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OPERATOR VALUED FUNCTIONS ON
A BANACH SPACE

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ABSTRACT
BANACH SPACE VALUED STATIONARY STOCHASTIC PROCESSES AND

FACTORIZATION OF NONNEGATIVE OPERATOR VALUED
FUNCTIONS ON A BANACH SPACE

By
A.G. Miamee

In this thesis the theory of Banach space valued stationary
stochastic processes and the problem of factorization of nonnegative
Operator valued functions are studied. The the thesis consists of
eight chapters and one appendix.

Chapters I and II are introductory. In Chapter III, Banach
Space valued stationary stochastic processes are systematically
studied. The results, such as Wold's decomposition, Cramér's
decomposition, Wold-Cramér concordance theorem, etc., which are
fundamental in this area are established. These include the
extension to the Banach space of most of the results of R. Gangolli.

In Chapter IV the factorization problem of Banach space
valued stationary stochastic processes which plays an important
role in the prediction theory of Banach space valued stationary
processes, is considered. Several theorems concerning this
factorization are given. These involve the analysis of quasi
square roots and their corresponding invariant subspaces. Con-
tinuing our study of the factorization problem, in Chapter V
several necessary and sufficient conditions for factorability of

these functions are given. The works of Chapters IV and V extend

A.G. Miamee

to the Banach space case, most of the result of R.G. Douglas and
the recent work of Yu. A. Rozanov as well as a certain result of
R. Payen on factoring a nonnegative operator valued function on a
Hilbert space.

Let f be a factorable nonnegative Hilbert space Operator
valued function, and let U be a unitary valued function. A
natural question is to see if the nonnegative operator valued
function UfU* is factorable. This problem is investigated in
Chapter VI. As an application of this study some results, such
as a Devinatz's type necessary condition and characterization for
the factorization problem are given.

In Chapter VII the important problem of finding a computable
algorithm for finding the optimal factor and the linear predictor
of a stochastic process is considered. An algorithm similar to
the one given by N. Weiner and P. Masani for the infinite dimensional
process is obtained. This involves the Fourier analysis of in-
finite dimensional matrix valued functions.

In Chapter VIII the problem of minimality and interpola-
tion of infinite dimensional stationary processes is studied,

Most of the results of H. Salehi for multivariate case are extended
to infinite dimensional case. Also a well known result of P. Masani
on minimal multivariate processes is extended to the infinite

dimensional case. In the appendix the construction of quasi square

roots of several operators is given.

BANACH SPACE VALUED STATIONARY STOCHASTIC PROCESSES AND
FACTORIZAIION OF NONNEGATIVE OPERATOR VALUED
FUNCTIONS ON A BANACH SPACE

By ‘..
v. ‘Jk 15 S 6' 1‘
A.GI Miamee

A’THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics

1973

TO l‘fl WIFE EFFY

ii

ACKNOWLEDGEMENTS

I am deeply indebted to Professor H. Salehi for his helpful
guidance during the preparation of this thesis. To say that I
appreciate all the time he has taken for me and all the kindly ad-
vice he has given me is certainly an understatement and I can only
express my deep gratitude.

I also wish to express my gratitude to Professors
V. Mandrekar and J. Shapiro for the interest they have displayed
regarding my thesis, and for the useful discussions I have had with
them. Thanks are also due to Professors H. Davis and R. Phillips
for serving on my guidance committee.

Finally, I am grateful to the National Science Foundation
and the Department of Mathematics, Michigan State University for

financial support during my stay at Michigan State University.

iii

Chapter
I
II

III

IV

VI

VII

TABLE OF CONTENTS

INTRODUCTION
PRELIMINARIES

ANALYSIS OF BANACH SPACE VALUED STATIONARY
STOCHASTIC PROCESSES

Introduction

Preliminaries

Time and spectral analysis

Subprocesses and spectral conditions for
factorability of the spectral density

uuww
J-‘UNH

FACTORIZATION OF NONNEGATIVE OPERATOR VALUED
FUNCTIONS ON A BANACH SPACE

Introduction
Ancillary results
Main lemmas

Main theorems

bat-Sb
war-I

NECESSARY AND SUFFICIENT CONDITIONS FOR
FACTORABILITY OF NONNEGATIVE OPERATOR VALUED
FUNCTIONS ON A BANACH SPACE

5.1 Introduction
5.2 Preliminaries
5.3 Main results

*
FACTORIZATION OF UfU

6.1 Introduction *
6.2 Factorability of UfU
6.3 Application

ALGORITHMS FOR DETERMINING THE OPTIMAL FACTOR
AND THE LINEAR PREDICTOR

Introduction

Prel iminar ies

Further Analysis of time and spectral domain
Determination of the generating function

and the linear predictor

\INNN
waH

iv

Page

27

36
36
37

39
43

53
53
53
54
61
61
61
67
78
78
89

101

Chapter Page

VIII MINIMALITY AND INTERPOLATION OF INFINITE
DIMENSIONAL STATIONARY STOCHASTIC PROCESSES 109
8.1 Introduction 109
8.2 Minimality and interpolation 109
APPENDIX 1 l 7

REFERENCES 0 120

CHAPTER.I

INTRODUCTION

The idea of Banach space valued stationary stochastic pro-
cesses has been recently introduced by S.A. Chobanian in [2 1.
Subsequently some basic results concerning these processes were
announced [24], [ 3]. In the Hilbert space case the basic
questions of regularity, Wold's decomposition, Wold-Cramér con-
cordance, factorability of spectral density, etc. have been studied
in detail [4 ], [8 ], [10], [12], [13], [16], [18], [19]. ‘However
in the Banach space case the study of stationary stochastic processes
and the related problems are in its early stages, and the results
obtained in this direction are not as yet complete. In particular
the important problem of factoring a nonnegative operator valued
function on a Banach space has not been investigated. The problem
of determining the Optimal factor of a spectral density plays an
important role in the prediction theory of stationary stochastic
processes. This problem was tackled by Wiener and Masani [26g and
later on by Masani [14] for the finite dimensional case. This prob-
lem.remains open for the infinite dimensional processes.

In this thesis we first study Banach space valued stationary
stochastic processes and prove some known results as well as
several new results. We then consider the question of factorability

for the Banach space case and establish several criteria for the

factorization problem. In particular we Obtain several comparison
type sufficient conditions and some analytic necessary and sufficient
conditions for the factorization problem. In the second part of
this thesis we provide an algorithm for finding the optimal factor
and the linear predictor for the Hilbert Space case. We also

study the problem of minimality and interpolation of Hilbert space
valued stationary stochastic processes. With this background we

now summarize the content of each chapter in more detail.

In Chapter II we recall some notations and terminologies
from [2 ] concerning Banach space valued stationary stochastic pro-
cesses. We also state some facts [1 ], [24] regarding these pro-
cesses which are needed in the later chapters.

In the first part of Chapter III we study Banach Space
valued stationary stochastic processes. Using a new technique
(to be made clear later) we will provide proofs for Wold's de-
composition, relation between regularity and factorization which
were announced in [243, [3 ]. We also prove several new results
such as a time domain and a Spectral domain decomposition as well
as moving average representation for these processes. In the
second part of Chapter III the idea of subprocesses is introduced
and most of the results of R. Gangolli [8 ] are extended to the
Banach space case. In particular a Wold-Cramér concordance theorem
for the Banach space valued stationary stochastic processes as
well as some sufficient condition for the factorization problem are
obtained.

In Chapter IV we consider the problem of factoring a non-

negative operator valued function f on a Banach space in the form

9*Q, where Q is a conjugate analytic Operator valued function.

We give several comparison type sufficient conditions for factoriza-
tion problem by extending to the Banach Space case most of the
results of R.G. Douglas [4 ]. In the Hilbert Space case, Jf, the
positive square root of f whose existence is known is used
frequently. When f is a positive operator valued function on a
Banach space I the existence of a square root in the ordinary

way does not make sense. Nevertheless we will prove (c.f. Lemma
4.3.1) the existence of a measurable function Q on 1, into some
auxiliary Hilbert space which behaves almost like a square root in
the sense that f -'Q*O. We will call this a quasi square root.

The quasi square root will play the role Of square root in this work.

The results of Chapters III and IV provide only sufficient
conditions for the factorization problem. In Chapter V we establish
several necessary and sufficient conditions for the factorability of
a nonnegative operator valued function on a Banach space. In
particular our main theorem of this chapter (Theorem 5.3.8) extends
to the Banach Space case the recent work of‘Yu. A. Rozanov [19]
and a certain result of R. Payen [18] on the factorization problem.
The notion Of quasi square root is basic in this chapter.

In Chapter VI we Study the following natural question raised
by MQG. Nadkarni in [16]. Given a factorable nonnegative operator
valued function f on a Hilbert space, to see if the nonnegative
Operator valued function UfU* is factorable, where U denotes
a neasurable unitary valued function. We apply these results to
prove some well known facts as well as some new results regarding

the factorization problem for the Hilbert space case.

In Chapter VII we consider the important problem of finding
an algorithm for determining the Optimal factor and the linear pre-
dictor of a Hilbert space valued stationary stochastic process. In
this chapter we will adopt the notations of [16] and employ the
technique of [14] in order to establish our algorithm.

In Chapter VIII we investigate the problems interpolation
and minimality of a Hilbert space valued stationary stochastic
process. We extend most of the results of H. Salehi [21], [22],
[23] to the infinite dimensional case. Using Salehi's technique
we prove infinite dimensional extensions of a result due to Masani
on minimal full rank processes.

Finally in the Appendix we give the construction of a quasi
square root for a particular nonnegative operator valued function

on a Banach Space.

CHAPTER II

PRELIMINARIES

In this chapter we introduce some basic terminologies
and state some known facts which will be needed in the latter
chapters.

2.1 Notation. The script letters I, and 14 will denote Banach
spaces and the script letters )1 and X will stand for Hilbert
spaces. If I, is a Banach space, 1* will denote the Banach
space of all conjugate linear functionals on I. For any two
Banach spaces 1 and u, B('L,u) will stand for the Banach space
Of all bounded linear Operators on X, into ‘4'

In this work all the Banach spaces are assumed to be
separable.

2.2 Definition. An operator f in B(I,I*) is said to be non-
negative if for each x E I, (fit) (x) 2 0. B+(I,,I*) will denote
the class of all such operators.

2.3 Definition. Let I be a Banach space and K be a Hilbert
Space. A sequence gn, -co < n < an of elements of B(I,7() is
called a B(I,7() -valued stationary stochastic process (SSP) if
§:§m depends only on mm. The Operators R(m-n) - §:§m is
called the covariance Operators of the process.

The following theorem is proved in [l ].

2.4 Theorem. Let R(n) , «a < n < an be a sequence of operators

'1:
on X, into I . Then R(n), -oo < n < an is the covariance
5 .

Operators Of some SSP 5n, -m‘< nr< a if and only if it can be
represented as
R(n) = i; E" e'chde).

where F is a B+TI,If)-valued measure and the integral is in the
weak sense. In this case F is called the spectral distribution
of the process En, -ai< nr< m. In case that F is a.c. with
reapect to (w.r.t.) the Lebesgue measure, its derivative f is
called the spectral density of the process.

If B is a subset Of some Hilbert Space X we will denote
by 6{B] the Smallest closed subspace of K containing 8.

Let us give the following definition.
2.5 Definition. Let 6;“, -co < n < an be a B(I,)()-valued SSP.

Then we need to define the following subspaces

Hgfin) :3 6{§kx, -m < k < on, x E I]

H§(n) -6{§kX. -oo< ksn. x 6 I}

H (‘00) 'DH

3 n 3“”

When there is no danger of confusion we will omit the index g
in the above definition.
The following definition is basic in the theory of Stationary
stochastic processes.
2.6 Definition. Let g“, -w‘< n.< o be a B(I”x)-valued SSP.
Then g“, -cn < n < an is called
(1) Deterministic (or singular) if H(-a) - H(n), for all n.
(ii) Nondeterministic if H(-o) # H(n) for some n.

(iii) Purely nondeterministic (or regular) if H(-m) = 0.

2.7 Definition. Let X' be a separable Hilbert space and let
L2(X) denote the Hilbert space of all X“-valued functions on
the unit circle which have a square summable norm. The L2(X)

inner product of two functions g1 and 32 is given by
2n
l;_ ie ie
2"] <31<e ). g2<e ))de.

The subspace Lg-(x) (Lg+(x)) consists of all functions g in

mesea-0 for all n<0 (n >0).

2n 18
L2(X) for which A g(e )e
2.8 Definition. A weakly measurable B(I,R9-valued function
A - A(ei9) is called analytic (conjugate analytic) if for each
x e r. A(eie)x e L‘ZHOO (A(e1°)x e Lg'oo).
2.9 Definition. Let f = f(eie) be a weakly summable B+(I,If)-
valued function on the unit circle. We say that f is factorable

if there exists a Hilbert Space X' and a conjugate analytic

B(IWKQ-valued function A = A(eie) such that
*
He”) = A (e1°>A(ei°).
in the sense that

(f(e19)x)(y) = (A(eie)x,A(eie)y), for all x,y E I.

CHAPTER III

ANALYSIS OF BANACH SPACE VALUED STATIONARY
STOCHASTIC PROCESSES

3.1 Introduction. The main aim of this chapter is to extend to
the Banach space the well known results of R. Gangolli [8 ] on
subprocesses, Wold-Cramér concordance and factorability. We also
extend to the Banach space a time domain decomposition due to
R. Payen [18]. In the course of our work.we will have occasions
to improve some of the results contained in [24], [3 ] as well as
providing proofs for some Others.

To accomplish our goal we will associate to our SSP an auxiliary
Hilbert space valued stationary process. This will make it possible
to utilize the available results for the Hilbert space case.

We settle preliminaries in §3.2. In §3.3 we develop some
of the theory Of Banach space valued stationary stochastic pro-
cesses by introducing a Hilbert space valued stationary stochastic
process which is relevant to our process. In this section we prove
some new results as well as most Of the results in [24], [3 ] by
using our Hilbert Space valued stationary process mentioned above.
In 63.4 we extend most of the results Of R. Gangolli [8 ] to.
Banach Space valued stationary processes.
3.2 Preliminaries. All the Banach spaces and Hilbert spaces con-

sidered here will be separable.

3.2.1 Definition. Let S C B(I,7(). By 6(3) we mean the
smallest closed (in strong sense) subSpace Of B(‘I,,)() containing
all the elements of the form SA, where S 6 S and A 6 B(I,I)
and by 6(3) we mean the smallest closed subspace of x con-
taining all the elements Of the form Sx, where S E S and

x E 1. One can prove the following theorem by an argument Similar
to [18], p. 335.

3.2.2. Theorem. With the notation of Definition 3.2.1, for any

collection 3 C. B(I,,7() we have

6(3) = Boa. 6(8))-

3.2.3 Definition. Let §n, -m < n < on be a B('L,7()-valued SSP.

Then we define the following subspaces
no.) . sgkx, -oo < k < on, x e I}, in.) = égk, -cn < k < a]

H(n) - egkx. m < k g n}, in.) =- égk, -.. < k s n}

n(-o) = n R(n) and iv...) = n Em).
II n

We remark here that by Theorem 3.2.2 it is clear that

m...) = {Ax, A 6 Eu), x 6 13,110..) = {Ax,A e Econ), x e I}

and
H(n) - {Ax, A 6 Km), x e 1}.

