = < F(An~)x,* >, A62, xcez}.

Define SzH-i] by

n 11
Sf = kglak T(Ak)xk where f() = 1‘21 ak < F(Akn-)xk,* > (H.

Since F is a 2—majorant of T there is a constant C with

"an2 = ”ii a T(A)x "2 = nil: T(A)(ax)I|2
I k k k] k=1 k kk }

k=l
n n
S C 1.21 1‘21 < F(AknAj)(akxk), ajxj >

11 I]
— 2
= o 3:21 1‘31 akaj < kanajukxj > = 0 ”run, ch

so S is bounded with "S" 5 JC
Extend S linearly and continuously (without increasing norm) to all of

H = H(F) and continue to call the extended Operator S. Let

20

E: 2 » L(H(F)) be the canonical finitely additive spectral measure in
H(F). Recall E is countably additive in case F is c.a..
Define R: .34 H(F) by Rx = < F(Qn-)x,* >, xc$
Then nRxan) = u < F(fln-)x,¢ > ”H(F) = < F(Q)x,x >1/2
5 u Howl/2 "x"; for each x in .2:

So Rename) with nan s ||F(fl)lll/2-
Let xefi A62. Then

SE(A)Rx = SE(A) (< F(fln-)x,¢ >) = S(< F(An-)x,¢ >)

= T(A)x. This proves (a).

To prove (b) simply take A A = E(A)R and B A = S. Then

AA: 3.. H(F). BA: H(F) -» y, IIAAII s MR” 5 ||F(fl)||1/2,

"BA“ = ”S” 5 JC— and T(A) = BAAA, A52-

Bgmarfllfi. A proof of Theorem 2.15 can also be given using the concept
of the square root of the positive Operator F(A) given in [25]. Also, it has
been pointed out by H. Salehi that the following corollary follows directly

from the definition Of a 2—majorant.

2.17 QQrQllary. Suppose {T(A): A62} g L(efiy) has a finitely additive
(c.a.) 2—majorant. Then T: 23 -v L(ezy) is a finitely additive (c.a)
Operator-valued measure.

We now consider the case 23‘ = {fl,¢}, T(¢) = 0, T(fl) = T, to obtain
a new characterization of when a single Operator TcL(.z}) will factor
through a Hilbert space. This characterization gives precisely the Hilbert
space through which the Operator T factors.

21

2.15 flfhwrem. Let TcL(..£}’) where 3:] are Banach Spaces. The

following are equivalent:
(a) There exists a positive Operator F: .3 -+ 3* and a constant C

such that ”TX”; 3 C < Fx,x > for all x in s:

(b) There exists a constant C with the following prOperty:
For all nxn unitary matrices (aij)’ we have for all xl,x2,...,xn in .X
3M2 jjTu2 02""2
x < x.
i=1 j=1y 1‘3
(c) T is Hilbertian.

REEL (a) => (b). Let F: .34 3* and C < 00 be such that
”TX”; 5 C < Fx,x >, X63

If A = (a is an nxn unitary matrix and x1,x2,...,xn 63’ then

ij)
3% .23 i‘: E
aij ij T a...x
11
(CE <F(2 amx),.E ai..x>
i=1 j=nllJJ j=1 1]]
cz‘i E 2
= ai. a <Fx., x
i=1k=1j=1jk 11‘
Il
=02: 2(2 a..a—.)<Fx.,x>
k=1j=1i=llJ 1k 1k
2 ""12“"?
=C <Fx.,.x><C F x.
j_1 J’J 15113

since A*A = I.
(b) =9 (c) is known (Theorem 1.7).
(0) =9 (a). Suppose T is Hilbertian. Let H be a Hilbert space and
A: H -» y, B: .3 -1 H be bounded linear Operators such that T = AB.
Define F = B*B. By the definition of B* we have,

< Fx,x > = < B*Bx,x > = < Bx,Bx >H = IIBxIIfi, xc3

and

22

< Fx,y > = < B*Bx,y > = <Bx,By >H = < By,Bx >H

 

= < B By,x > = < Fy,x >, x,y63 so
F: .34 3" is positive.
Moreover for X63
"nu; = ”12221;: "All2 IIBxllfi = "All2 < Fx,x>
so (a) holds with c = ||A||2.
We now give the relation these results have to the spectral dilation
problem. For this, we give the following extension Of the definition of

Rosenberg [28].

2.19 Definigion. Let 2 be an algebra (a—algebra) of subsets of a set 9
and 3] be Banach spaces. A set function T: 2 4 L(3y) is said to
have a finitely additive (c.a.) spectral dilation (S,H,E,R) if there exist a
Hilbert space H, a finitely additive (c.a.) spectral measure E in H and
bounded linear Operators R: .3 4 H and S: H 4 I such that for all
A62, T(A) = SE(A)R.

M. We have seen (Theorem 2.15) that if T: E 4 L(3y) has a
finitely additive (c.a.) 2—majorant then T has a finitely additive (c.a.)
Spectral dilation. We now note that T being 2—majorized is also
necessary for the existence of a Spectral dilation of an L(3}’)-valued

measure.

2.21 Theorem. Let 3] be Banach spaces. A finitely additive (c.a.)
measure T: 2 4 L(3y) has a finitely additive (c.a.) spectral dilation if and
only if T has a finitely additive (c.a.) 2—majorant.

23

REEL Let (S,H,E,R) be a finitely additive (c.a.) spectral dilation of
T: 2 4 L(.3,'}’). Then for xl,x2,...,xn in .3 and A1,A2,...,An in 2,
we have,

2
9’

n
u 2 T(Ajlxju
i=1

2 §SEARx
(7-Hj=1 (j) ju

< IISIIZIIE E(A-)Rx-Il2
- j=1 J 1H

_ 2 n n
_ ||S|| 3°21 1.31 < E(Aj)ij, E(Ak)ka >H

_ 2 n n ..
_ us” 121 1‘21 <R E(AjnAk)ij,xk >.

Let F(-) = R*E(o)R. Since <F(A)x,x > = ||E(A)Rx||12{ 2 o for all x
in H, F6L+(33"). Also F is finitely additive (c.a.) since E is. Thus
F is a finitely additive (c.a.) 2—majorant of T.

In view of Remark 2.20, this completes the proof.

2.22 Remflk. Theorem 2.21 was proved by Rosenberg [28] for the case
where .3] are both Hilbert spaces. A generalization Of Rosenberg's result
was given by Miamee [23]. Both their proofs depend on the fact that since
11 is a Hilbert space the kernel K(A1,A2) = F(Al n A2) — [T(Al)]* T(A2)
is non—negative definite. In our case this kernel is not well—defined unless

] is (conjugate) reflexive.