3.2.4 Definition. Let A and B be in B(I.,)(). Then by (A,B)
we mean the unique bounded Operator which is defined through

*
((A,B)x,y) - (Ax,By). It is clear that (A,B) = B A. Now if for

A,B 6 B(1,X), (A,B) - O, we say A .LB.

10

One can prove the following theorem
3.2.5 Theorem. Let A E B(I.7O and M = 8(8), Where 8 c: B('I,,}() .
Then there exists an Operator in B('L,7() denoted by (Ali) such
that (AW) 6 ii and A = A - (Hg) is orthogonal to i.
Page—f. Let (A‘M) (x) = (Ax‘M), where M - 6(3).
3.2.6 Definition. Let gn, -co < n < .0 be a B(I,,)()-valued SSP.
Then we call gn '- §n - (§n|H(n-l)), -oo < n < co the innovation
process Of gn, -a < n < on. We write G - (g0,go) and call it the
predictor error Operator Of the SSP gn, «no < n < no. If G is
boundedly invertible then the process is called of full rank. If
G is one-to-One then the process is said to be of nearly full rank.
3.2.7 RM. Let g“, -ca < n < on be a B(I,,)()-valued SSP and
let G be its predictor error Operator. Then it is easy to see
that g“, -oo < n < an is singular if and only if G = O and is
nondeterministic if and only if G 3‘ 0.

We give the following lemma for later reference.
3.2.8 m. Let gn, -cn<n < co and 1]“, -co<n <co be two
B('I,,)() -va1ued SSP'S with the same Spectral distribution or equi-
valently with the same covariance structure. Then
(i) g“, «a < n < an is regular iff ‘nn, -cn < n < on is regular
(ii) gn, -co < n < on is singular iff 1]“, no < n < no is singular

(iii) Gg " G“.

Proof. Proof depends on the fact that the operator V sending

gnx to 1}“): can be extended to an isometry on H (on) onto Huh).

§
It is clear that Hn(n) - V(H§(n)) for all n. Hence Hn(-co) -
V(H§(-co)). Thus (1) and (ii) are Obvious. For (iii) we further

note that

11
V(§0\H§(-l)) = (V§o\VH§(-1)) = (Tb\Hn(-1)).

and hence

on = (no - (Manon). lb - (Manon) =

(V110 - vmomnon). vno - vmomncm =

(so - (aw-1,01». so - (refine-1)) = cg.

Finally we give the following definition which we will
need later.
3.2.9 Definition. Let 5“, -co<n<ao and m, -oo<n<oo be
two B(I”XQ-valued SSP. We say §n, -w:< nr< m is dominated

by ‘qn,-o<n<co if H(n)c:H(n) forall n.

E 11

3.3 Time and spectralhgnalysis. In this section we first
associate to any Banach space valued SSP 5“, -w < n < w a Hilbert
Space valued SSP un, ~¢r< n < m (c.f. Lemma 3.3.1). We then
examine the close tie which exists between g“, -m‘< n < a and

un’ «n.< n‘< m. Using these new processes we can transfer the
information we know for Hilbert space valued processes to get the
corresponding results for Banach space case. By making use of

this technique we provide a proof for Theorems 3.3.5, 3.3.7, 3.3.10,
which are announced in [3 ]. However our moving average representa-
tion for the regular processes will have the natural form, which
prevails in the one dimensional case. Using the technique mentioned
above we also state and prove several new theorems which extends

the known results for Hilbert space valued stationary stochastic

processes.

12

The following lemma is essential.
3.3.1 m. Let En, ~09 < n < on be a B('I,,)()-valued SSP. Then
there is a Hilbert space ac: X and a B(d,)() -valued SSP un,
-m.< n < m Such that g“ I ungo for all n. Moreover g“ and
un have the same shift .
H.229 Define U on 6{§nx, x 6 I, -00 < n < +m] by ugnx =

§n+1x, then we have

(ugnx. ugny) = (Snflx. gnfly) = (tinx, guy).

80 U is a unitary Operator. It is now clear that g“ = Ungo.
Let a =- 6{§0x, x 6 I] and let una = Una, for a 6 d. Then url
is a B(d,7() -va1ued SSP and En = ungo. So un, ~aa < n < an is the
desired process. It is clear that un and gn have the same
Shift U.
In view Of the close relations between g“ and un the
following definition is appropriate.
3.3.2 Definition. Let g“ and un be as above. Then un,
~00 < n < an is called the associated process of g“, -oo < n < on.
The following theorem gives some relations between E“
and un.
3.3.3 Theorem. Let g“, «I» < n < an be a B(I,’() -va1ued SSP and
let un be its associated process. Then

(a) R (n) - :3 name,

a
*
(b) F, - :0 Euro

(c) F

is absolutely continuous (a.c.) iff Fu is a.c.,

5
and in this case we have f§(x.Y) ' fu(g0x’ EOY). where f(x.Y)

denotes the density of (F(d9)x)(y).

13

Proof. We Observe that

 

(R <n)x)<y) - (sax. Soy) = (unsox. 50x)

'5
a (sum (SOX))(§0y).

Hence R§(n) = ggRu(n)§o.

To see (b) we note that

11'

l 2 -ine
(R (n)x)(y) = —& e (F

g 2" (d9) X) (y) (1)

E

and

(cggku<n)§o><x)>(y) = (Rucn)(§0x))(§oy>

211 (2)
= l -in9 *
—211 e ((goFUNB) :0) (3‘)) (Y) -

Now (a), (1) and (2) imply that F (de) and gzruwemo have

§

the same Fourier coefficients and hence they are the same measure .

(c) Suppose Fu is a.c., then

2—9 (E§(de)X)(y) ‘3 g3 (53Fu(d°)§0x)(y))
d (3)
. 33 (Fu(de)(§ox)(§0)’))
9..
so de((F

g(de)x) (y)) exists and is equal to %9- (Fume) (tox))(§oy) -

Now to see the other way suppose F is a.c., then

S
%6((F§(dB)X) (3‘)) 8818128 for all x E 1, hence

(Fu(de) (50x) (gox)) exists for each x 6 1,. Therefore

(Fu(de)a) (a)) exists for each a 6 Igor. Thus by [ 9], §66

(D 0

21-9- (Fu(de)a)(a)) exists for each a 6 4- Therefore I"u is a.c.

Now in this case we have

l4

{iguana - (Eganxxy) = <<§3Fu<a>go)x)<y)

- (quencgoxneoy) - {fucgom gowde.
Hence for all x and y in I, we have
f§(X.y) = fu(§ox, goy) for almost every e.

The following lemma reveals a strong tie which exists
between the two processes g“, -co < n < co and un’ -co < n < co.
3.3.4 Lemm. Let gm, -a° < n < an be a B(I,}()-valued SSP with

un’ -ao < n < on its associated SSP, then

(a) (n) = Hum), H («0) -= Hump).

H
§ E
(b) E“, -oo < n < on is regular iff un, -oo < n < on is regular.

(c) g“, -oo< n < an is singular iff un, -oo< n < on is singular.

Proof. For the proof of (a), (b) and (c) it is sufficient to

 

Show that H§(n) = Hu(n) , for all n, -oo < n < co. Consider

H (n) =5{§kx, x E I, kSn] =6{u

g x,x€1,ksn]. Then

150

H§(n) 6(ngox, x 6 I, k s n] (l)

where U is the shift Operator in leuma 3.3.1. We note that

6(U An!) " 5(U 6(Aa)) , (2)
01 a

where {Ad} is any collection of subsets of X. Combining (1)

and (2) we get

ago) - S{S(ngox. x e x. k s m . (3)

Now we observe that for any subset of K we have 6(UA) - U(6(A)),

because U is unitary. Using (3) and this Observation we get

15

H§(n) - 6{Uk(5(§01)), k s n] = kam), k s n]

= 5{uk(d) . k S n] = Hu(n)

Now we give a proof for the Wold decomposition theorem.
3.3.5 Theorem. For any B(‘L,)() -valued SSP g“, -oo < n < on, there
exists two B(I,,,() -valued SSP 1%, -oo < n < co and gn, -oo < n < m

such that
(i) §n=m+gn,forall n,-oo<n<oo.

(ii) 1]“, -o < n < an and g“, -oo < n < on are dominated by

§n, -co < n < an and have the same shift as gn, -oo < n < co.

(iii) TL, —o < n < on is orthogonal to Q“, -m < n < co, i.e.

no IQ“ for all m,n.
(iv) 1]“, -oo < n < so is regular and g“, -oo < n < on is Singular.

Proof. Let un be the associated process of g“, «D < n < on.
Then by the usual WOldeecompOS ition for the Hilbert Space valued
process un, -o < n < co [8], p. 899, we have un I vn + wn
where V“, -oo < n < co and wn, -co < n < on are B(a,7() -valued
SSP'S satisfying similar conditions as (1)-(iv). Now let

“n - vngo and Cu - wngo. Then Obviously g“ I 11“ + Q“. It is
clear that 1]“, -ao< n < co and g“, -w < n < on are SSP'S with

the same shift as gn, -o:! < n < on. Now
“L3" any) = (”350", ““50” = (vnmox). wnmoy» = 0

because v .Lw . This means fin .L gm. Now Observe that
n m
n
n“ vngo (U vo)§o. Because vn, -oo < n < an is dominated by

u“, -o0< n < co and H (n) = Hu(n) (c.f. Leanna 3.3.4), it follows

E

16

that fin, -cn < n < an is dominated by g“, Io: < n < co. Similarly one
can Show that g“, -ao< n < so is dominated by g“, -co < n < on.

It remains to verify (iv) . For this we observe

H,“(n) I 5[1\kx, x 6 I, k s n] I 6{vk§0x, k s n, x E I] .
Thus
H,“(n) c: Hv(n) for all n.
Hence H,“(-co) I 2 Hn(n) r: rt‘]Hv(n) = Hv(-m) . Since vn, -co < n < on
is regular therefore 1]“, Ion < n < an is regular. Now to prove

that g“, -u: < n < o: is singular, similarly we can get Hc(n) :
Hw(n) . But Hw(n) I Hw(-o) for all n, Since wn, -a < n < on is

singular. Hence for each n, H (n) c Hw(-ao). Now we know that for

S
the Hilbert space case Hw(-m) I Hu(-oo) (c.f. [8], p. 899) SO we
have H (n) I Hu(-co). To complete the proof it suffices to Show

C

that

H§(n) = Hu('co), for all n. (4)

Suppose (4) is not true, i.e. suppose Hu(-oo) iHcm) for some n.

Then there exists h, 0 I h E Hu(-¢) 9 Hc(n). Hence
higkx for all x61 and all kSn . (5)
On the other hand

hlnkx forall x61 andall kin: (5)

because nkx I vkgox I vk(gox) , h e Hu(-oo) I Hw(-m) and the fact

that vn .Lwn. Since :1“ I “n + g“ for all n using (5) and (6)

17

we get h ngx, for all x E I, and all k s n. SO h Iuk§ox
for all x 6 I, and all k s n. Hence h .Luka for all a E 501

and all k s n. But since go}. is dense in a we see that

h J. uka for all a E a and all k s n. Thus h iHu(n) which
implies h .L Hu(-co). But h 6 Hu(-co) by the choice of h, hence

h I O which is a contradiction to the choice of h. This completes
the proof.

3.3.6 RM. The Wold decomposition is unique. To see this let

11“, -m < n < co and Cn’ -co < n < an be two B('I,,7()-valued SSP'S

satisfying (1)-(iv) of the last theorem. Then we claim

H§(-w) = HC(-I) - (1)

Granting (1), since §nx = nhx + gnx we get
manages») = (mung-e» + (gnxmgcen = (mung-en
+ (CUXIHg(-~))-
Thus by (iii) and (iv) we get
manages) - Cnx and nnx - Snx - (SnX\H§(-w))

which means the uniqueness. Now we will verify (1). By (i) we
have H§(n) I ngkx, x E I, k s n] I emkx + gkx, x c I, k s n].

Thus H§(n) c Hn(n) 6 Hc(n). But g is singular and hence

Hg(n) c Hn(n) @ HC(-a) . We know Hn(n) and H§(-m) c: H

hence H§(n) I H“(n) @ Hc(-oo) , H“(n) .L H§(-oo) so we have

g(“)’

n(-~)—nu<n) =00!

-a a -ee . 2
g n g n “(106116 )) H€( ) ()

18

“(n) I {0] .

In the next theorem we give a moving average representa-

The last equality in (2) follows because 0 H
n

tion for a regular B(‘I.,)() -valued SSP. A moving average representa-
tion was also given in [24]. We remark that our representation is
exactly in the form that one Obtains for the finite dimensional case.
Our proof is natural and based on the associated process un,

~00 < n < on.

3.3.7 Theorem. Let g“, -oo < n < on be a B(I,70 -valued SSP which

is regular, then we have the representation

Q
§ ' 2 3 _ (1)
n 10.0 n k Alt
where 81's are orthogonal partial isometries and Ak 's are in B(I,,)() ,

and the convergent in (l) is in the weak sense.
Proof. Let un, -eo < n < co and a be as in learns 3.3.1. Then by
the correSponding theorem for the Hilbert Space valued process

u -au< n < co (c.f. [18‘], p. 359) we have

n,

uIZS_B, (2)
n k.0an

where 81's are partial isometries on K and Bk 6 B(a,)().

Applying both sides of (2) to go we get

a
gn . “ago - (kgosn-kBkmo . 2 Sn-kBk§O°

be A then we have

Letting B k’

150
g‘n . kEOSn-k Ak °

3.3.8 Rflrk. The coefficients Ai's in the moving average repre-

sentation (1) is not unique, however 81A are unique for each i

.1

l9

and j. For if there is An's and Bn's satisfying in the

hypothesis of the last theorem, then we have

a to
En " kfosn-kAk "' kEoSn-knk '
We multiply both sides Of the last relation by sis: to get
313*(gs A) -ss*(;s a)
ik_0n-kk iik‘on-kk
Hence SiAn_1 I SiBn_1 for all n and 1. Thus 31A]: 81Bj

for all i and j. Now we can prove the following theorem.
3.3.9 Theorem. Let g“, Io: < n < an be a B(I,)()-valued SSP.

Then the following are equivalent:

(1) g“, -ao < n < an is regular.

CD
4'6)
(ii) g“ a REOSP'RAI" Ak E B(I,d) and {Sn]ns_m is a sequence
of mutually orthogonal isometries in B(d,)().

(iii) a (-e.) - {0}.