CHAPTER 3

QRTHOQQNALLY SQATTEBED DILATIQN
A N E T

In [26] Niemi showed that every countably additive Hilbert space valued
measure Of finite semivariation has a c.a.O.s. dilation. Subsequently, Chatterji
[8] proved that every finitely additive Hilbert space valued measure Of finite
semivariation has a f.a.o.s. dilation. The key to these results is the fact,
essentially due to Grothendieck [13], that every bounded linear Operator
<I>: Com) 4 H is 2—summing.

We consider the case of a finitely additive measure T which takes
values in a Banach space y. In case T: 2 4 y is of finite semivariation
we consider a bounded linear Operator QT: B(2,C) 4 i which, when I is
a Banach Space of cotype 2 is 2—summing. This fact is seen to be equivalent
to the Operator <I>T factoring through a Hilbert space H. Using this and
results from Chapter 2, we produce a factorization T(A) = S {(A), A62
where 5 is a finitely additive orthogonally scattered H—valued measure.

In case i = L(H,K) where H,K are Hilbert spaces, Makagon and
Salehi [19] constructed a continuous linear Operator
‘iT: (S(2,H), ||-|| co) 4 K (where again T has finite semivariation) which
when 2—summing implies T: E 4 L(H,K) has a finitely additive spectral
dilation. We Show the condition of (BT being 2—Summing implies QT is
2—summing which, in view of our result, shows T has a finitely additive
orthogonally scattered dilation.

We remind the reader that prOperties concerning 2—summing and

Hilbertian Operators and Banach spaces of cotype 2 are contained in Chapter

24

25

1. For completeness, we review the case T: 2 4 H where H is a Hilbert

Space.

2,1 Definition l8]. Let 2 be an algebra Of subsets Of a set (I and H
be a Hilbert space. A finitely additive set function 5: 2 4 H is said to be
a finitely additive orthogonally scattered (f.a.o.s.) measure if

< ((A), {(B) >H = 0 whenever A,B are disjoint elements of 2. If, in
addition, 5 is countably additive on 2 then f is called a countably
additive orthogonally scattered (c.a.o.s.) measure. Moreover,

||£(A)||I2I = g(A) is a non—negative finitely additive (c.a.) measure on 2.

2.2 DefiniLion [8]. Let 2 be an algebra of subsets Of a set 0 and H
be a Hilbert Space. A finitely additive set function T: 2 4 H has a f.a.o.s.
dilation if there exists a Hilbert Space H containing H, and a f.a.o.s.
H-valued measure { such that for each A62, T(A) = VP§(A) where P
is the orthogonal projection of H onto a closed linear subspace M of H
and V: M 4 H is a unitary isomorphism. In case 2 is a a—algebra and

f is countably additive on ((2,2), T is said to have a c.a.o.s. dilation.

2.2 flfomrem l8]. Let T: 2 4 H be a finitely additive measure of finite
semivariation. Then T has a f.a.o.s. dilation. If T is countably
additive, then T has a c.a.o.s. dilation.

We now extend Definition 3.2 (see Remark 3.8).

2.4 Definition. Let 2 be an algebra Of subsets of a set a and I be
a Banach space. A finitely additive set function T: 2 4 I has a f.a.o.s.

dilation if there exists a Hilbert space H, a f.a.o.s. measure 5: 2 4 H and

26

a bounded linear Operator S: H 4 I such that for each A62, T(A) = S((A).
In case 2 is a a-algebra and 5 is countably additive on ((1,2), T is
said to have a c.a.o.s. dilation.

Following ([10],p6), given a I—valued finitely additive measure T of
finite semivariation, we define a bounded linear Operator (PT: B(2,C) 4 I.

We recall (Chapter 1), B(2,C) is the uniform closure Of S(2,C), the scalar

n
valued 2-simple functions on 9. For f = k2 aklAk 68(2,C) where

A1, A2,...,An are pairwise disjoint elements of 2 we define

I1
(DT(f) = 1.21 akT(Ak).

Then (PT: S(2,C) 4 I is linear and if

n
a = "f” = sup {|f(w)|: W60} = max [0 | where f = 2 a l
sup 15kgn 1‘ k=1 1‘ Ak

then ||<I>T(f)ll}.= "1:1 amour diffs) T(Ak)||,s an'rnsv
so <I>T is bounded on S(2,C) with ||<I>T|| 5 ”Tllsv’ In fact,
II<I>TII = "Tu,v since
ll<I>T|| = sup “was", 2821:). llfllsup s 1}
= sup "1:1 camp", = IITIISV.

We extend QT, without increasing norm, to all Of B(2,C) and continue to
denote this extended Operator by (PT. We call QT the Operator associated
with the bounded vector measure T.

It will be useful, in the following, to consider (PT as a bounded linear
Operator defined on a space of continuous functions with domain some
compact set .3. Using an argument as in Chatterji [8], there is a compact

Hausdorff space .3, called the Stone space associated with ((2,2), such that
0(3) and B(2,C) are isometrically isomorphic (as algebras). Let

27

r: 0(3) 4 B(2,C) denote this isomorphism. Defining <I>Tz C(fl) 4 I by
<i>T(i) = arm) we see

1121.6)", = II<I>T(rr}n,s ||<I>Tll 1}ersup = urns, urnsup
so QT is bounded.

Let I be a Banach space. Throughout the remainder of this chapter
we will make the following identification. Given a I-valued set function on
(9,2) define the L(C,I)—valued measure T by
T(A)a = aT(A), A62, net. Since

~ I] ~ 11 ~
n'lrn,,=.supuk21ekT(Ak}}}L(¢u,,= suplsuplnkgl akT(Ak)all}‘- Ia} s 1}}

= sup{sup{llkEl (skew/2k)", Ia} .<. 1}} = "Tu,v
this identification defines an isometric isomorphism between the Space Of
I—valued measures Of finte semivariation and the space of L(C, I)-valued
measures Of finite semivariation.

We are now ready to prove our main result.

2.2 Theorem. Let I be a Banach Space, T be a finitely additive
I—valued measure Of finite semivariation on (9,2) and (PT: B(2,C) 4 I
be the Operator associated with T. If QT is 2—summing then T has a
f.a.o.s. dilation. In case T is c.a. on ((2,2), then T has a c.a.o.s.

dilation.

13ml. Let .3 be the Stone Space associated with ((2,2) and suppose
QT: B(2,C) 4 I is 2—sumrning. Again letting 7 denote the isometric
isomorphism between B(2,C) and C(JK), we have for all f1, f2,...,fn6C(fl),

28

|| 21 $1.61. }II,= II 2 <1» T(rrpnf 0 SIM 212=Ierrkll ‘PfBB(gc)*}

n
= C SUP {REIIWRHZ : 053(C(m)*}

for some constant C independent of f1,f2,.. .,fn. Thus, by definition, (IT

is 2—summing. By the Pietsch Factorization Theorem (Theorem 1.2), there

exists a probability measure )I' on .3 and a constant C1 such that
||<I>T(f)||;,5 c1 33/ |f(k)|2 x'(dlc) for all item).
Let g6B(2,C). Then r-lg6 C(fl) SO
nelson},= ||<i>T(r_ s}II}< c I Ir’1gIkll2 A'Idk}
= c1 )2 l g(wlI2 MdW) (3.6}
where ,\(A) = 25/ (F1 1A)(k) A'(dk), A62 is finitely additive and positive.

Here we have used the fact that 7, being an isomorphism between the

complex algebras 0(3) and B(2,C), satisfies Ir-lg]2 = r—llglz, g6B(2,C).

n
Taking g = k2 lAkak6B(2,C) in (3.6) yields
=1

11 n
IIsTsII; = "12 T(Ak)akll2 5 01 [I k2l aklAk(w)|2 AIdw}
n

=C 2 2 a a.A(A.nA)
lk=1j=l kaj jk

so since A is positive, A is a finitely additive 2—majorant of T. By
Theorem 2.15, _

T(A) = SE(A)R, A62 where R: C 4 H(A), E is the canonical
finitely additive Spectral measure in H(A) and S: H(A) 4 I. But since
R: C 4 H(A) is bounded and linear there is a unique x0 in H(A) such
that Ra = axo ,a6C. Thus T(A): SE(A)R = SE(A)xO, A62.

Since E(A1) E(A2) = 0 for all disjoint A1,A262, setting
£(A) = E(A)xo, A62 gives the result.

29

Note that we have used the identification of T: 2 4 I and T: 2 4 L(C, I)
described above.

We now suppose T is countably additive on ((2,2) where 2 is a
a—algebra. The fact that A can be taken to be countably additive in (3.6)
was proven by Chatterji ([8],p273). We include his proof here.

Let A = A1 + A2 be the Hewitt—Yoshida decompositon of A,
([12],p163), i.e. A1: 2 4 [0,oo) is countably additive and A2: 2 4 [O,oo) is
purely finitely additive (finitely additive and Singular with respect to all
countably additive measures). Let p: 2 4 [0,oo) be the countably additive
control measure associated with the countably additive measure T, ([10],p27),
i.e. ”(123130 ||T||(A) = 0 where "T" (-) is the semivariation Of T
([10],p2). Then there exists B1162 such that A2(Bn) 4 0 and
A1(B;l) + ”(BI'I) 4 0 as n 4 do where B1; = Q\Bn.

Since

ll¢T(ngl'l)ll , s IIeIIsup ||T||(BI'1) . o for every seBI2t}
we have from (3.6),

II<I>.1.(san}II}scl [I IBn(W)lg(W)l2(A1(dW) + A2Idw».

By letting n 4 do,

II<PT(g) )II;< c1 )2 IS( w)| A1(dw)

Thus A can be taken to be countably additive in (3.6). Following the
remainder of the proof, we see A is a countably additive 2-majorant of T
which yields, by Theorem 2.15, that the canonical spectral measure in H(A)
is countably additive and hence {(A) = E(A)x0 is c.a.o.s.

2.7 Corollary. Let I be a Banach space Of cotype 2 and T be a

finitely additive I—valued measure of finite semivariation on ((2,2). Then

30

T has a f.a.o.s. dilation. In case T is c.a. on the a—algebra 2, T has a

c.a.o.s. dilation.

M. As in the beginning Of the proof Of Theorem 3.5, we regard the
associated Operator QT: 0(3) 4 I where .3 is compact. Then since I
is Of cotype 2, (PT is 2—summing (Theorem 1.5). By Theorem 3.5, the
result holds.

2.2 Remerk. As noted before (Remark 2.12), H(A) is isometric to
L2(fl,2,A) and the isometry U is onto. Furthermore, by the definition Of
U, the canonical spectral measure E in H(A) is mapped to multiplication

by indicator functions, i.e. E(A) = U‘IMIAU, A62.

Since T(A) = SE(A)x0, xO6H(A), T(A) = so"1 M1 Uxo = S'E'(A)x'

A
where x' = Uxo, E'(A) = M1A in L2(fl,2,A) and S': L2(fl,2,A) .. I is
a bounded linear Operator. In case I = H, a Hilbert space, we can identify
H with an apprOpriate subspace of L2(Q,2,A) and consider 8' as a
projection. This is precisely the orthogonally scattered dilations of [8] and
[26].

In general, in Theorem 3.5, we do not know a simple condition for
(PT: B(2,C) 4 I to be 2—summing. However, we know that (PT being
2—summing is equivalent to QT being Hilbertian. To see this suppose
QT = AB where B: B(2,C) 4 H and A: H 4 I are bounded linear
Operators and H is a Hilbert space. Then (again identifying B(2,C) and
C(10) B is 2—Summing ([27],p54), so since A is bounded, T = AB is

2—Summing ([27],p9). Conversely if (PT is 2-summing then <I>T factors

31

through a Hilbert space (see discussion following Corollary 1.3). In view of

this Observation, we have the following (see Theorem 1.7).

2.9 ijhmrem. Let I be a Banach space and T: 2 4 I be a finitely
additive measure Of bounded semivariation. The following are equivalent:
(a) T has a f.a.o.s. dilation.

(b) There exists a constant C with the following prOperty:

For all nxn unitary matrices (aj) we have for all f1,f2,... ,nf 63(2,C),

21ill2 a M¢T(f)lli< c 2 ”Mini,p

We now consider the Special case I = L(H,K) where H,K are
Hilbert Spaces. Following Makagon and Salehi [19], we consider
T: 2 4 L(H,K) by using an associated Operator (see Chapter 1)
«PT: ISI2.H}, ll-Ilm) .. K,

.. n
QT(f) =j—-E-1 T(Aj)xj, f = j211A].jx 6 S(2, H).
Then 2T is continuous if T is Of finite semivariation with

||<I>T|| = IITII,v

2.12 fIthrem. Let H,K be Hilbert spaces, 2 be an algebra Of subsets
Of a set 0 and T: 2 4 L(H,K) be f.a. with finite semivariation. If

 

(PT: (S(2,H), ||.||m) -» K is 2—summing then <I>T: B(2,C) -» L(H,K) is

2—summing and, hence, T has a f.a.o.s. dilation.