6
Proof. (1) =9 (ii) by Theorem 3.3.7. (iii) a (i) follows by the

uniqueness Of Wold's decomposition. It remains to Show (ii) =9

(iii). Now suppose (ii) holds. Then for each x we have
u: “2_ <§s A>x2-n§s Axn2=§us Axnz (1)
0x “k‘Ion-kk“ k_on-kk b0 n-kk' <°°°
Letting R(n) I $18,9an it follows that
— °° - 2 °° 2

whence from (1) \\(§0|R(-n))(x)“ .. 0, as n _. on for each x e I.
But from (ii) H(-n) C R(m) and “(§O\H(-n))x“ s H§O\R(-n)“ and

hence

20

H(§0\H(-n))x“ I O as n.~ m for each x 6 I

“(gox‘H(-n))“~ 0 as n I w for each x E I .
Similarly we can Show for each k s 0 and each x E I

\\(§kx\H(-n))“—e 0 as n -0 a)

hence

“Calm-n)“ ‘° 0

“(a\H(-n))“ » 0 for all a 6 {KER}, X E In k s O) =
(2)

the linear manifold generated by {gkx, x 6 I, k s 0].
Now given a >'0 and b E H(O) then there exists a 6 JK§Rx,
x e x. k s 0) Such that ua-bn < g. Then “(b\H(-n))“ <

“((a-b)\H(-n))“ + “(a\H(-n))“. Hence for all n >rN we have
WNMmDHSWIH+WnMIDH<§+§=e-

Hence “(b‘H(-n))“ a 0 as n.~ a for all b 6 H(O). Now using

this and the fact that P Strongly we see that

H(-n) " P11(-...)
H(-a) . 00

Now we prove the following theorem, part (b) of which was
announced in [3 ].

3.3.10 Theorem. Let gn, -m‘< n1< m be a B(Iaxj-valued stationary

process with spectral distribution ~F. Then

21

(a) if 5“, -cn < n < on has a two-sided moving average representa-
41»

tion g“ I 10.1.)"n (Pu-kAk with (mnmpn) I Oan. K 9‘ 0. Ak E BCIJO

and q)" 6 B()(,7(). Then its Spectral distribution F is a.c. and

we have
"3'6 (F(d6)X) (5')) = (§(eie)x, New)”
where
“e16”, I k: e.ike JR Akx

(b) g“, -co< n < on is regular iff F is a.c. and

fig cr<de)x>(x)) = uueienn"

. Q
where §(ele)x of the form §(eie)x I E e-ikeA x.
k
kIO

Proof. (a) Consider

+oo 41::

(Meiek. mien) = ( r. e‘ikexk Akx. E aim/K Akx)
kfi-O k=-oo

+ee +oo ,
- E E e'1(k'k)°UkAkx./K Aw“)

k'I-ca RI-co

4m -ine “I'm
'3 E e 2 (JR Akx, A, Ak-nx)
nII-oo its-..

80 the n-th Fourier coefficient of (6(e19)x, Q(eie)x) is

he
2: (JR Akx. JR ARmX)

kI «a

0n the other hand the n-th Fourier coefficient Of (F(de)x) (y)

is (R(n)x) (x) which is equal to

22

‘+a +m
(R(n)x)(Y) a (Sax: gox) a (k:§¢ Th-KAKX’ k;¥° T-KAEX)
'hn‘hn

2 E (cpn_kAkX. cp_k.Ak.x)

RI-a k'I-a

B 2 (CPD A X, cP-k'Ak'X)

k and k' “k 1‘
n-kI-k'
+m
= A
kg-” (fK Akx, /l( k-nx)

Hence (F(de)x) (x) and (9(eie)x, 9(eie)x) have the same Fourier
coefficients and hence the proof is complete.

(b) Necessity. Suppose gn, -ca< n < on is regular, then by
Theorem 3.3.9 it has a moving average representation. Now apply
part (a) to conclude factorability.

Sufficiency. Let ((90):.0‘, C B(7(,7() be any seque-nie such that
((9,), (an) I bum]: and consider the new SSP gr: I 1‘31; (Pn-kAk’ then

by theorem 3.3.9, :5, Ion < n < on is regular, hence by part (a) ,

F is a.c. and we have

g.
"3'6 (Pg.<de)x)(x)) = (mien. Mambo
hence
g-(1i (doom) = 1- 0? «one»
de §' do g °

Hence g“, «no < n < an and gr", I» < n < on have the same Spectral

distributions, and so by lemma 3.2.8, gn, ~04< n.< a is regular.
AS a consequence of the Wold decomposition theorem we have

3.3.11 Theorem. ‘Let g“, .¢.< n < m be a non-deterministic

B(I”x9—valued SSP with F as its spectral distribution. Then

23

F = Fn'+ FQ’

where F is a.c. and gs-(F (de)x)(y)) = (@(eie)x,§(eie)y),

i n
e) as before

n
9(e

= I . .
Proof. Let g“ n“ +V€n be Wold s decomp051tion of g“,
-m4< n < m. Then by part (iii) of theorem 3.3.5 we get

1 2"

—g e'i“9(<F(de>x>y>)

2n (R

g(n)X)(y) = (SnX. éoy)

(nnx +. gnxa Soy + C’OY)

(“nx’ Roy) + (gnx, goy)

2n .
1 .-
358 e 1n9[(Fn(de)X)X) + (Fg<de>x>(x>1-

Thus F(de) = Fn(de) + F (de). Now by theorem 3.3.10 FT} has the

C

required properties.

3.3.12 Theorem. Let F be the Spectral distribution of a B(I,X)-
valued stationary process §n, -m < ni< m. Then En, -m < n < w

is regular of full rank iff F is a.c. and g6'(F(d9)X)(X)) =

o 2 a -
“0(ele)xn , where §(e16)x e z e ikfiARx, with An 6 BCI’X9 and
k=0
*
ADA0 being invertible. Furthermore if we assume that F' is

bounded operator valued then 6 is also a bounded Operator valued

function.

Proof. Because of theorem 3.3.10, part (b) it suffices to Show

i ” -ike
that, for the function 6(e 9) = E Ake ,
k=0 m

*
= i =
have C AOAO. To see this, set 5“ REOIh-kAk’ where

in that theorem we

Ih.€ 3(I:X9 With (Sh:¢h) = 5nm1' Then consider the SSP g],

-w.< n.< a, defined by QOX = §6X ‘ (56X\H§'('1)) = quOx.

Now for each x and y in I we have

24

*
(G§.x)(y) . (camcay) .. (cpvox,<pvoy) = (AOX’AOY) = (AOAOX:Y)-
*

Er], I» < n < co have the same covariance structure. Hence by

But one can see that g“, -co < n < co and

ienna 3.2.8 0,; - cg“ Thus c;g . AEAO. The proof of the last
statement Of the theorem is clear. In fact one can Show more.
See remark 4.2.1.

The following is an extension of a result due to Payen
[18], pp. 371-372.
3.3.13 Theorem. Every B(I,)() -valued SSP g“, -cn < n < on is the

sum of three processes
1 2 3
an an + an + §n .

which are mutually orthogonal, the first one being regular, the
second one being deterministic with a.c. Spectral distribution
and the third one being deterministic with Singular Spectral dis-
tribution.
3%. Let un and a be as in lemma 3.3.1. Then let un =
u: + u: + u: be the correSponding decomposition of the process
un’ ~09 < n < o, given in [18], pp. 371-372. Let g: I nigo,
i I 1,2,3. Then gt, I: < n < on are mutually orthogonal SSP 'S
because uni, m<n<¢ are so. Since ut, m<n<oo is
regular as in the proof Of Wold's decomposition theorem, we can
see {'1', «a < n < a is regular. Now to Show that 5:, -co < n < on
is deterministic with singular Spectral distribution it suffices
to Show F 3 is singular. (By Wold's decomposition theorem.)
But (F 3(de)a) (a) .L d9 for all a 6 4- Hence

u

((F 3(de))(§ox))(§ox) J- de for all x E I which implies that
u

25

(F 3(de>x)(x) - (:31? 3(de)§0)(x))(x)- Hence (F 3(dense) a de
fog all x E I, which means F 3 is singular. Finally by a
sflmilar argument one can Show that F 2 is a.c. using the fact
that F 2 is a.c. SO we just have to show that §:, -m«< n < m
is detetministic. Suppose this is not the case, i.e. suppose
there exists 0 I a 6 H 2(0) 9 H 2(-l). SO a i.§:x for all

x 6 I and all k s -1. Also a i.§:x for all x 6 I, all k
and i = 1,3. (Because a E H 2(0) and §:, -ml< n < m is
orthogonal to ti, -mi< n<< m gand 5:, «n.< n < m). Hence

a l.§kx, for all x 6 I, and all k s -1. Hence a l.H§(-1).

SO we get
a .LH§(-co) . (1)

On the other hand H 2(0) G H 2(0) I H 2(-m), because ui,

-¢.< n < w is deterministic.u 80 H 2t(0) :‘H 2(-a0 CiHu(-m), by
the choice of un, -m‘< ni< m, see [18], pp. 371-372. Hence

H 2(0) CIHu(-m) I H§(-a0 by lemma 3.3.6, part (a). Thus

§

a E H (-a0 because a E H 2(O), by the choice of a. But this

E
and (1) implies that a I O, which is a contradiction.

The following corollary gives an extension of Cramér's
decompos it ion theorem .
3.3.14 Corollagy (Cramér's decomposition). Let F be the spectral
distribution of a B(I”x)-valued SSP 5“, -m < ni< a then we can
decompose F as
(a) F I F1 +F2 +-F3, where F1 is a.c. and spectral distribution

of a regular process, F2 and F3 are Spectral distributions of

a deterministic process with F2 being a.c. while F3 is singular.

26

(b) F = Fa +-F8, where F8 is a.c. and F3 is singular.

Proof. (a) Let §n I g: +'§: +'§: be the decomposition of

fin, -o«< n«< o as given in theorem 3.3.13. Then Since these are
mutually orthogonal processes by the standard computations one

can inmadiately see F I F + F + F . We can take F, = F
:1 :2 :3 1 S

i
for i I 1,2,3.

(b) Let Fa I F1 +F2 and FS = F3, then obv10usly F8 18 a.c.
and FS is singular.

We conclude this section with the following theorem which
gives a sufficient Devinatz's type condition for the factorability
of a the spectral distribution of a Banach Space valued SSP.
3.3.15 Theorem, Let gn, -m < n < m be a B(I”Y)-valued SSP with

a bounded Spectral density f satisfying

S

211 .
A long;1(e19)u-1de > -m . (1)

Then f is factorable or equivalently g“, -m.< ni< m is

S

regular.
211
Proof. We have (gox,§0x) = (Rox)(x) a it] (f§(eie)x)(x)de.

Hence for each x E I, we have
2T! 211' _ .-
Héoxnz ' '21:; (f§(eie)")°‘)d9 2 fl uxnznfeleihu 1..

2n .
-- uni; i"; \\f;‘<e1°>u“-de = uxnzx.

211
where I I %:'.1[ “f;1(eie)u-1de, Obviously O < I < on. By theorem

3.3.6 we have (fgx)(y) I fu(§ox,§oy) for all x,y E I. So

\fu(eie)(a,b)\ . \(f§(eie)§61a)(§61a)\ s “f§(eie)““§61a“

s new”)nugaluzuauz . r ‘1\f§<e“’>\mau2.

27

for all a E 4. Hence fu(a,b) is a bounded bilinear form and
hence there exists an operator valued function fu(eie): 4.. a

such that (f§(eie)x)(y)I (§ofu (e1 e)gox)(y). ‘Now (1) means that

2n (e e)X)(X)
A log inf __§_, ]de > -m . (2)
0 61 quz

 

Hence we get

 

 

 

ie * is
2n (fu(e )x)(x) 2n (éofu(e )§OX)(X)
logfinf 2 ]de I log{inf 2 ]de > -m.
mixer uxu 0+x€i HXH
Thus
2n (f u(§QX))(§ x)
£0 log inf d9 > -m .
61' “go“ flux”
Hence
f (f “a)(a)}d
log in de > -co .
A0£a6501\\anz

But Since (:01 is dense in a we get

2n (f a)(a)
log]: inf __l_1__2__} > -oo .
866? “8“

Hence the associated process un’ -m < n < m is regular.

By lemma 3.3.7 we see that our process gn, -or< n < m is regular.

3.4 Subprocesses g d Spectral Conditions for Factorability of the
Spgctral Density. In this section we extend to the Banach
space case the results of R. Gangolli on subprocesses and their
relation to the process itself and to the factorability of its
Spectral density. Making use Of the results of §3.3 the technique

employed by Gangolli can be used to establish our results.

28

3.4.1 Definition. Let En, -oo < n < on be a B(I,7() -valued SSP,
and let 9 be a subspace of I, then the SSP gnw, -co < n < co
is called a subprocess Of gn, -co < n < on.

Note that in case that 9 is complementary, gn‘e can
be identified with §nP where P is the projection on 9. Hence
in the Hilbert space case this definition coincides with Gangolli's
definition. Since we will be mostly working with finite dimensional
subspaces, which, are complementary we sometimes use gnP
instead of §n|9° Hence gn‘g E B(I,9() .
3.4.2 1:913:33. Let gn, -oo < n < an be a B(I,)() -valued SSP and 9

be a subspace of I. Then

G9>G,

i.e. (69x) (x) > (Gx) (x) for all x E 9, here G and G9 are
the predictor error operator of gn, -m < n < co and §n\9,

-co < n < co respectively.

PM. Since H

(n) C H (n), for each x E 0 we have

§|9 E
(0920 (x) = (Sox - (gammy-1)). S02: - (gox‘H§o\9('m' Hence

(69x) (x) 2 (ng - gox‘H§(-1), gox - §0x\H§(-1)) I (Gx) (x). Hence
(69x)(x) 2 (Gx)(x) for all x 6 9 .

3.4.3 Notation. Denote by AM?) I inf{(G0x) (x) , “x“ I l, x E 9].
The following theorem will be useful later.
3.4.4 Theorem. Let gn, -oo < n < on be a B(I,7() -valued SSP, then

En. '69 < n < I is of full rank iff

inf{L(G0)\0 a finite dimensional subspace of I] 2 c2 > 0. (l)

29

Proof. If §n, -I'< n <:m is of full rank then inf (Gx)(x) I
HXH=1

c2 > 0. Hence by lemma 3.4.2 Am?) 2 c2 for all 9, a sub-

Space of ‘I. 'Now suppose that (1) holds. We must Show G 2 c2.

Suppose not. Then there exists x 6 I with “x“ I 1 Such that

(Gx)(x) I c'2 < c2. But (Gx)(x) I (gox - (§0x\H§(-l)), gox -

(§Ox\H§(-l)). Hence “gox - (§0x\H§(-1)H I c' < c. Thus, the
distance of gox from H§(-l) I c' < c. Therefore there exists

numbers aik’ j I -l,-2,...,4N, k I 1,2,...,N and vectors

xk E I, k I 1,2,...,N such that

-N N
Hgox - j=§1 kglajkgjxj“ < c.

Letting 9 I €5[x,x1,x2, . . . ,N} we then have

-N N
MG?) s Héox - (€0Px\H§P(-1))l\ s “sox - . E 5181*:

x.“ < c.
jI-l k— J

J

This is a contradiction to (1).
3.4.5 Remark. The finite dimensional subspaces are essentially
multivariate SSP in the sense of Weiner and Masani [25] and in
this case hm?) is the smallest eigenvalue of the matrix 69'
As was noted in [17] there are errors in theorem 5.3 and
5.4. Because the proof Of theorem 7.3 depends on theorem 5.3
this theorem is also in doubt. Using the result of [17], p. 405,
we extend correct versions of Gangolli's result to Banach space.
The next theorem gives a concordance between these two de-
compositions.