m. We Observe that if X6H and f6B(2,C) is a uniform limit Of
2—Simple functions {fn} then

I < m.fx > - < m, inx > I s "x” IImII,v Ilf — inns“, .. o

32

for every m6(M(2,H*), ||-||sv) = (S(2,H), ||~||m)* (Lemma 1.14). By

 

definition, fnx 4 fx in (S(2,H), ||o||m). Hence functions of the form

 

fX6 (S(2,H), Il'llm) provided f6B(2,,C) X6H. Let (a,,) be an nxn unitary

matrix and f1,f2,..f ., nmore), X6H. Since °T< 2 a1, J)(x) <I>T(j_ 2 4a,,f,x)

and (PT is 2—summing, there is, using the Pietscjh Factorization Theorem, a

constant C and a probability measure A on B = {m: 2 4 H: ||m||SV 5 1}

such that
D
Mfl2amwwx<cgl<, a,,¢dflx>H|AWM
s c 2 n2 a,,-<I>m(f,)IIH IIxIIH AIdm)
1:1
<wn22 2 In m(fk) Aw)
= x < (Pm <1) > m
H j=l k=l all‘ _2 H
so since (aij)n is unitary,

2
2 (II
i:l" T(jgl aijfjlllMH K)

$C222aj.a <<I>f<I>f>Adm
j_1k_1i=1 m2 In(,) m(k) H()

— —c 2 2 II» mII,)IIH AIdm)
But for ||m||3v 5 1, J||<I>m(fj)||H< _ ”fjl'sup so since A is a probability
measure on B,
21]," 21a..<1>T(I.)||2 < C 2 ||f.||2 .
i=11jTJ L(H,K) - j=1 J 811])
By Theorem 1.7, 10.1.: B(2,C) 4 L(H,K) is Hilbertian, hence 2—summing.
Also by Theorem 3.9, T has a f.a.o.s. dilation.

HAPTE
H NIZABLE AND P E

Let {th tell} be a stochastic process taking values in a (complex)
Hilbert space H. It is well known ([17],p483) that if {Xt: tell} is

m .
stationary then it admits a representation, Xt = I e“Its {(ds) where 5
"' on

is a c.a.o.s. H-valued measure on all). It is also well known ([1],[24]) that
{th tell} is harmonizable if and only if it is the projection of some
stationary process taking values in a larger Hilbert space K. Our main goal
in this chapter is to obtain a corresponding result for stochastic process,
{Xg,ch}, G a separable locally compact abelian group, taking values in a
(complex) Banach space I. We apprOpriately extend the definitions of
Bochner V—boundedness and harmonizability to fvalued stochastic processes,
show their equivalence and obtain a stationary dilation theorem. We note
that in contrast to [19], who show that every harmonizable L(H,K)-valued
process {thtcll} can be represented as Xt = JYt where Yt is a
stationary L(H,K)—valued process, we obtain a representation X8 = SYg
where the stationary process {Ygz ch} takes values in a Hilbert space.

We motivate our definitions by first considering y—valued processes where
y is a Banach space of cotype 2. Any symmetric a stable (SaS) process,
1 < a 5 2, takes values in a cotype 2 space. For such a process the
concept of harmonizability has been defined [7]. In this chapter we also
obtain a more detailed Spectral representation for SaS processes than that
obtained by Makagon and Mandrekar [18].

We recall some results concerning Hilbert space valued stochastic

processes.

33

34

4.1 Dflinitign. Let H be a Hilbert space. By an H—valued stochastic
process we mean any function {th tell} from I! = (— ao, + 00) into H.
An H—valued stochastic process {th tell} is said to be
(a) stationary [17] if there exists an H—valued c.a.o.s. measure g on

all), the Borel sets in R, such that for every tcll,

xt = £ e’its {(ds)

(b) harmonizable ([26],[24]) if there exists an H—valued (c.a.) measure

Z on all) such that for every tell,

Xt = £ e—lt'8 Z(ds)

(c) Bochner V—bounded [19] if Xt is strongly continuous and there exists a
constant C such that for every ch1(R,.flR),dt),
where the integral on the left hand side is in the sense of Bochner

([10],p44) and f is the Fourier transform of ch1(R,1R),dt).

W. We note that {th tell} is Bochner V—bounded if and only
if Xt is strongly continuous and the linear mapping 7]: 1:10!) a H defined
by

r,(f) = & f(t)Xt dt

is continuous where £101) = {in rcL1(R,.z(a),dt)}.

4.3 flfhmrem. ([26],[24]) Let H be a Hilbert space. For any H-valued
stochastic process {th tcR} the following are equivalent:

(a) {Xt: tell} is harmonizable.

(b) {th tell} is Bochner V—bounded.

35

(c) There exists a Hilbert space K, an isometry J: H -» K and a

K—valued stationary process {Ytz tell} such that for every tell,

Xt = J*Yt.

As mentioned, we will first consider the case of a y—valued stochastic
process where y is a Banach space of cotype 2. In this case, every
y—valued vector measure T of finite semivariation has an orthogonally
scattered dilation (Corollary 3.7). In addition, the associated Operator
QT: B(2,C) 4 I factors through a Hilbert Space. Our proof for y-valued
harmonizable and V—bounded processes, where y is of cotype 2 will
motivate the definition of harmonizability and V—boundedness for the case of
a general Banach space.

For integration with respect to y—valued measures we follow Kluvanek
[15]. The integral I de of a scalar-valued function f with respect to a
y—valued measure Z is defined to be that element of ] satisfying
< y*, I de > = I f(s) < y*, Z(ds) > for every fey.

4.4 Ddim’tign. Let [I be a Banach space and G be a separable locally
compact abelian group. By a y-valued stochastic process we mean any
function {ng ch} from G into I.

We offer the following definitions in case ] is a Banach space of

cotype 2.

A

W. Let G be a separable locally compact Abelian group, G
denote the group of continuous characters of G, dg be the Haar measure on
G and y be a Banach space of cotype 2. A y—valued stochastic process
{ng ch} is said to be

36

(a) harmonizable if there exists a y-valued c.a. regular measure Z on

$6?) such that for any ch,

 

Xg = 1 < 7.3 > Z(dv).
G

(b) Bochner V—bounded if the mapping g -+ X8 is strongly continuous

and there exists a constant C such that for every ch1(G,.flG), dg),
< A .
Here f is the Fourier Transform of feL1(G,.flG),dg) and the integral on

the left hand side is taken in the sense of Bochner.