3.4.6 Theorem. (Concordance theorem). Suppose the B(IHX)-valued

SSP g“, -m‘< n < I has full rank” Let F I Fn‘+ FQ and

30

F I Fa + F8 be Wold's and Cramér's decomposition of F, the Spectral
distribution of gn, -c° < n < co. Assume that F'(eie) is bounded

and has a bounded inverse for almost every 9. Then

10

Proof. Take some Such that F'(e 0) is bounded and boundedly

e0
invertible. Then by a lemma in [ l], p. 21, there exists a Hilbert

10
space a and a bounded operator T: I I a such that F'(e 0

190

) ==

*

T T and range of T dense in 4. Now Since F'(e ) has a

bounded inverse T is onto and has a bounded inverse. Now define
-1 A

the process un, -co < n < on by una gnT a, a E d. Then uni

-co < n < on is a B(d,7() -valued SSP and we have gm I unT. One can

Show that g“ and un and T satisfies most of the prOperties

we proved about gn, un and go in Section 3.3. In particular

the results 3.3.3, 3.3.4, 3.3.5, 3.3.7, 3.3.13 and 3.3.14 hold.

Now let vn, wn and 'nn, gn be the components of Wold's
decomposition of the processes On, I» < n < co and g“, -ao < n < co
respectively, as in theorem 3.3.5. Let “3’ -oo < n < co and
g:, -co < n < on, i I 1,2,3 be as in theorem 3.3.13. Now as in the

I 19 * I is .
proof of theorem 3.3.15, we have F (e ) I T Fu(e )T and
*- -
F&(eie) . T 1F'(eie)T 1. Thus 3&(e19) is bounded and has a
*
bounded inverse a.e. We also note that GgI T GUT and hence un,

«9 < n < I is of full rank. From the results on page 405 Of

[17] we get
(F) 3F - (1)

i .
But we have “n I vnT and g: I unT for all n and i I 1,2,3

31

(c.f. theorems 3.3.5 and 3.3.13). An argument similar to the proof

of theorem 3.3.3 may be used to show that

* *
F = T F T and F = T F T, i = 1,2,3. (2)
n v i 1
§ u
We also have
(Fu)a = F 1 + F 2 and F8 = F 1 + F 2 . (3)
u u g g
By (3) and (2) we get
*
Fa = T (Fu)aT . (4)

Now by (l), (4) and (2) we get Fa = F,n which is the concordance.
Now as a corollary we have the following theorem.
3.4.7 Theorem. ‘Let F and g“, -m < n < m be as in the previous
theorem. Then the SSP g“, -m < n < m is regular iff the follow-
ing two conditions hold
(1) F is absolutely continuous
(ii) there exists constant c > 0 such that 3196) 2 c, for all
finite dimcnsional subspaces 0.
2522;. If g“, -w‘< n < m is regular of full rank, then by theorem
3.3.10 F is a.c. and (ii) follows from theorem 3.4.4.
Now suppose (i) and (ii) hold. Then from (ii) and theorem
3.4.4 it follows that the process is of full rank and hence by the
concordance theorem we get FC = F8. But FB = 0 by (1), hence
Fc - 0, so P 8 F“, i.e. gn, -m‘< n'< m is regular.
3.4.8 Notation. Let R be an n X n positive matrix with eigen-

values *1 5 k2 <...s x“. Following Gangolli for a 6 [0,1] we

1 1 1
let [“1’°'Z"'°’°’n] - 0&3, ; ,..., H] + (1 - o)[1,0,0,...] and

32

a1 U2 (In
define A(o.R) = A1 , )‘2 ,...,\1 . Note that A(0,R) is the

smallest eigenvalue of and A(1,R) is the n-th root of deter-
minant of R and for a fixed R, A(a,R) is a continuous increasing
function of a on [0,1]. We also note that P*FP is the spectral
distribution of the finite dimensional subprocess §nP. Next
theorem deals with the evaluation of 3‘06) in terms of F.

3.4.9 Theorem. Let gm, -m.< n < m be a B(IuX)-valued SSP with
spectral distribution F, which has a bounded derivative F' so
that F(d9) = F'(eie)de + F8(d9). Suppose that 9 is a finite

dimensional subspace of dimension n then there exists a unique

09
o = 0(60 in [0,1] such that

1 2" * '
53g 108 A(o(P), P F P)de = log MG?)-

211 *
Proof. Define f(a) = %;'3 log A(a,P F'P)de then f(0) =

1 2“ * fl *
55g log A(O,P ,F'P)de = log MP F'P)de. Hence

f(0) = log .MG'9) .
0n the other hand

n
1 2" *' 1 2" 9"‘7‘3—
f(1 = 3:4; log M1,? F P)de = fig 10g ./dec p F P de

1 2 * l
= — ' g - = .
2 ! log det(P F P)de n log det G9 log 1(60)

Now f(a) being a continuous, increasing function on [0,1]
(see p. 907, [ 8]) and since 13(0) 5 log 1(69) 5 f(1) we see
that there exists a unique a = aCéD such that

2n

f(o) = -§; I log Mam), P*F'P)de = log M69).
0

33

3.4.10 Theorem. Suppose fin, -mi< n < m is a 3(15x9-valued SSP.
Then g“, -¢«< n < m is of full rank iff each finite dimensional
subprocess is of full rank and

Zn *

g log A(a(P)), P F'P)de 2 -c > -m,
where F' is the bounded Spectral density of F, any g(P) is
as in theorem 3.4.9. P is finite dimensional.
3.4.11 Theorem. 'Let gn, -o < n < m be a B(I”X)-valued SSP with
distribution F. For gn, -m < n < m to be of full rank it is

necessary that for all a, S a S 1 we have

“4
2n *
F log A(a, P F'P)de 2 -c > -m

and it is sufficient that for some a, 0 s a s o_ we have

2n *
8 log A(o,P F'P)de 2 -c > -m,

where P is any finite dimensional projection and F' is the

bounded derivative of F. Here ¢r+ and a- are 1.u.b. and
g.u.b. of the set {a(P), P finite dimensional projections}.

' 19 + *
3.4.12 Theorem. Let f(e ) be a B (1,1 )-valued function on

the unit circle. Then (f(eie)x)(y) = (@(eie)x, Q(eie)y), where

19 ” -1ke *
§(e ) 8 z le with QOQO invertible iff for each finite
' k=0
* *
dhmensional P, P fP admits a factorization with Q Q in-

O,P 0,P
. * _1
vertible. Furthermore in this case H(QO’PQO’P) H s c < a, where

c is a constant independent of P.

Proof. (a) Clearly if we have (f(eie)x)(y) = (Q(eie)x,§(eie)y)
* 19 1.9 19 *

then ((P f(e )P)X)(y)) - ((§(e )P)X. (¢(e )P)y)- Hence P fP

*
is factorable. Now as in the proof of theorem 3.3.12, QOQO is the

34

predictor error operator G, of the corresponding SSP and hence

*
the invertibility of Q0 PQO P follows from lemma 3.4.2.
*
(c) Since Q0 PQO P have bounded inverses then by lemma 3.4.4, the

corresponding process which has density f is of full rank and

hence by the concordance theorem Fa = F so F = F1 = F“, i.e.

f = fu, but by theorem 3.3.10 fu, and hence f is factorable. Now
* *

invertibility of QOQO follows since QOQO = G as above.

3.4.13 Theorem. Let q+ and a be as in theorem 3.4.11. Then

. * . .
for f to be factorable as f(ele) = Q (e19)Q(ele), where

19 m -ine . * . .
Q(e ) = 2 e Q , with Q Q invertible it IS necessary that
n O 0
n=0
for all a, a s a S l we have

+

2n *
g log Mo, P fP)de 2 -c > -0.

and it is Sufficient that for some a, 0 S a s a_

211 *
lg log Mo, P fP)de 2 -c > -o:,

where P is any finite dimensional projection and c is inde-
pendent of P.

3.4.14 Remark. If we put a = O in the second part of the last
211 *
theorem we get that the condition g log L(P fP)de 2 -c > -m is

ie *

* *
sufficient for factorability of f as f(e ) = Q (e e)Q(e 9),
. w _. * p
where Q(ele) = 2 Q e Ike, with Q Q invertible. This is an

improvement on theorem 3.3.15.
3.4.15 Remarks. (a) The proof of lemma 3.3.4 can be simplified

considerably. Note that H (n) CZH (n) and for a 6 a3 a = lim gox
g uk k m’“ m
for some sequence {xm}<: 1, hence U a = lim U goxm = lim §kxm

which gives Hu(n) c H (n). Also the proof of theorem 3.3.5 can be

§

35

directly obtained by using projections in B(IWXQ (see theorem 3.2.5)
and standard methods.

(b) Throughout this work we shall work with the assumption
that K' is separable. In case xn, -m‘< n«< m is a stochastic pro-
cess taking values in a separable Banach Space I then the relevant
Hilbert space is X = 5{x*(xn), x* E 1*, n E Z}. It can be shown that

2
under the condition Euxou < m (in particular where x is Gaussian)

O
*

X’ is separable. We note that we do not use here separability of I .

3.4.16 Remark. Suppose the covariance operators Rn’ -m«< n < w

is given. Let g“, -mm< n < m be the SSP given in theorem 2.4. In

the next chapter we assume that the condition of I under the norm
|\|x\“ = (R(0)x)(x) is separable. In this case we can show that
H§(n) = 6{§kx, x 6 IO, k S n}. Thus for the study of predict ion prob-
lem the relevant factorization problem can be studied with I and

X' being separable. AS remarked before, this assumption is satisfied

in several cases.

CHAPTER IV

FACTORIZATION OF NONNEGATIVE OPERATOR VALUED
FUNCTIONS ON A BANACH SPACE

4.1 Introduction. The main purpose of this chapter is to extend
most of the results of R.G. Douglas [ 4] on factoring nonnegative
operator valued functions on a Hilbert space to nonnegative operator
valued functions on a Banach space. As we mentioned before the
problem of factoring nonnegative operator valued functions on a
Banach space plays an important role in the study of Banach space
valued stationary stochastic processes (c.f. Theorems 3.3.10

and 3.3.12).

We remark that our definition of "factorization" is exactly
what Douglas called "conjugate factorization". However all our
results have dual statements and hence we have the extension of
Douglas' results.

When f is a positive operator valued function on a
Hilbert space, J}, the Square root of f whose existence is known
plays a significant role. But when f is a positive operator
valued function on a Banach Space I the existence of a square
root in the ordinary way does not make sense. Nevertheless we
can prove (c.f. lemma 4.3.1) the existence of a measurable operator
valued function A on I into some auxiliary Hilbert space which
behaves almost like a square root in the sense that f = A*A. The

operator valued function A, called a quasi square root, enables

36

37

us to extend to the Banach space case a lemma of Helson [10],
p. 117 and the main lemma of Douglas [l+].

In §4.2 we set up necessary terminologies and state some
known results. Section 4.3 includes the proof of existence of a
quasi square root and two lemmas on the characterization of
factorability of a positive operator on a Banach space. The re-
sults of §4.4 extend in a natural way most of the work of R.G.
Douglas [4 ] to the Banach Space case. In establishing these re-
sults we make use of our fundamental lemmas proved in §4.3 and

Douglas' techniques employed in [4 1.

4.2 Ancillary results. In this chapter all the Banach spaces
and Hilbert spaces are separable. We recall that if f = f(eie)
is a weakly summable B+(I,i*)-valued function on the unit circle,
then we say f is factorable if there exists a Hilbert space X'
and a conjugate analytic B(I,X)-valued function A = A(eie) such
that f(eie) = A*(eie)A(eie), in the sense that (f(eie)x)(y) =
<A<e1°)x. A(eiems x,y e 1.
4.2.1 Remark. With the notation of the last paragraph we can
show that

m

A(eie) = z Ake
k=0

-ike
where Ak's are bounded, in fact we can Show that
Q

kgouAkxn; s c M2, x e x .

for some finite constant C.

To prove these we proceed as follows. It is easy to see

1 °° -in
that A(e 9) - E A e 9’ where A

k are linear operators
k=0

I
k S

38

” 2 2“ 19 2
defined on 1. We observe that z “Alec“ = (E “A(e )x“ d6 =
211 19 k=O X X
{g (f(e )x) (x)de < on. To complete the proof it is sufficient to

i
Show that the operator T: I -» L200 defined by Tx = A(e 9)x
is bounded. By the standard method [11], p. 85, one can Show
that T is closed, and being everywhere defined, it follows that
T is bounded.
i9 + * ,

Let f = f(e ) be a B (1,1 )-valued function on the
unit circle. Theorem 2.4 shows that there exists a stationary
process 5“, -co < n < co, gm: 1 - K, for some Hilbert space X,

whose spectral density is f(eie). Letting U(§nx) for

= §n+1x ’

all x E 1 and -co < n < on, we obtain a unitary operator on
5(§n1, -°° < n < m)‘

Let una = Una for each a E d = 6(§01). It is clear that un,
~00 < n < co, is a stationary process on a into K. Call Fu
the Spectral distribution of un. It is easy to see that Fg =
T*FuT, where T = :0 and F§ is the distribution of gn.

4.2.2 m. With the above notation, the distribution Fu is

a.c. w.r.t. the Lebesgue measure, and its density fu is a bi-

linear functional on a X 4. Moreover we have
(fX)(y) = fu(Tx,Ty); x,y 6 I . (1)

M. Let a B '15: for some x 6 1. Since (F§(de))x) (x) is

a.c. w.r.t. the Lebesgue measure and (F§(de)x) (x) = (Fu(de)Tx) (Tx) =
(Fu(de)a) (a), it follows that (Fu(de)a)(a) is a.c. w.r.t. the
Lebesgue measure for all a’e T(1) . By §66 [9 ] we have that

(Fu(de)a) (a) is a.c. on a. Now (1) easily follows.

39

4.3 'Main Lemmas. In this section we will Show the existence of
a quasi square root for a B+(1,1f)-valued function on the unit
circle. We then extend a lemma due to Helson [10], p. 117, and
the main lemma of Douglas [it] to the Banach space case.

Our first lemma is on the existence of a quasi square
root.
4.3.1 Lgmma, Let f be a weakly summable B+(1,If)—valued func-
tion on the unit circle. Then there exists a Hilbert space X
and a measurable B(1,7() -va1ued function Q = Q(eie) on the unit

circle such that

19

f(e )=Q*(eie)Q(eie) a.c., (4.3.2)

in the sense that

(f(ei9)x)<y) = «2(e19>x, Q<e‘°)y), x,y e x .

Proof. By lemma 4.2.2 there exists a Hilbert Space an an Operator
T in B(1,d), and a bilinear functional g on a X a with

a = 6(T1) such that
(f(eie)X)(y) = g(eie)(TX.Ty); for x,y 6 I. (1)

Let {xi, 1 s i.< a} be a countable dense subset of 1. Consider
Q Q
[Txi}1=1 and let {ei}1=1 be the Gram-Schmidt orthogonalization

of {Tx Set

131=1'
s1j<eie> = g(e‘°)<ei. ej>

It is clear that gij(eie) defines a nonnegative matrix (not

necessarily bounded). The result of p. 112 [10] can be applied

40

to Show the existence of a Hilbert Space X’ and a sequence

{F of functions in L2(X) such that (e19) =

111=1
(F1(e19). Fj(e‘°)>

gij
X” Following [10], p. 113, we obtain an operator
A on the finite linear combinations of {ei]:=1 by

N N
A( B a.c.) = 2 a F. - (2)
i=1 1 1 i=1 i 1
It is clear that
u N 2 N ' 19
A( 2 a e )H = Z a a 8 (e ) - (3)
i=111 X i,j=lijij
N
Let x 6 (xi, 1 S i < a]. Then Tx = 1Elaiei, for some at,
1si$N<oo. Thenby (3) wehave
N
2 - ie
ATx = , 4
H “x' 1’§=181 jgij(e ) ( )

By (1), (4) and bilinearity of g we have

i, 1 S 1 < m}. (5)

2 i
“ATxHX = (f(e e)x)(x), x e {x
Because f is weakly summable by (5) we have that the operator
AT: 1 ... L200 is densely defined. For almost all 9's we de-
fine B(eie) on {xi, 1 s i < m} by B(eie)x = (ATx)(e16).
Extend B(eie) to 10, the finite rational linear combinations of
I

x1 8 through linearity. From (5) we obtain

(B(e19)x, B<e1°)y) = (f(eie)x)(y), x,y e 10' (6)

This shows that for almost all e's, B(eie) is well defined and
bounded on 10. Hence a.e. B(eie) has a continuous extension
to 1. Call this extension (2(eie). It follows by (6) and the

continuity of Q and f that

41
Q*(eie)q (e19) = f(eie).

Furthermore from the continuity of Q(eie), and (2) it follows
that Q(eie) is weakly measurable. This completes the proof.

Lemma 4.3.1 allows us to make the following definition.
4.3.3 Definition. The function Q ==Q(eie) in lemma 4.3.1 is
called a quasi square root of f.

In the absense of the square root, a quasi square root
has almost all the desired properties. One of its applications
is demonstrated in the following lemma where we extend to the
Banach Space a result due to Helson [10], p. 117. First we
introduce the following definition.

4.3.4 Definition. Let f be a weakly sunlnable B+(1,1*)-valued
function on the unit circle. Let Q be a quasi square root of

f with values in B(1,7(). 77((0) will denote the smallest sub-
Space of L2(X) invariant with respect to U (U is multiplica-
tion by e-ie) that contains the functions {Q(eie)x : x E 1}.

N
A function p(eie) - Z e-1n9x

n=0
conjugate analytic trigonometric polynomial and the set of all

with xn 6 1 is said to be a

such polynomials will be denoted by u. It is clear that 7K0)
is the norm closure of Q(u) in L200 .

4.3.5 Lemme, Let f be a weakly summable B+(1,1f)-valued func-
tion on the unit circle. Then f is factorable iff for any Q,
a quasi square root of f, WKQ) contains no non-trivial reducing
subspace of U.

M. Suppose 77((Q) C L200 contains no non-trivial reducing

subspace. Then, [10], p. 61, it has the form V(Lg-W)), where

42

V is a measurable isometry Operator on fl' into x2 In particular

0.
Qx 8 VGX, Gx 6 L2 0V). Therefore

<fx><x> = (QX.QX),( = (vex. V9.97. = (ex, (33%. <1)

Define the operator Q(eie) on 1 into K’ by
Q(eie)x = Gx(ele), x E 1 a.e. (2)

It is clear that Q(eie) is linear, and moreover by (l) and (2)

we have

uueihxui = ucx<e19>uf<
(3)

_ ie 16 2 = iQ

-“V(e )Gx(e )“y (f(e )x)(x),
Hence Q is bounded. Then by (2) and (3) and the weak summability
of f it follows that Q is a conjugate analytic B(1dK)-valued

function. Hence

i9

f(e ) = Q*(eie)Q(eie). (4)

By (4) f is factorable.

Now assume f is factorable, say f = Q*Q, where Q is
a conjugate analytic B(15XQ-valued function. Let Q be a B(1JV)-
valued function which is a quasi square root of f. We can compare

Q and Q as follows. Define
V(§P) =QP, P E u - (5)

We have (V(Qp) . V(Qp)) = (Q1). Qp) =
211 L26”) L200
%fi'£ (f(eie)P(e19)(P(eie))d9 = (9?, Qp)L2(x3. Hence we can extend

V to an isometry on 77((Q) onto 77((Q), where 77((Q) = 6(Qp, p 6 u).

43

This mapping commutes with multiplication with ede. Now 7/((Q)
contains no non-trivial reducing subspace of the shift U, be-
cause it is a part of Lgnoo. Hence its image 77((Q), under V
cannot contain a non-trivial reducing subspace.

Now we can extend the main lemma of Douglas as follows.
4.3.6 LSEEE: Let f be a weakly summable B+(1,1f)-valued func~
tion on the unit circle. Then f is factorable iff for each non-
zero function g 6 77((Q), the measure of Zg = {eie, g(eie) = 0}
is positive.
2322;. This follows from lemma 4.3.5 and the fact that an in-
variant subSpace of the shift U contains a non-trivial reducing

subspace of U iff it contains a non-zero function g for which

the measure of 28 is positive.

4.4 ‘Main Theorems. In this section we extend most of the results
of Douglas [4'] to the Banach space case. Lemma 4.3.6 is repeatedly
used in the course of the proof of our theorems.

4.4.1 Theorem. Let f1 and f2 be weakly summable B+(1,15)-
valued functions on the unit circle and. Q1 and Q2 be B(I,X)-

valued quasi square roots of f1 and f respectively Such that

2
(a) f2(eie) 2 f1(eie) a.e.,
* 'k
(b) 72(Q2) 2 7((QI) a.e-.
(c) o(e‘9)\h1(eie)x\\x 2 \\Q:(ei°)02(eie)xu ,, a.e..
'1

where m is some nonnegative scalar valued function. If f1

is factorable then f2 is factorable.

44

Proof. Let u be the set of all conjugate analytic polynomials

in 1, Suppose g E WKQZ). Then there exists a sequence {pa}:=1

in 1.1 such that {Q2pn}n=l

converges to some g in L2(X).
Now f (e16) s f (e19) a e implies that {Q p )m is a Cauchy
1 2 ° ' 1 n n=1
sequence in L200. Therefore there exists some h 6 722(Q1) such
Q a
that fillpn}n=l converges to h in L2(x9. We choose a sub-

sequence of pn, denoting it again by pn, such that

Q1(eie)pn(ele) converges a.e. to h(ele) in X'

(1)
Q2(eie)Pn(eie) converges a.e. to g(eie) in xx
By (a) and (1) we have
\\h<eie>\\,( s \\g(eie)\\x a.e.
Hence
the measure of 22,2h is zero. (2)

It follows from (c) that

\p’iceihgeiemf 11m \p:<ei°)c22<e19>pn<e19)\\1,

n—oa

IA

mammal \\Q(e19)pn<ei°>\\,(

n—OQ

cp<e19>\\h<ei°>u,(-

Hence by (b), for almost all e's we have the following implica-

tions.
Me”) = o = Q:(eie)g(eie) = o =5 Q:(eie)g(eie) = o

=9 g(eie) E ’IKQZ).

45

But on the other hand by (1) we have g(ele) 6 closure of range
of Q2(eie). Hence g(eie) - 0 a.e., because closure of range

*
Q2919) is a subset of 72L(Q2(eie)). Therefore
Z§ Zg has zero measure. (3)

(2) and (3) imply that Zh and Z8 are a.e. equal. Now apply-
ing lemma 4.3.5 we conclude that f2 is factorable.
4.4.2 m. In case 1 is a Hilbert-Space with Q1 = ff;
Q2 -/F2', (a) and (b) imply that 72(f1) = 72(f2). Also in this
case, condition (c) is the same as cp(e19)f1(eie) 2
Q2(eie)f1(eie)Q2(eie) a.e. Hence our result 4.4.1 extends the
main theorem of Douglas [4 ].

The following theorem does not seem to follow from theorem
4.4.1. However we provide a direct proof of it based on lama
4.3.5.
1 and f2 be weakly summable B+(1,1*)-

valued functions on the unit circle such that

4.4.3 Theorem. Let f

f2(e19) 2 g(e“) 2 ¢p(eie)f2(eie) a.e.,

where (90316) is a positive scalar valued function. If f1 is
factorable, then f2 is factorable.

111133;. By lemma 4.3.5 it is sufficient to prove that for a non-
zero g in 7R(Q2), the measure of Z8 is zero. Let QZPn «g
in L2(7(2), then f1 5 f2 implies that there exists h E L2(,(1)
such that len .. h in L2(K1). Choose a subsequence of p“,

denoting it again by pn, such that

46

Q2(eie)pn(eie) converges a.c. in Xi to g(eie)

(1)

Q1(eie)Pn(eie) converges a.e. in *3. to h(eie).

By (1) we have, a.e., “h(eie)“x,s “g(eie)“x,. Hence the measure
1 2
of zgxzh = 0. Similarly by (l) and assumption for almost all

e's we have

 

1 n .. 19 ie
“g(e 9>\\,(2 3:102“ ”g(e >sz
1 1 1
s 16 lim W21(e e)pn(e 9)“

@(e ) n-cco K1

1 ie
“h(e ) .
(6,9) “*1

 

Hence the measure of Zfi\zg is zero. So we have shown that

Zh - Zg a.c. Now by lemma 4.3.5 and factorability of f1 it

follows that f2 is factorable.

The following theorem is a slight extension of theorem

4.4.1.

4.4.4 Theorem. Let f1 and f2 be weakly summable B+(1,1f)-

valued functions on the unit circle such that

(a) f2(eie) 2 m(eie)f1(eie) a.e. where m(eie) is non-
211
negative scalar valued and 5 log m(eie)de > -m,

(b) m2) :2 moi). a.e..'
(c) =p<e1°>an<eie>xnx 2 \hZceiemleienn ,, me-
I

where m is a nonnegative scalar valued function.

If f is factorable, then f

1 is factorable.

2

47

Proof. The proof is a combination of a standard method and theorem

4.4.1. Let t(eie) B 1 A m(eie). Then 0 s t(eie) s l a.e. and

Zn 16 16

g log t(e )de >t-w. By Szego's theorem there exists k=k(e ) in the
i6 ‘

Hardy class H2 such that t(e ) = ‘k(eie)\2. Assuming f1(e19) =

Q*(eie)Q(eie) we have
c(ei°>f1<eie> = (R(eihueie)>*(1'<<e19>s(e19>).

Applying theorem 4.4.1 to f1(eie) and t(eie)f1(eie) we conclude
the factorability of f2.

We now state the following extension of theorem 4.4.3,
whose proof is omitted.

*
4.4.5 Theorem” Let f and f be weakly summable B+(I,I )-

1 2

valued functions on the unit circle such that
16 19 19 ie 19
f2(e ) 2 m(e )f1(e ) 2 m(e )f2(e ) a.e.,

where m and m are positive scalar valued functions, with

is factorable, then f is factor-

211 19
11 log m(e )de > -co. If f 2

I
able.

Now we shall give some Devinatz' type theorems. First
we introduce the following definition.
4.4.6 Definition. Let f be a B+(1,1f)-valued function on the

unit circle, we say that

(a) f has a "conjugate analytic null function" if

(i) ‘n(f(eie» is complementary a.e. and

(11) “Weigh“: -= (s(ei°)x, Q(eie)x)x; x e I.

where P is the projection into the complement of n(f(eie))

48

along ‘fl(f(eie)), and Q is a conjugate analytic B(1”x3-valued
function.
(b) f has a "quasi conjugate analytic null function" if condi-

tion (a) (1) holds and (a) (ii) replaced by
' i i 2
(Q(ele)x, ¢(e 9>xzx s HP<e as“I

s o<eie)(s<e‘9)x, s(eie)x> . x e x,

K'

where m is a positive scalar valued function.

We remark that in the Hilbert space case the termonologies
"conjugate analytic null", "quasi conjugate analytic null" and
"conjugate analytic range" are all equivalent.

The following result is a generalization of theorem 2
of [ 4] to the Banach space case.

4.4.7 Theorem. Let f be a weakly summable B+(1,1f)-valued
function on the unit circle such that
(a) f has a quasi conjugate analytic null function,

(b) <f<e19>x><x> 2 n<eie)nr<e‘°)xui,

2n .
is a positive valued function with I log n(ele)d9 > -w.

0

where n(eie)
Then f is factorable.

2322;, Let t(eie) ' l A n(eie). Then as in the proof of theorem
4.4.4, t(e19) = ‘p(eie)‘2, with p(eie) 6 H2. Since f -=Q*Q,

71(f) - 71(Q), and (f(x))(y) = 0 if either x or y is in 71(Q).

We then have
n(eie)uP(eie)x“i s (f(e1°)x)(x) s \f(e19)\2\\p(e19)xni. (1)

By (1) and our assumptions we have

49

(1(eie>x, 1(e19)x) s (f(eie)x>(x> s “f(ei9>uo<eie>(9(2i9>x.1<eie)x>

X K
where m is as in definition 4.4.6(b). Hence we have
(p(e19)s<ei°>)*(p<e19)1<eie)> s f(eie) (2)

s iceie)<6<e19>1<e19>>*<5(ei°>s<eie)).

where w = \f‘qflt. Hence by theorem 4.4.3, f is factorable.
Using this result we can prove the following theorem
which generalizes Devinatz' theorem [10], p. 119 to the Banach
space case.
4.4.8 Theorem. Let f be a weakly summable B+(1,1f)-valued func-
tion on the unit circle such that f-1(eie) exists a.e. and is
11’

bounded. If & log[“f-1(eie)“-1]de > «m, then f is factorable.

Proof. Let us denote Hffl'l(eie)um1 by n(eie), so we have

211
n(eie)“x“2 s (f(eie)x) (x) and g log n(eie)de > -m. (1)
1 2" 1 2
Denoting the positive quantity Ea-g n(e e)de by N from (1)
we obtain
211
2 2 1
N m s 2171-] (f(e e)x)(x)de . (2)

By [1.] there exists a Hilbert Space X' and an operator T in

B (1 9K) 8 UCh that
211
* 1
(T Tx>(x) = %;-j (f(e e)x)(x)de. x e x. (3)
0 ..
By (2), (3) and boundedness of T we have

NHx“ s “Tqu s MHxH for all x e I, (4)

50

where 0 < N S M < on. We note that 92(f(e19)) = {0}, so that

the projection Operator occurring in the last theorem is identity.
Then (1) and (4) guarantee the validity of the hypothesis Of
theorem 4.4.7. Hence f is factorable.

4.4.9 Remark. As we have seen above the condition
2" -1 1 -1
g log[“f (e 6)“ ]de > -w implies the existence Of a Hilbert

Space X’ and a bounded linear Operator T on 1 onto X’ which
is one to one. This means that the topology of 1 can be Obtained
through an inner product. Hence one could also Obtain our theorem
4.4.9 by appealing directly to the Hilbert space case.

It is useful to know under what condition the finite sum,
limit and series of factorable B+(1,1f)-valued functions is
factorable. Having our main lemma 4.3.6 available we can prove
the following theorems.

*
4.4.10 Theorem. Let f and f be weakly summable B+(1,1 )-

1 2

valued functions on the unit circle. If f1 and f2 are factor-

able, then f = f1 +f2 is factorable.