4.§ Bamark. In case G = R = (- 00, + co) and y = H a Hilbert space
Z: all) -i H. Since Z is c.a. and all) is a a—algebra, Z has finite
semivariation (Theorem 1.13). By [8], there is a positive, finite, c.a. measure
is on silk) with "Z(A)”I21 5 ,u(A), AcflR). Since p is regular [4,p7], Z
is regular. Thus our definition of harmonizability reduces to Definition 4.1.
We note that {Xg: ch} is Bochner V—bounded if and only if X3 is
strongly continuous and the Operator 1]: L1(G) -i I defined by

”(f) = g; f(g) ngg, ch'(G) is continuous. Here

111(6) = if: feL1(G,aG).dg)}.

4.7 fIthrem. Let I be a Banach space of cotype 2, G be a separable
locally compact abelian group and {X g: ch} _C_ y be a stochastic process.
The following are equivalent:

(a) {ng ch} is harmonizable.

(b) {X g: ch} is Bochner V—bounded.

37

M. (a) => (b) Suppose {Xg: ch} is harmonizable so

Xg = I < 7,g > Z(d7), ch where Z: flG) -+ y is c.a..
G
Then X8 is norm—bounded and strongly continuous ([15],p269).

Let (DZ: B(flG),C) -i y be the Operator associated with Z. Since Z is
Of finite semivariation (Theorem 1.13), <I>Z is continuous.
Let u L1(G) -+ y be defined by
17(f) = (I; f(g) ngg, ch1(G).

We claim "(b = <1>Z(i) for all ch1(G).
Let chl(G). Since X8 is strongly continuous and norm-bounded and G
is separable I f(g)Xg dg exists since, in this case, f(g)Xg is separably
valued and weakly measurable, hence, strongly measurable.
For y*c}"‘, as in [15], we have

< y*, é f(g)Xg dg > = g} f(g) < y*,Xg > dg

= g} f(g) (I < 7.3 > < y*,Z(d7) >) dg
G
= 1 (é f(g) <72} <13) < y*. Z(d‘r) >
G

= l %(7) < y*.Z(d7) > = < y*, 1 f(v) Z((17) > -
G G

Thus W) = é f(g) ngg = 1 f(7)Z(d7) = <I>Z(f).
G
Since <I>Z is continuous, so is 17 from which {ng ch} is Bochner

V—bounded.

(b) => (a). Suppose {Xg: ch} is Bochner V-bounded i.e. suppose

r]: L1(G) -» y is continuous. Since L1(G) is dense in CO(G), ([29],p6),
we can extend 17 continuously to CO(G). Now I is of cotype 2 so 7;

is 2-summing and, hence, Hilbertian. There exists a Hilbert space H and

38

bounded linear Operators B: CO(G) -» H, A: H -i I such that r) = AB.
Since H is reflexive, B is weakly compact ([12],p483) so 17 = AB is
weakly compact ([12],p484). Thus {n(h): ”hump g 1, thO(G)} is a
weakly compact set in 11. By ([15],p264) there exists a unique c.a. regular
measure Z: .flG) -+ f such that n(h) = I h(7) Z(d7) for every

(3
tho(G). Let ch1(G). Then chO(G) so

so = I f(g) xgds = 1 its) Z(d7)-
G
Hence for y*cr,

< y*, I: f(g) ngg > = 1 f(7) < y*, Z(dv) >
G

= I I; f(g) < 7.3 > dg < y*, Z(d'r) >
G

= I f(g) I; < 7,g > < y*, Z(d'y) > dg for every chl(G)
G

< y"‘,Xg > = I < 7,g > <y*,Z(d7) > for every ch1(G), y*cfl‘.
G
Thus X8 = I . < 7,g > Z(d7) almost everywhere on G. But g —. X g
G

strongly continuous implies this equality holds for all ch. Thus

 

X8 = I < 7,g > Z(d7), ch i.e. Xg is harmonizable.
G

4.§ Thmrem. Suppose {X g: ch} is a harmonizable y—valued process
where y is Of cotype 2. Then there exits a Hilbert space H, an

H—valued stationary process YS and a bounded linear Operator S: H -+ }'

such that for every ch, Xg = SYg.

39

215291. By assumption, X8 = I < 7,g > Z(d7) where Z is a regular c.a.
. G
measure. Let <I>Zz B(B(G),C) -» y be the Operator associated with Z.

Since Z has finite semivariation and y is of cotype 2 we have, by
Corollary 3.7, Z has a c.a.o.s. dilation Z(A) = S((A) where 5 is a

c.a.o.s. measure on some Hilbert space H and S: H -» y is bounded. Let

 

Yg = I < 7,g > ((d7). Then Yg is stationary and for y*cfl“,
G

< y*,SYg > = < y*, S I < 7.3 > €(d7) >
G

= < S*y*. I < 7.3 > €(d7) >H
G

I < 7,8 > < 3*y*a {(d‘f) >H

G
= I < me > < y*,S€(d7) > = < y*, I < 7s > S£(d7) >

.G G
=<y*,I<7,g>Z(d7)>=<y*,Xg>
G
X = Y.
so g Sg

Let 1 < a < 2 and (0,3,?) be a probability space. All random
variables in the ensuing discussion will be considered as being defined on
(9.5:?)-

A real random variable X is symmetric a stable, SOS, if its
characteristic function tp is of the form tp(t) = Eexp{—c|t|a} where c
is a positive constant. The real random variables X1,X2,...,Xn are jointly
SOS if their joint characteristic function tp is of the form
gp(tl,t2,...,tn) = exp { — I | 131 thkla d1"x (xl,x2,...,xn)} where I‘X

n
is a symmetric measure on the unit sphere Sn of Euclidean n—space Rn

40

uniquely determined by the joint distribution of X = (X1,X2,...,Xn). A
complex random variable Z = X + iY is SCYS, [5], if the real random
variables X,Y are jointly SOS and its characteristic function is

Eexp(itX + isY) = exp (- I Szltx+sy|a dI‘X,Y(x,y)). The Schilder

norm, [31], of the complex SOS random variable Z = X + iY is defined
by uzn, = [Fx,yeznl/‘t

Following [7], by a 808 process we mean a family of complex random
variables {Xu: ucI}, where I is an arbitrary set, such that for each finite
set i g I, the collection Of random variables {ReXu, ImXu: uci} is SOS.
If {Xuz ml} is a complex SaS process, then by 4X) we will denote
the closure with respect to the Schilder norm Of the set of all finite linear
combinations of {Xu: ucI}. If 1 < p < a, the Schilder norm ||~||a is
related to the Lp(fl,3;P) norm by (B(uzupnl/P = c [|le where c is
a constant depending on p and a [6]. Thus (4X), Illla) is a Banach
space which is a subspace of Lp(fl,3§P) for l < p < a and, hence is Of

cotype 2. From Theorem 4.8 we immediately Obtain.

4.9 Corollgy. Every harmonizable SaS process {X g: ch}, with
1 < a < 2, has a representation of the form X3 = SYg, ch where S is
a continuous linear Operator on a Hilbert space H into .2(X) and
{Ygz ch} is an H-valued stationary process.
We now Obtain the precise form of the Operator S.

Suppose {Xg: geG} is a SOS harmonizable process, 1 < a < 2.
Then there exists a regular c.a. 4X)—valued measure Z on (G,cflG))

 

such that Xg = I < 7,g > Z(d7).
G
Fix 1 < p < a. Since .2(X) as a subspace of Lp(fl,5;P) for some

41

probability space (0,3?) is of cotype 2 then the Operator

(DZ: B(flG),€) -+ Lp(f2,r9,'P) factors through a Hilbert space H. Thus
dz = SR where R: B(flGM) a H and s: H -+ LP(O,.9;P). We apply
the following result, due to Krivine to the Operator S ([27],p100).

Meg. Let H be a Hilbert space and A be a Banach lattice
([27],p97). Any Operator u: H -l A satisfies: there is some constant C

II n
2 l 2 2 l 2
"((31 Ikal )/ "A s C(kg1 |ka" ) / .

Since A = LP(O,.9;P) is a Banach lattice ([27],p97), s: H -» Lp(fl,.9;P)

satisfies
n n
2 1/2 2 1/2
”(EISXI) IISC(3||X||)-
k=1 k P k=1 1‘
But by a result due to Maurey ([22],pl6-18) this is equivalent to the

existence of a real measurable function f, such that

A |f(y)|r P(dy) S 1, r = 2%, and a bounded linear Operator

V: H -+ L2(Q,.9;P) such that S = MfoV where Mf denotes the operation
of multiplication by the fixed function f.

Consider VoR: B(aG),C) -» L2(O,5;P). Since L2(fl,.9,'P) is a Hilbert
space, using Chatterji [8] we Obtain a Hilbert space H containing
L2(Q,5,'P), a projection P of H onto a closed linear monfold M of H,
a unitary onto isomorphism U: M -» H and a bounded linear Operator
T: B(flG),C) -» H such that VR = UPT.

In particular, for AcflG),
VR(1A) -.- UPT(1A) = UP Em) where a are) -+ H
is orthogonally scattered.

42

Thus Z(A) = oz (1 A) = SR(1A) = MfVR(1A) = Mf UP g(A), AccflG).
Since Z is c.a.,g is c.a.o.s.

From Theorem 4.8 we immediately obtain the following.

4.19 Theorem. Every SOS harmonizable process {ng ch}, 1 < O < 2,
has a representation Of the form X g = MfUPYg, ch where for any fixed
p, l < p < O, chr(fl,.9;P), r = fig for some probability space (0,5;P),
{Yg: ch} is a stationary process taking values in a Hilbert space H
containing L2(Q,3,’P), P is a projection Operator from H onto a closed
linear manifold M Of H and U: M .. L2(Q,.9,’P) is unitary.

In case G = R, a similar result has been derived independently by
Houdré [Technical Report NO. 258, Center for Stochastic Processes, University
of North Carolina].

In case G = R or Z (the set of integers) it was proved in [18] that
every bounded SOS process is a limit of a sequence of SOS harmonizable

processes. For G = R they obtained:

4.11 Theorem. For any continuous bounded SOS process {Xt: tcR}
there exists a sequence of harmonizable SOS processes
WED) = I_+m°° exp(-its) Zn(ds) such that for each ftelfll),

lIilrn I + °° f(t) w?” (it = I + °° f(t) xt dt.

00 -(D

Here MR) is the space of rapidly decreasing functions on R with the
usual tOpology [14] and {th tell} being continuous and bounded means
{Xt: tell} is continuous in probability and sup "XtIIO < + co.

tell

From this and Theorem 4.10 we immediately have the following.

43

4.12 flfheerem. For any continuous, bounded SOS process

{Xt: tell}, 1 < O < 2, there exists a sequence of stationary processes
Yén) = I_+m°° exp(—its) {H(ds) taking values in a Hilbert space Hn

containing L2(Qn,.9;l,Pn) for some probability Space (01141311), such

that for each fceflll),

I+mf()Xd l MUP 1+°°i()v(")d
t 13: im t I.
-00 t n-H-oo g1]1111 --00 t

where (Pn) is a sequence of projection Operators on HD into a closed
linear manifold MD of Hn’ (Un) is a sequence Of unitary Operators on
Mn into L2(Qn,.91'l,Pn) and (gn) is a sequence Of functions in

r _ 2
L (Qn,.9;l,Pn), r — 23% for each fixed 1 < p < O.

4.1 mark. In [18] a result analogous to Theorem 4.11 was proved for
the case G = Z. Our result in Theorem 4.10 could be used to Obtain a
theorem similar to Theorem 4.12 in this case.

We recall that the proof Of Theorem 4.7 relied mainly on the fact that
certain Operators factored through a Hilbert space. We propose the

following definitions.

4.14 Definition. Let G be a separable locally compact abelian group, j!
be a Banach space and {Xg: ch} be a y—valued stochastic process.
(a) {ng ch} is harmonizable if there exists a regular c.a. measure

Z: 3(G) -» 3! whose restricted associated Operator (DZ: CO(G) a y is

 

Hilbertian and for every ch, X = I < 7,g > Z(d7).

g
G

44

(b) (XS: ch} is V-bounded if g -» Xg is strongly continuous, the
linear operator if. L1(G) -l ] defined by 0(f) = (I‘ f(g)Xg dg is

continuous and the continuous linear extension of n to C0(G) is

Hilbertian.

4.1§ Bema_rk. In case I is of cotype 2, this definition coincides with
Definition 4.5. In particular, for y = H, a Hilbert space, this definition
reduces to Definition 4.1.

4.15 flfheerem. Let y be a Banach space and {ng ch} be a

y—valued stochastic process. Then the following are equivalent:
(a) {Xg: ch} is harmonizable.
(b) {ng ch} is V-bounded.

Prggt. The proof follows along the lines of Theorem 4.7.
(a) =9 (b) Suppose {Xg: ch} is harmonizable so

Xg =.J < 7,g > Z(d7), ch where Z: .flG) -i y is c.a.
G
and regular. Then g -+ X3 is strongly continuous and norm—bounded [15],

so, since G is separable, f(g)X g is strongly measurable for every

ch1(G). Thus (I‘f(g)Xg dg exists.