Proof. Let Q1, Q2 and Q be quasi square roots Of f f

1’ 2
and f respectively, and X1: Kk and X' be the corresponding

Hilbert spaces. Let g E 77((Q), then there exists a sequence

pnéu such that limen-g in L200. Since fzf ,j-Il,2,

J
M
giving similar argument as in the proof of theorem 4.4.3 we can

show the existence Of a subsequence Of p“, denoting it again by

pn, such that

f
11".“
(1)

4 “A”
limen=g in X a.e.; limijn=g

3"“ m
K n

j in X} a.c., j = 1,2.

 

51

From (1) it follows that

Hale‘s)“; + Hezeihuiz = “30219)“; <2)

Since fj (j = 1,2) is factorable, by lemma 4.3.6 either gj

(j = 1,2) is a zero function or the measure of Zgj is zero. In
any case from (2) it follows that either g is a zero function

or the measure of Zg is zero. Hence by lemma 4.3.6 the proof

is complete.

4.4.11 Theorem. Let {f1}:;1 be an increasing sequence Of factor-

*
able B+(1,1 )—valued functions on the unit circle and m ‘be a non-

negative scalar valued function such that

(8) lim fj(eie) = f(eie) a.e.,
j—uo
211' 19
(b) 1im g (£j(e )x)(x)de < a, for all x e x,
n—m
(c) “f(eie)“ s m(eie) a.e.

Then f is factorable.

M. Let {Qj}:=l and Q denote quasi square roots Of
{fj}:=1 and f respectively. By (b) and (c) f is weakly
summable B+(1,1f)-valued. If f is not factorable then there
exists a sequence p11 6 u and a function g E L2(X) such that
lim Qpn = g in L200 with g non-zero and the measure Of 23
Esmpositive (c.f. lemma 4.3.6). AS in the proof Of the last
theorem, there exists a subsequence Of pn, say pn, so that

:1: ijn = gj, for each j, in L209) norm, and ‘11:: qun =

gJ a.e. in K}. (These limits are uniform w.r.t. j because

f dominates all fj's.) Since lim f (e19) - f(eie) in the

j-OQ

52

strong sense we have lim (fj(eie)x)(x) = (f(eie)x)(x), x 6 1,
j—oa

and hence for almost all e we have

1:: us,<e“’>\\,(j = Haeihnx -

Thus the measure Of Z8 is pointwise positive for some j which
implies by lemma 4.3.6, gj = 0 a.e. and hence g = 0 a.e. This
contradiction completes the proof.

4.4.12 Theorem” Let {f1}:;1 be a sequence Of factorable B+(1,1#)-
valued functions on the unit circle and m be a nonnegative scalar

valued function such that

(a) z fj(eie) = f(eie)
1‘1
a 2n

(b) 2 (f (eihxxxwe < e, x e x,
i=1 3

(C) “f(eie)“ < m(eie) a.e.

Then f is factorable.

Proof. Apply theorem 4.4.11 to the increasing sequence Of partial

N
sums { 2 f
1'1

11;;1’ whiCh are factorable by theorem 4.4.10.

CHAPTER V
NECESSARY AND SUFFICIENT CONDITIONS FOR FACTORABILITY

OF NONNEGATIVE OPERATOR VALUED FUNCTIONS ON A
BANACH SPACE

5.1 Introduction. In this chapter we continue to study the
important problem Of factoring a nonnegative Operator valued func-
tion on a Banach space. In Chapter IV we were able to extend to
the Banach Space the work of R.G. Douglas [4 ] on factoring non-
negative operator valued functions. However these results pro-
vided only sufficient condition for the factorization problem.

Our purpose here is to establish some necessary and sufficient
conditions for factorability of nonnegative Operator valued func-
tions on a Banach Space. This extends to the Banach space the re-
cent work.of Yu. A. Rozanov [19] and a certain result of R. Payen
[18] on necessary and sufficient conditions for the factorization
problem. It also reveals the close connection which exists between
these characterizations.

In §5.2 we set up necessary terminologies and state some
known results. In Q5.3 we prove our main theorem on characterizing
factorable Operator valued functions on a Banach space. In establish-
ing our main theorem we make use Of quasi square roousand technique

employed in [19].

5.2 Preliminaries. In this chapter all Banach Spaces and Hilbert
spaces will be separable.

53

54

Let f 8 He”) be a weakly summable B+(1,1*) valued
function on the unit circle. Then by lemma 4.3.1 a quasi square
root Of f, Q -Q(eie) , with values in B(1,K) exists. Let
go“ ‘ emeQ(eie)- Then En, -co < n < an, is a B(1,L2(70)-valued SSP
whose spectral density is f. From here on gn, -oo < n < as, re-
presents this process.

In §5.3 we need a lemma due to Rozanov. Because of its
importance and for ease of reference we state this lemma here.
First we introduce some notations (c.f. [19]). Let B be a
linear manifold in L200 and S -= {gn(e19) }:=1 be a complete
orthonormal system of functions in B. We denote by BS(eie) the
linear manifold in the Hilbert Space K generated by all values

g1(eie),g2(eig),... . Obviously the closure B(eie) = BS(eie) ,

does not depend on s in the sense that is (em) = is (e19) a.e.
l 2
e, if 31 and 32 are any two complete orthonormal systems in B.

In case B = Q(eie)1 it easily follows that
= i 1
B(e 9) =Q(e an 2.2.,

where Q(eie)1 denotes the closure of the range Of the operator
n(e”).

5.2.1 m (Rozanov). Let B be a linear manifold in L200.
Then the subspace 6(e1neB, -cn < n < co) generated by eineB,

-oo < n < co, consists Of all functions g 6 L200 such that
i =
g(e 9) E B(e ) a.e.

5.3 Main results. In this section we prove our main results. 1

is a separable Banach Space and K is a separable Hilbert space.

55

f is a weakly summable B+(1,1*)~valued function on the unit circle.
Q will denote a quasi square root of f with values in B(on7o
5n = eineQ(eie) , -co < n < on, is a B(1,L2(7())-valued SSP with the
spectral density f. Let H = 6(§nx, x c3 1, -co < n < co) and
H(n) = 5(§nx, x 6 1, -¢o s n).

We shall be interested in the structure Of the subspaces
B = H(T) 9 H(S), where T, S are some sets Of integers and for

any set T
H(T) =6(§nX. x 6 I. n E T).

One can say that B is the innovation of H(T) in comparison
with H(S).

LT will denote the linear space of all If valued in-
tegrable functions q{for each x E 1, m(eie)x is summable} with

Fourier decomposition of the form

. . *
m(ele) ~ g anenle, an e 1 (5.3.1)
n€T
i.e.
m(eiek ~ 2 aux enie. x E 1
nET
such that
i *. i
cp(e 9) 6Q (e 9W (53-2)
and
Zn *
- i i 2
“I: 1(e 9).,(e 9)“ de a a, (5.3.3)
*-1 16 * 19
where Q (e ) is the inverse Operator from Q (e )X onto
'L - orthogonal complement Of the null subspace Of Q*(eie).

aka“)

56

5.3.4 Lemma. Let S be the complement Of T in the set of all

integers and B '3 H(T) O H(S). Then

T
*-
Br =Q 1LT, (5.3.5)
Proof. Let (g(ei 9,1,.)EB Define Y(e16)n=Q*(eie) m(eie). From
211
the relations gm; (e E’)tp(e E’))(x)de=:gfl (m(e 9,) Q(e 9)x)de
211

s (gflntfieiefl‘z d9) 15(51‘12 (eie)x“ 2d9)$2 < as, it follows that

2n

14916) (X) is 801111131318, X E I.- Also we have that 0 = g (3.189 19

(Q(e ).

2n
Q(eiehdde = g e‘iseche‘e).cp<e1°)><x)de. s e 8. Since o e u.

by lemma 5.2.1 and the paragraph preceding it m(eie) 6 Q(eie)1 a.e.

*_ .
Hence (9(ei9) -Q 1(emyueie) and Y(e19) 6LT. Thus

*- 16 19 _
BT :Q 111,. Now let Y(e ) be in LT' Set m(e ) —
9:-
Q 1(eie)‘l’(eie). We will show that m(eie) 6 3T We note that
for each x E 1, (m(eie),Q (eie)x) = Y(e19)(x). Hence
2n .
[E e-ise(cp(ele) ,Q (eie)x)de = 0, s E S, x 6 1. It suffices to show
that (p G H = 5(e1n9Q(eie)1, -cu < n < co) . But m(eie) E range Of

*-1 19 ie
Q (e ) a Q(e )1. Hence by lemma 5.2.1 and the paragraph pre-

ceding it we have that m(eie) 6 BT. This completes the proof of

Q*'1 c B
LT T.

5.3.6 Corollary. The relation B(emeBT, -ao < n < co) = H holds

if and only if

 

 

Q(eien - Q*’h.r(eie) a.e. (5.3.7)

Proof. Since H = 5(eineQ(e 19)11, «:9 < n < so), by lema 5.2.1
H ' 5(emeBT, -co < n < on) if and only if BT (e16) - Q(e16)1

But by the last lemma BT (e19) - Q TILFRW) .

57

We note that this result in particular is useful for
regularity if we take T to be the set Of all nonnegative in-

tegers. In this case the Space L is included in the Hardy

T

class Of functions

19 m 1ne
m(e ) ~ 2 a e
n

0
More precisely they are so that for each x 6‘1, the scalar valued
function m(e19)(x) is in the classical Hardy class H1. We also

note that in this case the relation
ingB
6(e T’ -oo<n<ao)=H

Of corollary 5.3.6 is equivalent to the regularity Of the process
1 i
gn-en%(e 9), m<n<¢h
We are now ready to prove our main result.
19 + *
5.3.8 Theorem. Let f = f(e ) be a weakly summable B (1,1 )-
valued function on the unit circle. Then the following statements
are equivalent:
1. f is factorable

2. there exists a conjugate analytic B(1wx)-valued function Y

such that

(1) 1*(e‘°)x e q*<e‘°)x

 

(u) Q*'1<e‘°>v*(e‘°>x-Q<e‘°)x a.c.

2n _
(111) [E “0* 1(cej'Le)~4r"'(e”)knzde < e, k e x.

3. The process einQQ(eie), -w<< n < m, is regular.

58

i
4. There exists a sequence {qh(e e)]:=1 of measurable xzvalued

functions such that

19 o
(a) a.e. e, {¢h(e )]n=1 is an orthonormal basis for
1
Q(e 9n,
19 * ie 19 . .
(b) For each n, gn(e )x ==Q (e )¢h(e )x 18 1n the usual

Hardy class H2 for all x 6 1.

*
Proof. (1) = (2). Let f = Q Q, where Q is conjugate analytic.

Then

W2(eie)xu2 = “Q(eie)x“2, x E 1. (5.3.9)

Define
' 1
V(e19)q (eie)x = Q(e 9)x, x e 1. (5.3.10)

By (5.3.9) - (5.3.10), V(eie) can be extended to an isometry

on Q(e19)1 onto Q(eie)1. Then the Operator valued function

Y = VQ satisfies the conditions (i) - (iii) of (2).

(2) =1(3). Let Y be a conjugate analytic B(1”X)-valued function
satisfying (i) - (iii). By corollary 5.3.6 and the paragraph

following this corollary, it suffices to Show that
i *- i
Q(e 9)I ==Q 11.1.(e 9) a.e.

Clearly the right hand Side is a Subset of the left hand side.
The other inclusion follows from (ii) - (iii) and the fact that
for each R E K, Y*k€ L1,.

(3) =>(4). Since einQQ(eie), ~w«< n < a is a regular process,

H(O) = 6(eineQ(eie)1, n s 0) does not contain a non-trivial

59

doubly invariant Subspace Of K. Hence by [10], p. 61 it is of
the form V Lg-(fi(), where V is a measurable isometry operator
19 on

on some Hilbert Space K into X. Let {¢n(e )]nml be an

orthonormal basis for H(O) 9 H(-l) . An argument Similar to one

used in [18], p. 380 and [10], p. 61 may be used tO Show that for
19 on

almost all e, {(pn(e )]““1 forms an orthonormal basis for

i i * i i
Q(e 9)1. Let x E 1. To show gn(e 6) (x) =Q (e e)tpn(e 6) (x)
2
is in the Hardy class H , we Observe that for each n,

(pm 1 eikeQ(eie)x for x E 1 and k s -1. Hence

2n _ 2n _
g e ikes,..<e“’><x>de = S e ikelake”)we”)<x>de =

211 _
{E e 1kg((pn(eie),Q(eie)x)de = 0. Thus gnx E H2.

(4) =3 (1). Let (a) - (b) hold. Let {en}:=1 be an orthonormal

basis for X. We define the Operator valued function Q by

 

1<e19>x = z 8n(eie>(’"ea’ x e 1- (5.3.11)
n
We note that for all x,y E 1
(fx)(y) = (Qx.Qy) = z (QX.cpn)(Qy.<pn)
n
"In?“ * ——
= 2 (Q onXHQ cpny) = 2 (ans)(gny) (5-3-12)
n n
= (a (snx)en. 2 (any)en) = (MAY).
n n

where the second equality follows by (a). Hence Q(eie) 6 B(1,)() .
Because Of weak summability Of f it follows that for each
x E 1, Qx 6 L200. Using (5.3.11) and (5.3.12) it is not hard

to Show that the conjugate analytic B(1,X) -valued sequence

N
§N(eie)x - zgn(eie)(x)en converges to Q(eie)x in L200.

n=l

6O

- *

Therefore Q E L? (x). Since by (5.3.12) f = Q Q the result

follows.
2n i

We remark that if one assumes I “f(e e)“de< m, then

0

part (b) Of (4) in the above theorem can be replaced by (b'):

*
For each n, gn(eie) =(2 (eie)¢h(eie) satisfies

infisnwihuzde < e-

CHAPTER VI

*
FACTORIZATION OF UfU

6.1 Introductigg, Let f = f(ei 8) be a B+ (X',X)-valued function
on the unit circle. Suppose U(ei 9) is a B(xgfl) unitary valued
function. Then UfU* is also nonnegative Operator valued func-
tion. Suppose f is factorable, then the natural question raised
by M.G. Nadkarni in [16] is to investigate the factorability of
UfU*. We study this problem here in this chapter. Using the
results Of Yu. A. Rozanov on factorization problem [19] (c.f.
theorem 5.3.8) we give several necessary and sufficient conditions
for the factorability of UfU*. AS a natural application Of our
theorems we Obtain a result similar to the one given in [18],

p. 381 on the factorization of a nonnegative Operator valued func-
tion, involving the eigenvalues Of f. We also Obtain a Devinatz's
type necessary condition for the factorability Of nonnegative
operator valued functions. In connection with these results we
conSider the problem of factoring a 2 X 2 nonnegative matrix
valued function of rank one which has been discussed earlier by

N. Weiner and P. Masani [27].

6.2 Factorability Of UfU*. Let X' and fi' be two separable

Hilbert Spaces. A B+1x3X)-valued function f - f(eie) is called
uniformly summable if g Mufleie)“ d9 < a, and weakly sumable if
En(f(eie )x,x)de < m, for all x E 1. A weakly summable B+ (K',X)-

61

62

valued function is called factorable if f(eie) = Q*(eie)Q(eie)
where Q(eie) is a conjugate analytic B(x3x9-valued function
(c.f. definition 2.9).

TO Stay in the framework Of the standard results on
Hilbert space valued stationary stochastic processes in the rest
of this thesis we take the following definition Of factorability
which in the Hilbert Space case is equivalent to our earlier
notion of factorability given in definition 2.9.