Also (DZ: CO(G) 4 [I is Hilbertian. Since, (DZ = 17 on L1(G) and

(DZ: CO(G) -l y is continuous (Z has finite semivariation), r]: L1(G) -o y
is bounded. Hence <I>Z and 17 are continuous linear Operators which agree
on a dense subspace of CO(G) so i) is Hilbertian.

(b) =9 (a) Suppose {Xg: ch} is V—bounded. Then 1;: CO(G) -+ y is

45

Hilbertian. Then there is a Hilbert space H and bounded linear Operators
B: C0(G) -o H and A: H -v I such that n = AB. Using exactly the
same arguement as in (b) => (a) Of Theorem 4.7, we obtain a c.a. regular

measure Z: c1G) -» y such that for every ch

 

X8 = I < 7,g > Z(d7).
G A A
Also, as above, the continuous extension Of r): L1(G) -+ y to C0(G)

agrees with the continuous Operator <I>Zz CO(G) -i y. Since 1r. CO(G) -i y
is Hilbertian, so is (DZ. Hence {ng ch} is harmonizable.

4.17 mm . If Z: KG) -» y is a regular c.a. measure and the restricted
associated Operator (DZ: CO(G) -» y is Hilbertian then Z has a c.a.o.s.
dilation.

M. There is a Hilbert space H and bounded linear Operators
B: 00(G) -» H and A: H .. 1 such that <I>Z = AB. Since
B: CO(G) -l H is linear and continous there exists ([10],p152) a c.a. measure
M: flG) -+ H such that
Bf = I {(7) M(d7) for all cho(G).

C)

Here M(A) = B**(tpA) where goAc[Co(G)]** is defined by
cpA(p) = g(A), pc[CO(G)]*. Using this identification, the fact H is
reflexive and <I>Z = AB we Obtain

Z(A) = AM(A), Add).
Using [8], M(A) = UP {(A), AcflG) where 5 is c.a.o.s. and U and
P are bounded. Thus Z(A) = AUP {(A), A62 so Z has a c.a.o.s.

dilation.

46

4.18 flfheerem. Let {ng ch} be a harmonizable y—valued process.
Then there exists a Hilbert space H, a stationary H-valued process

{Yg: ch} and a bounded linear Operator S: H - I such that for every

ch, X3 = SYg.

Pgeef. By assumption, Xg = I < 7,g > Z(d'y) where Z is a regular
. G
c.a. measure on .SXG) into ] with restricted associated Operator

oz: CO(G) -» y Hilbertian.
By Lemma 4.17, Z has a c.a.o.s. dilation. There is a Hilbert space H, a
c.a.o.s. measure 5: EKG) -i H and a bounded linear Operator 8: H -l ;/

 

with Z(A) = S((A), AcflG). Let Y8 = I < 7,g > {(d'y). Since 5 is
G
c.a.o.s., {Yg: ch} is stationary and for every y*cy",

 

< y*, SYg > = < y*, SI < mg > £(d7) >
G

= < S*y*, I < 7.3 > {(dv) >
G

< 7.8 > < S*y*,£(d7) >

< 7.3 > < y*, SE(d'r) >

< 7.3 > < y*,Z(d7) >

y*, I < 73g > Z(d’y) > = < y*,Xg >
G

A ca»— Qw— Cl»—

80

47

Here we have examined the connection between harmonizability and
V—boundedness by relating it to the factorization of certain Operators through
a Hilbert Space. In [19] a concept of harmonizability and V—boundedness was
discussed for L(H,K)—valued measures T by using the spectral dilation of

such measures considered as continuous Operators (PT: (S(2,H), ||-|| ) -l K.

su
We have established (Theorem 3.10) a relationship between 2—summin1g)
Operators <I>T on (S(2,H), |- [00) into K and Hilbertian Operators on
B(2,C) into L(H,K). The natural question that arises is what is the
structure of the I] ollsup-continuous Operators mentioned above. We examine
this problem in the context when K = C. In this case, these Operators turn

out to be in a smaller class than the collection of 2—Summing Operators. We

reserve Chapter 5 for this examination.

Recently, ([19[,[23],[28]), there has been considerable interest in the
spectral dilation problem for Operator-valued measures. The most significant
results are Obtained in [19]. There, Makagon and Salehi studied measures T
taking values in L(H,K) where H, K are Hilbert spaces by considering
Operators <I>T defined on the space S(2,H) of H—valued E—Simple
functions taking values in K (see Chapter 1). In particular, they obtained
two interesting sufficient conditions for T: 23 -i L(H,K) to have a Spectral

dilation.

flfhmrem 5.1 ([19], Theorem 5.4). Let T be a finitely additive
L(H,K)-valued measure of finite semivariation defined on an algebra B. If

éT: (S(2,H), ||~||m) -+ K is 2—summing then T has a f.a. Spectral dilation.

Ihgzrem 5.2 ([19], Theorem 5.18). Let T be a finitely additive
L(H,K)—valued measure defined on an algebra 2. If ‘iT: (S(2,H), ”'"sup) -i K
is continuous then T has a f.a. spectral dilation.

The question was raised by the authors of [19] as to the connection
between the continuity of the Operator <I>T with respect to the
||-||8up-norm and the 2—summability of <1>T with respect to the
ll-llw-norm. We investigate this question in the Special case K = C.

Although in this case T is an .5—valued measure, we will regard (IT as a

linear functional on S(2,.2). Here .2” is a Banach space.

48

49

5.3 flfhmrem. Let T be a finitely additive measure of finite semivariation

on an algebra 2. If (3T: B(2,.Z) -i C is "'"sup— continuous then

 

«iT: (S(2,s), Ilillm) .. c is 2—summing.

Em. By Theorem 1.10 there is an isometric isomorphism

v
J: B(2,C) 0 $4 B(2,.Z) given by

n n
J(kE1 fk O xk) = k=21kak, f1,f2,...,fn (B(2,C), x1,x2,...,xnc.£

.. - v
Since QT: B(2,.Z) -i C is ||-||sup—continuous, (PT 0 Jc(B(2,C) e .2)*
which gives by Grothendieck's Theorem (Theorem 1.8) that for ch(2,C),

xcex
<I>T o J(f e x) = I I < m,f > < x*,x >H dA(m,x*)

where A is a regular Borel measure on B1 x B2 with Bl being the
unit ball of (B(2,C))* and B2 being the unit ball of .3". Moreover,

"(FT 0 J": "Tllsv =|’\| (Bl x 32)-

k v
For k = 1,2,...,N, let gk = 2 fl; 0 xi (B(2,C) G .27
° 1

fi

Then
" n {It k 2
|<I>T o J(iE1 iOxi)|

=1Il IN- 2: <m,i1.‘><x*,x’i‘ >H dA(m,x*)|2

_ <||T||8v I1 I] i231 < m,fk > < x* *,xli‘ >H|2 d|A|(m,x*)

50

11 ~ 11
2 c 235x15 2
k=l I T(i=l‘ 1“

<||T||2 s I; g fk * k 2- * <1
- 3,, up{k I,1<m.i><x.x,>|.||m|ls,,.llx||_ -}

=1:

1:

11
Taking hk = .2 lAk x‘i‘ tS(2,H), k=1,2,...,N yields
i

Elm )I2
<I>h
k-l Tk

 

< ”Tu2 su { >15 Iii mm“) < x*x“ > |2° Ilmll ||x*u < 1}
- sv Pk=1i=1 i ’i H’ sv’ -°
For mc(B(2,C))*, x*crfi", “m"sv 5 1, ||x*|[ S 1 define
Mm,x*(A) = rn(A)x*, ACE.
Then ||Mm,x*||sv g um"8v ||x*|[ g 1 and so
I; I“ ( )I2
(D h
k=1 T k
2 N 2
S IlTllsvsup {1‘21 | I < hkidM > I = M6M(>3.u%), IIMII8V S 1}
obtaining ‘i’T (S(2,.z), ||.||m) .. C is 2-summing.
fimgk 5.5. Returning briefly to the question of the structure of the

Operator <I>T on (S(2,.3), Il'llsup)’ we note that in case K = C and <I>T
is H - "sup—continuous we have that QT generates an integral bilinear form
which implies that <I>T is ||~||0° 2—summing. For the case when K is a
general Hilbert space we have been unable to isolate the form of <I>T and

leave the examination of this problem for later work.

[1]
[2]
[3]
[4]
[5]

[6]
[7]

[8]

[9]

[10]

[11]
[12]

[13]

[14]

[15]

FE N

Abreu, J .L. (1970). A note on harmonizable and stationary sequences.
BOl. Soc. Mat. Mexicana _15, 48—51.

Abreu, J .L. ( 1976). H—valued generalized functions and orthogonally
scattered measures. Advances in Math. 19, 382—412.

Aronsajn, N. (1950). Theory of reproducing kernels. Trans. Amer.
Math. Soc. 55, 337—401.

Billingsley, P. (1968). Convergence Of Probability Measures, John
Wiley and Sons Inc., New York.

Cambanis, S. (1982). Complex symmetric stable variables and
processes. Contributions to Statistics: Essays in Honour of Norman L.
Johnson. Sen, P.K., (Ed.) North Holland, New York, 63—79.

Cambanis, S. and Miller, G. (1981). Linear problems in pth order and
stable processes. SIAM J. Appl. Math 41, 43—69.

Cambanis, S. and SOltani, A. Reza (1984). Prediction Of stable
processes: spectral and moving average representations. Z. Wahrsch.
Verw. Gebiete 55, 593—612.

Chatterji, SD. (1982). Orthogonally scattered dilation of Hilbert Space
valued set functlons. Lecture notes in Math. 945, Springer—Verlag,
New York, 269—281.

Chatterji, SD. and Mandrekar, V. (1978). On equivalence and
singularity of Gaussian measures and their applications. Probabilistic
analysis and related tOpics, Academic Press V51. 1, 169—197.

Diestel, J. and Uhl, J.J., Jr. I1977). Vector Measures, Math. Surveys
15, Amer. Math. Soc., Provi ence, R.I..

Dinculeanu, N. (1976). Vector Measures. Pergamon Press, Berlin.

Dunford, N. and Schwartz, J .T. (1953). Linear Operators Part I,
General Theory, Interscience Publ., New York.

Grothendieck, A. (1956). Resume de la théorie métrique des produits
tensoriels tOpologiques. Bol. Soc. Mat. Sao Paulo 5, 1—79.

Hérmander, L. (1983). The analysis of linear partial differential
Operatorskl, Distribution theory and Fourier analysis, Springer-Verlag,
New Yor .

Kluvanek, I. (1967). Characterization Of Fourier—Stieltjes transforms
as vector and Operator valued measures. Czech. Math. J. u, 261—277.

51

[16]

[17]

[18]

[19]

[20]
[21]

[22]

[23]

[24]

[25]
[26]
[27]

[28]

[29]

[30]
[31]

52

Lindenstrauss, J. and Pelczyl’lski, A. (1968). Absolutely summing
Operators in Lp—spaces and their applications. Studia Math. 22,

275-326.

Loéve, M. (1960). Probability Theory. D. Van Nostrand Company,
New Jersey.

Makagon, A. and Mandrekar, V. (1988). The spectral representation of
stable processes: harmonizability and regularity. To appear in
Probability Theory and Related TOpics.

Maka on, A. and Salehi, H. (1987). Spectral dilation Of Operator
value measures and its appllcation to infinite dimensional
harmonizable processes. Studia Math. 85, 257—297.

Masani, P. (1970 . Quasi—isometric measures and their applications.
Bull. Amer. Mat . Soc. 7 , 427—528.

Masani, P. (1978). Dilations as prOpagatorS of Hilbertian varieties.
SIAM J. Anal. 2, 414-456.

Maurey, B. (1974). Théorémes de factorisation pour les Operateurs
linéares a valeurs dans les espaces Lp, Astérisque 11.
Miamee, AG. (1989). Spectral dilation Of L(B,H)—valued measures

and its application to stationary dilation for Banach Space valued
processes. Indiana Univ. Math. J. 88, 841-860.

Miamee, AG. and Salehi, H. (1978). Harmonizability,
V—boundedness, and stationary dilation Of stochastic processes, Indiana
Univ. Math. J. 21, 37—50.

Miamee AG. and Salehi, H. 1977). On the square root of a positive
B(fiefl") valued function. J. ultlvariate Anal. _7_, 535—550.

Niemi, H. (1977). On ortho onally scattered dilations Of bounded
vector measures. Ann. Aca . Sci. Fenn. Ser. AI Math. 8, 43—51.

Pisier, G. (1984). Factorization of linear Operators and eometry of
Banach Spaces. CBMS no. 50, Amer. Math. Soc., Provi ence, R.I.

Rosenberg, M. ( 1982). Quasi—isometric dilations of Operator valued
measures and Grothendieck's inequality. Pacific J. Math., m3,
135-161.

Rudin, W. (1962). Fourier Analysis on Groups. Interscience
Publ., New York.

Rudin, W. (1973). Functional Analysis. McGraw—Hill, New York.

Schilder, M. 1970). Some structure theorems for the symmetric stable
laws. Ann. ath. Stat. 41, 412—421.

"‘lllllllllllllll“