We say that a weakly summable B+CK3X)-valued function
f = f(eie) is factorable if there exists an analytic B(xgx)—

19

valued function Q = Q(e ) such that

f(eie) = Q(eie)Q*(eie)

The following theorem is a natural extension of Devinatz's
theorem [10], p. 119.
6.2.1 Theorem. 'Let f = f(eie) be a unifonmly summable B+(K3K)-
valued function which is a.e. invertible and
Eulong-1(eie)ude < m . (1)
Then the B+OIM -valued function UfU-Ar is factorable.

Proof. We Observe that for each x E X’ we have
(cafv*)'*x.x) - (Uf'1U*x.x> = (I'1u*x.u*x) s Hf'luuxuz-

SO we have “(UfU*)-1“ s “f-l“. Hence by (l) we see that
211

* -
A logu(UfU ) 1“de< a. ‘But by Devinatz's theorem we see that

*
UfU is factorable.

63

The following theorem is an immediate consequence Of
theorem 6.2.1.

6.2.2 Corollary. Let f = f(eie) be a weakly summable non-
negative finite dimensional matrix valued function which is of
full rank. Then f is factorable if and only if UfU* is
factorable.

Recently Yu. A. Rozanov [19] gave a necessary and suf-
ficient condition for factorability Of a weakly summable B+(xgx0-
valued function. We extended his results to the Banach space
case in Chapter V.. However since we are going tO use his result
in this chapter, we will state his theorem in the context of the
notations of this chapter. Before doing so we recall the follow-
ing necessary notation.

6.2.3 Notation. As before L2(X) is the Hilbert space Of all

measurable X' valued functions k(ei9) such that

£fl“k(eie)“2de < m, with the inner product Of any two elements

k1 = k1(e19) and k2 -= k2(eie) e 1200 defined by (k1,k2) =
2

£7"- gflacleie) .k2(e1°)de.

6.2.4 Theorem (Rozanov). Let f = f(eie) be a weakly summable

B+(x3x9-valued function. Then f is factorable if and only if

there exists an analytic operator valued function Y such that

(a) Y(e19)7(: £3(e19)x a-.e.

 

(b) f-)5(eie)‘f(eie)7(- fk(eie))( a.e.

2n _
(c) g Hf *(e‘hwe‘ennzde < o

64

-1 1

where f (e19) is the inverse Operator from f (eie)X' into

the“ )K.

6.2.6 Theorem. Let f be a nonnegative finite dimensional

matrix valued function such that M(ele)/m(ele) is in L , where

1
19 i9) denote the largest and smallest non zero

19

M(e ) and m(e

eigenvalue Of f(e ). Suppose f!5(eie )X' is an invariant sub-

space Of U(eie).

Then f is factorable if and only if UfU*

is factorable.

2529f, Suppose f is factorable as f(eie) = Q(eie)Q*(eie),
with Q(eie) being analytic valued function. We will Show that
conditions (a) - (d) hold for UfU* with Y(eie) being

1
Q(e 6). By [5 ], p. 413 we have

the i">K= Q(e 19m - (2)

But by hypothesis U£%x'= fix} Hence by (2) we have

=f1x= nth-=1: f""=<u:o (Ufkuhx,

*
so we have QK = (Uf’iU )K for a.e. 9 which is (8). Now by defini-

-% we have f-%f%K = ngI SO we have

Uf’kf1x = U£3x . (3)

Now fHK' is invariant under U. Hence £3x’ is also invariant

tion of f

*
under U . Thus from (3) we get

11ka - 0595“?” = Uf”‘(u*£"yo = Uf”‘u*r”)(.

SO by (2) we get U£%x'- Uf-%U*f%x'= Uf-%U*Qx2 Hence we get

Ukafx'B Uf-%U*QX’ which is (b). TO see (c) we have flif’i - QQ

 

65

BY 1 5], p. 413 there exists an Operator valued function C(eie)

%

with Banach norm one Such that Q(eie) = f (eie)C(eie). SO we have

i"“Uf-ku*§“2de s E"\\f"iu*o\\2de s Efi\\f-%U*f-%uzde
S Enuf'guzflfg’fizde s Enmeiewmeiende < on.

Now suppose UfU* is factorable. Note that UfU* and f have
the same eigenvalues. We also note that Ufkufix' is invariant
under U*, because U*(Uf%U*x9 = U*U(f%U*X) = f%U*X'= f%K’=
U(f%x) = (Uf%)(x) = (Ufk

* *
ment we gave above, with UfU and U instead Of f and U

* gU*
)(U K) = Uf X. SO repeating the argu-

reapectively we can Show that f is factorable.

We now state the following infinite dimensional extension
Of the last theorem whose proof is exactly as in the last theorem
and hence is omitted.
6.2.7 Theorem. Let U be an unitary B(x3fl3-valued function.
Let f(eie) be a uniformly summable B+(XQX)-valued function such
that R(eie) = f(eie)X' is a reducing subspace (closed) of U(eie).
Suppose that m(eie)1 s f(eie) s M(eie)1 on R(eie), with
H(eie)/m(eie) being in L1. Then f is factorable if and only if
UfU* is factorable.

As a consequence tO theorem.6.2.7 we Obtain the following
corollary.
6.2.8 Corollagy. Let f be a uniformly summable B+(K3X)-valued
function satisfying m(eie)I s f(eie) s M(eie)I, with M(eie)/m(eie)€L1.
Let U be a unitary B(XM -valued function. Then f is factor-

*
able if and only if UfU is factorable.

66
19 16
Proof. From M(e )/m(e ) E_L1, and
m(ei 9)I s f(ei 9) s M(ei 9)I,

it follows that f(eie)X'= X'a.e. We now apply theorem 6.2.7 to
conclude the proof of this corollary.
The next two theorems provide some necessary and suf-
ficient conditions for the factorability of UfU*.
6.2.9 Theogem. Let f = f(eie) be a B+(X',X)-valued function
11’

such that g “f(e e)ude < a. Then UfU* is factorable if and

only if there exists a partial isometry valued function V(e ie),

 

with initial range in fl' and terminal range U(ei e)f W(ei0)U (e 19)” fl,

*
such that U(e ie)f%(eie)U (eie)V(eie) is an analytic B(fl3fl3-
valued function.

gua-
Proof. Sufficiency. Let Q = Uf V. Then Q is an analytic
* %U* * %U* *

operator valued function and QQ = Uf VV Uf = UfU .

* * *
Necessity. Suppose UfU is factorable of the form UfU = QQ ,

where Q is analytic operator valued function Operator valued

function. We then get

(Uka*)(Uf%U*) = 11*. (1)
So for each x e x' we have

ucvf“u*>"‘xu2 = 16x12 . <2)

19 %U* * *
Then we can define W(e ) on (Uf ) X' into Q X’ by
i gua * *
W(e e)((Uf ) x) B Q x and by (2) we can extend it to an isometry

%U* *
W on Uf K onto QX. We then have

67
WUf%U* = Q* . (3)

* *
Taking adjoint on both sides of (3) we get Q = Uka W and
* 50* *
letting W to be V we get Uf V = Q. Thus Ufliu v is analytic
Operator valued because Q is SO.
The following theorem gives a relation between the factors

*
of f and UfU .
6.2.10 Theorem. Let f be a B+(K3X)-valued function such that

2n
g “f(eie)“d9 < "a Suppose that Q is an analytic factor Of f.

*
Then UfU is factorable if and only if there exists a partial

isometry valued function V(eie) with terminal range being

 

19 16 * ie *
U(e )Q(e )U (e )fl' such that UQU V is an analytic Operator
*
valued function. In this case UQU V is the analytic factor Of

*
UfU .

6.3 Application. In this section we establish some new results
and prove some well known facts using the materials of §6.2.

The following is a special case Of a result due tO Weiner
and Masani [25].
6.3.1 Corollagy. 'Let f be a nonnegative finite dimensional
matrix valued function such that M/m G Ll, where ‘M(eie) and
'm(e19) are the largest and smallest eigenvalues of f(eie)
reapectively. Then f is faCtorable if and only if log det f 6 L1.
2222;, Let U(eie) be the unitary Operator valued function which

is measurable and diagonalizes f. Suppose we have

 

 

68

where 11 s 12 s 13 s...s kn are the eigenvalues Of f. We
know by corollary'6-2-8 that f is factorable iff UfU* is factor-
able. But Obviously UfU* is factorable iff log 11 6 L1 for
all i = l,2,...,n. Hence f is factorable if and only if
lOg 11 G Ll. Thus f is factorable if and only if log det f 6 L1.

A well known sufficient condition for factorability Of a
weakly summable B+(7(,7() -valued function f is

Enlog “f-1(eie)“-1de > «a .

The following theorem shows that under some extra con-
ditions the above condition is also necessary. First we prove
the following lemma.
6.3.2 nggg, Let f be a uniformly summable B+(X3X)-valued
function which is factorable. Suppose m(eie)1 s f(eie) S M(eie)l
with M/m 6 L1. Let Meie) be an eigenvalue of finite multi-
plicity for f(eie) a.e. Then

Zn
3 log 1(eie)de > -m.

giggf, Let V(eie) be a measurable unit eigenvalue Of f(eie)
correSponded to 1(eie). (For the existence of such an eigen-
function one may give a proof similar to the one in the proof
of part (b) Of theorem 6.3.5.) Let U(eie) be a measurable
unitary operator valued function such that UfU* = QQ*, where

* * * *
Q is analytic. Hence (UfU ,a) = (QQ a,a) = (Q a,Q a). Thus

we have

69

1(219) = (f(ei9)v(eie>.v<e1°>) = <f<e19>u*<eie>e1.U*<eie>e1>
= (U(eie)f(eie)U*(eie)e1,e1) = (Q*e1,Q*e1).
* _ m *
Since Q e1 - E (Q el’en)en we get
n=l
1<eie) = z \<1*el.en)|2. <1)

k=1

' m * -
19) = 8 Que ins
n=0

*
But since Q (e we see that

* 19 = m * -in9
(1 (e )el.em) z (one1.em>e
n=0

2“ * 1e 2
SO 5 log \(Q (e )e1,em)\ > -m for some m. Hence by (1)

2n 19
g log 1(e )de > -w .
6.3.3 Theorem, Let f be a B+(x3x)-valued uniformly summable
19 19 19
function such that M/m 6 L1, where m(e )I s f(e ) S M(e )1.
Suppose f(eie) has at least one eigenfunction Of multiplicity
one. Then f being factorable implies that
2n
g log Hf 1(e19)\\‘lde > -m.

Proof. Since M(eie) 2 1(e16), using lemma 6.3.2 we get
211 2n .
-m1< 3 log 1(eie)de s g log M(ele)de . (1)

We also know that

2n . . 2n
0 S g (logIM(ele) - log m(ele))d9 = A log(M(eie)/m(eie))d6

(2)

2n
= & (M(ei°>/m(eie>>de < e .

70

From (1) and (2) we see that log m(ele) is summable, because
we have log m(eie) = log M(eie) - (log‘M(eie) - log m(eie)).

2n 19
Hence we get log m(e )de > -m, which means

2n
-l i —l
& log “f (e 9)“ d9 > -m.
The next theorem is an interesting consequence of corollary
6.2.8. We shall need the following lemma first.
6.3.4 Lemma. Let f be a measurable B+(xng-valued function
whose spectrum consists only Of the eigenvalues, each eigenvalue
16 ie 19

having finite multiplicity. Let L1(e ) 2 52(e ) 2 L3(e ) 2...

denote the eigenvalues of f(eie

) listed according to their
multiplicity. Then 41,L2,L3,... are measurable and there exists
a measurable unitarily valued function U(eie) Such that

Q

1 1 * 1 1 °
U(e ems 9w (e 9) = z g(e eM2102“).
1'1
where Qj's are constant one dimensional projectors.

Proof. Here is an outline Of the proof.
By a similar argument as in [ 6], p. 653, one can Show

q
that Z Lj(eie) is measurable for each q 2 1. Hence each
1'1

Lj(eie) iS measurable. Following the proof of [7 ], p. 391, we
can Show the existence Of a complete orthonormal sequence
{uJ]:_1 Of eigenvectors Of f which are measurable. Let
{e1}:-1 be a complete orthonormal sequence of vectors in x:

19)u

Define the unitary operator valued function U by U(e j = ej.
We then see that U has the desired properties.
+
6.3.5 Theorem. Let f be a uniformly summable B (x3x9-valued
16 16 19
function such that m(e )I s f(e ) s'M(e )I, with M/m being

summab le . Then

71

(a) Let the Spectrum of f consist Of only eigenvalues, each
one Of which being of finite multiplicity. Then f is factor-
able if and only if for each j, l s j < m, we have

211 19

8 log {g(e )de > -oo,
where L1(ele) 2 12(219) 2 L3(eie) 2... are the eigenvalues of
f(eie) listed according to their multiplicity.

(b) If

f(eie) = 2 p (eie)P (e19),

j=l J J
PJ's are measurable one-dimensional projection valued functions
such that Pj(eie) are mutually orthogonal. Then f is factor-

2n
able if and only if § log pj(eie)de > -m, for all j = 1,2,3,...

Proof. (a) Let U(e 9) be the unitary Operator valued function
Obtained in lemma 6.3.4. Then we have
1 1 * 1 ° 1
U(e ewe 9w (e 9) = z t (e °>Q .
H J j

where Q is the orthonormal projection on e . Now by corollary

J J
*
6.2.8 we know that UfU is factorable if and only if f is
factorable.
Now suppose that for each j = 1,2,3,... we have

2n

3 log Lj(eie)de >’-m .
Then there exists scalar valued conjugate analytic functions
qB such that Lj(eie) a ‘q3(eie)\2, i = 1,2,3,... . Let x 6 X2

Then we Observe that

72

N 19 1e 2 N
u z cpj(e )Qj(e M = < r. ojce 9m .x. 2111ij 9)Q x)
j=l j=1 J=1
N
= ( 214.103i e)ij X) 1 ( g Lj(e e)ij,X)-
J=1 J 1
Hence we have
N 19 2 is ie * 19
\\ zloj<e )ij\\ s we >f<e )0 (e mo (1)
j:
Similarly we get
M M
2
u z: oj(eie)czjxn = ( z: Lj(e19)qjx .x). (2)
J“N J‘N
By (1) and (2) we get wj(e e)ij converges and defines a
J= -1 m
bounded Operator, say Q, through Qx = th jx We also have
j= —l
N
HE cpji<e 92mg“ = < 2 L (e 9m x ..x> <3)
J=1cpj J=1

Now letting N -m in (3), we get

X. Q(e 6)Q X)

(Ufu*x,x) = u: cpj‘<e em 1sz = < z mlojeiem
J21 ‘Pj 1

J=1CPj J=1j

* a
which means that UfUm = QQ . Now clearly for each m = 1,2,3,...

19

we know that me = Q(e )ij, has only nonnegative Fourier

3]
J31
coefficients. We also Observe that
Zn M 2n M
g H E mq3(e 92)ijn de= SH ( s L (e e)QJx.x)de
2n M is
( 21Lj(e )ij,x)de
J=1
2n N

-g(2 Lj(ei e)Q jX,X)d9 .
j= -1

73

Hence by the dominated convergence theorem we see that the func-

'M
tion 2‘ (pj(eie)ij is Cauchy in L20!) . Hence we see that
J81

Q(eie)x has only nonnegative Fourier coefficients. Now for each

n s 0 we let gnx be the n-th Fourier coefficient of Qx, then
0
i
Qx = 2 e n9
n=0 *
with the weak integrability of UfU and closed graph theorem

* * *
gnx. But we have UfU = (Q x,§ x), which together

*
we see that Qn's are bounded (Remark 4.2.1). Hence UfU is factor-

able, which means that f is factorable. Then by lemma 6.3.2
2n .
& log(Lj(ele))de > -m, for all j = 1,2,3,...

(b) For each fixed j = 1,2,3,... let u (e19) be a unit vector

J

in P (e19). Let {e ]° be an orthonormal base for H2 Then
j m m=l

16 = i i -
for all j 6 2+, m.6 Z, Pj(e )em (em7uj(e e)uj(e 9) is measur
able . We can divide the unit circle as the disjoint union of

countable sequences of {E such that Pj(eie)em is different

mj}m=l

from zero on Em and zero on En for all n >‘m. Then obviously

J J

we have

P1(819)e

 

19
u (e if e E E
j “Pj (e19) em“ mj

(e19

j )

Now since each of P (eie)em is measurable, we see that u

J

are measurable. But Lj(eie) = (f(eie)uj(eie),u (eie)), so

J

Lj(eie) is measurable for each j = 1,2,3,... . Now one can

define the unitary operator valued function U through

U(eie)u (eie) = 6

J J

Now U is measurable and the rest of the proof is exactly similar

to the proof of (a).

74

Based on theorem 6.2.9 we give a proof of a result due
to Weiner and Masani. However we point out here that for the proof
of sufficiency we make use of an argument contained in [27].
First we introduce some notation and state a lemma from [27].

We denote by L0+ the boundary values of the functions

6
in Hardy class H6 and by' Qg+ the class of all functions f
. 0+
such that f = h1/h2 a.e. With hl and hz e L6 .

6.3.6 Lemma, Every function in Qg+1 on the unit circle,

0 s 61< m, such that \f\ = l a.e.k is in ng' and admits a
factorization f = $162, where $1,¢2 E LET and 111‘ = 1121 = 1.
6.3.7 Theorem, Let f = [€i;1:,j=l be a weakly summable 2 X 2
(non zero) nonnegative matrix valued function such that det f = O
a.e. Then the following two conditions are equivalent.

(a) f(eie) = Q(eie)§*(eie), where O # Q E Lg+.

(b) For i = l or 2, log fii E L

f
. ji 0+
1 and for 1#j, f,, 6Q6 .

11

Proof. Assuming £11 * O a.e. we have

19 i9 -
f11(e ) f12(e ) l V
f(eie) = = £11 ,
19 19 2
f21(e ) f22(e ) J 111..
f21
where W = E—-» Clearly the eigenvalues of f are zero and
11
2 1
£11 +f22 = f11(1 +-\¢\ ). Let a(e 9) be a scalar valued func-

tion, then the vector (-;a,a) and (;,v5) are eigenvalues of

f corresponded to zero and f11(1 + \w‘z) respectively. If we

let a = ‘—-lL-—-- then the unit vectors (-§a,a) and (£,¢£)
,/1 + ”‘2

are eigenvectors of f corresponded to zero and f11(1 + \W‘z)

respectively. Hence the unitary valued function

75

 

a -a V(eie)

U(eie

 

La ¢(eie) 3

sends the vectors (1,0) and (0,1) to (a,aw) and (-a$,a)

reSpectively. In other words we have

 

 

f11 E21 3 ‘31 f11 + f22 a 81
£21 £22 aw a L0 0 Liaw a
Let us denote by
£11 + £22 0
f' =
0 0

*
Then by theorem 6.2.9, f = Uf'U is factorable iff there exists
a partial isometry valued function V with terminal range

'%U* ’iu"
Uf X’ such that Uf' V is analytic. Now since

 

 

 

a J; /f11+f 22 0 Pa 6a
or? = . .
ta aJ 0 0 L-wa a
P _ (1)
1 ‘J‘
= a2 /f
1+ 2 2
1f2 * \W‘
L

 

*
Hence Uf'gU is the subspace generated by the vector (l,¢).

So the operator V(eie) has to send some vector, say

(s<eie>,t<e19)> co <——-—1——,——L—2-> and (-t(ele) s<e19>>
/1 +‘l¢\2 /1 *‘lJJZ

to (0,0), where s(eie) and t(e19) are some scalar valued

 

76

o 0 2
functions with the property \s(ele)\2 + \t(e19)\ = 1. 80 V
must be of the form
5 E
V(ele) = ——-—1-— , (2)

f1+ M2 193 w?
where \s‘2 + ‘t‘2 = 1. If we define V(eie) by (2) then from

(1) we get

an
H

25 * —
' =
Uf U ff“ - _ (3)
31¢ ti;
2 2
where \s‘ + \t‘ = 1. Now suppose (a) holds, then we have

~1- 2 0+ 2 2
f=§§ ’0*§=[q’ij]i,j=1€L2 . So we get f11=\¢11‘ +|cp12\ .

f22 = M) \2 + JCPZZJZ' Thus

log f11 > 2 108 “911‘: 2 108 \‘P121
log f22 > 2 log 1(9211’ 2 log JCPZZJ

Since Qi‘O then #0 for some i and j. Hence

‘91:]
log £11 6 L1 or log f22 E L1. Without loss of generality we

assume log fll E L Now by (3) and theorem 6.2.9 there exists

1.
functions a(eie) and t(eie) such that

'2: — s t 8 [£11 t ffll 0+
Uf V affll - _ = - _- - _— 6L2 .
sv t1; 8* [£11 ti ffu
‘3 fll—H - —— - —— 0+
Now since t= -_-——:- and 8 [£11 4‘: s ffll 6 L2 , we see that
a [£11

0+ 0+
1 EQZ , hence £21/f11 6 Q2 .

77

We now show that (b) implies (a). Suppose (b) holds.
For definiteness we assume that i = 1, so that we have
0+
and f21/f11 6(26 . If £22 0 then obviously

log f 6 L

11 1
f is factorable. Assuming that £22 > 0 on a set of positive
measure. Then as shown in [27], p. 306, the condition

0+ -
f21/f11 6 Q6 implies that log £22 E‘Ll. Let £11 - ¢l¢1

and f22 = ngb be the analytic factorization of £11 and £22.

f22 11
Apply lemma 6.3.5 to the function E—-'$- to get
11 2
f22 ‘91 11 0+
—--—=—.where \¢\=1 and $.6L for i=l,2.
£11 $2 $2 1 1 m
19 19 L2. (P1
Let s(e ) = t(e ) = 2 tz'f::: . Then we have
fill
5 '3 W J <9
2 1 2 l
* .___
UfJEUV =/£u _ _ =[-:-
$1 W 11‘92 11‘92

*
80 Ung V E L0+

0+ 0+
2 , because $1 E'L0° and mi EL2 for i - 1,2.

CHAPTER VII

ALGORITHMS FOR DETERMINING THE OPTIMAL FACTOR
AND THE LINEAR PREDICTOR

7.1 Introduction. The theory of multivariate stationary stochastic
processes as deve10ped by Wiener and Masani [25], [26], [14],
essentially consists of two parts. Part one deals with the analysis
of time and spectral domain. This part has been studied by several
authors and has been extended to the infinfl:e dimensional case

(c.f. [ 4], [ 8], [12], [16], [18], [19]). Part two is concerned
with the important problem of determining the generating function,
namely given a nonnegative Hermitian q X q, (l s q.< a) matrix
valued function on the unit circle, such that f(eie) is weakly
summable and log det f summable, to find a q X q matrix valued

function Q such that
1 *
f(e 9) = a(eie)¢ (e19).

where Q is an analytic optimal factor.
An iterative procedure which yields an infinite series
for 9 in terms of f has been given there by Wiener and Masani

[26] under the following assumptions

c I s f(eie) s czl, (l)

1

where 0 < c 5 c2 < a.

l

78

79

In [14] Masani was able to improve the result he and
Wiener give in [26] by assuming in lieu of condition (1), that
(i) f is a weakly summable hermitian matrix valued function.
(ii) f-1 exists a.e. and f-1 is weakly summable.

(iii) if V(eie) and n(eie) denote the smallest and largest
eigenvalue of f(eie), then u/v is summable.

The problem of determining the opthmal factor was also
the subject of discussion by Salehi [21], where some improvements
were made in the field.

The problem of determining the optimal factor for the
infinite dimensional case has not been discussed in the literature.
In this chapter we wish to obtain an algorithm for determining
the optimal factor and the linear predictor for the infinite
dimensional case. As seen from Wiener and Masani's work, it looks
as though one has to assume that the spectral density is bounded
away from zero. On the other hand a trace class operator on an
infinite dimensional Hilbert space is not bounded away from zero.
Hence processes with finite trace will not satisfy the stipulation
and purpose of this chapter. This suggests the adoption of nota-
tions and terminologies provided by MQG. Nadkarni. In doing so
we can extend the algorithm.given by Masani [14] for multivariate
process to the infinite dimensional case. Section 7.2 is devoted
to preliminary results. In §7.3 the relation between the two
sided predictor error matrices of a process and its subprocesses
is studied. Using this relation we show that under some bounded-
ness condition our process is minimal full rank. We then show the

1 -0+

crucial fact that for the optimal factor Q, Q- is in L2 In

80

§7.4, under some extra conditions, we obtain an algorithm for
finding the linear predictor.

We would like to mention that our method of attacking
the problem of determination of the generating factor and the
linear predictor is in the spirit of the work of N. Wiener and

P. Masani [26] and Masani [14].

7.2 Preliminaries. In this section we shall set down the notations
and preliminaries which will be needed in the next sections. Al-
though some of these notations have been introduced in Chapter II,
since we will sometimes deal with unbounded Operators, this re-
introduction is necessary. ‘Most of the notations and results of

the first half of this section are given in the work of MLG. Nadkarni
[16]. In the second half we prove some results on the Fourier
analysis of infinite dimensional matrix valued functions which

will be needed later.

7.2.1 Definition. Let k' be a complex separable Hilbert space.

We denote by $7 the collection of all g = [gn], n E Z+ of
elements in 71. Clearly 17 is a linear space and we give if
the product tapology, i.e. gm.» g if for each k 6 2+,

§:- gk in V. Let gm 6;. We denote by (§,T1) the Gramian
of g and n to be the matrix whose (i,j)-th entry is

(g1,nj). Clearly (§,§) is nonnegative and (§,§) = 0 if and
only if g = 0. The Gramian has the following properties:

(i) gm.» g, nh'” n implies that (gn,nh) a (§,TD, elementwise.
(ii) g“ «10 if and only if (gn,§n) a 0.

(iii) If A and B are infinite dimensional matrices such that

* *
A; and En are defined, then (A§,Bfl) = A(§,n)B , where B is

81

the adjoint matrix of B.
A closed subset H of i; is called a subspace if H is closed
under addition and Ag 6 H for any matrix A and any 5 G H
for which Ag is defined. We say § In if (gm) 8 0. A
vector g is called normal if (5,5) = I, where I is the identity
matrix. A sequence {gm}:_m is called orthonormal if (gn,§m) =
6nmI'

For any 6 cy, 5 C17, we write
(1) 6(5) - subspace of y spanned by 6
(ii) 6(5) - subspace of L7 spanned by 5
(iii) 6(5) = subspace of V spanned by coordinates of vectors

in 5
(iv) 666) a. the set of all vectors in 17 with coordinates in B.
It is easy to see that for a subspace B of V, 6(5) is a sub-
space of E and if E is a subspace of If then we have
5 '- 56(5)). Hence for any subspace E of 17 we have 5 . 5(5) ,
for some 6. Let g E 17. We write (g‘E) to denote the vector
whose i-th coordinator is given by (g‘Ef = (§1\B).
7.2.2 Definition. A sequence 5“, -a: < n < on of elements of
i is called an infinite dimensional stationary stochastic process
(SSP) if the Gramian (gm,§n) depends only on m-n. It is easy
to see that there exists a unitary operator U on l’ such that

(Pg; . 5:. Let U be its inflation operator defined on 1?.

Hence we have g“ - Ungo. Let g“, -ao < n < an be a SSP. We write
H(n) - 5(gk, k s n), {f(s) = 6(gk, k < m)

1'11-..) . n H(n) and in.) =- s(gk, k + n).
n

82

7.2.3 Definition. Let “n = E,“ - (§n\H(n-l)) . One can see that
T1“ = Unno. We call m, ~09 < n < co the innovation process of
§n, -oo < n < on. We write G = (1103b) and call G the predictor
error matrix of 1;“, -a < n < on. We say gn’ -co < n < on is of
full rank if its predictor error matrix is of full rank, i.e. if
G 2 k1 for some positive number 71-

7.2.4 Definition. Let gn = gn - (§n]R-(n)), the processes Qn,
-m.< n < o is called the two sided innovation process of g“.

We write 2 = (g0,g0) and we call the prOCESs En, -co < n < a:

to be minimal if I: i‘ 0, and minimal full rank if t, 2 Al, for
some positive number x.

7.2.5 Definition. Let u be an infinite dimensional nonnegative
matrix valued measure on the Borel subsets of the unit circle
[16], and let g = [gk], k 6 2+ be a row vector valued function
such that gk = 0 for all except finitely many k, and for these

k, gk is a trigonometric polynomial. Let L501.) be the set of

all such g's with norm given by

 

211‘ on
2 g _1__ 19 19
H8“ 2" m,§-1gm(e )umL(de)8L(e >de .

No elements of L2'(u) are identified if their difference has

zero norm. The inner product of two elements 3 and h is given

by
211 a:
= L 19 " 19
(g,h) 2" g m f—1gm(e )umL(de)hL(e )de .

7.2.6 Definition. Let L201.) denote the completion of L2'(p.).

83

In case of f being 3 Spectral density' we denote by
19
A = .
L2(f) the space L2(u), where umL( ) {me(e )de Hence the

inner product in this case is given by

 

211’ on .
_ l_' is ie 19
(g.h)f - 2" g m,§=18m(e >fm,(e >h,(e )de

2n . .

1 i *

2n 1 g(e‘9>f(e °)h (e19>de.
0

7.2.7 Definition. Let f be the spectral density of a SSP
gn, «m'< n.< c. There is a natural isomorphism between L2(f)
and H§(¢9 which can be obtained by linearity from the mapping

S: g: a Y:, where Y: = [e-ineé We now state the following

kL1‘
leuma (c.f. [16], p. 152).
7.2.8 Lemma. Let f(eie) be an infinite dimensional positive

matrix valued function which satisfies
19 19 19
0 < m(e )I s f(e ) s M(e )I a.e.

Then L2(f) consists of all L2 valued functions g = [g1,g2,...]
' *
with measurable entries such that “g./f“2 = g(eie)f(ele)g (e19) =
N
lim 2 gm(eie)fmc(eie)gL(eie) exists a.e. and the resulting
N-oco m,.(,=1
function is summable.
Now we give the following two definitions.
1
7.2.9 Definition. Let f(e 9) be a positive infinite-dimensional

matrix valued function which satisfies
0 < m(ele)I s f(eie) s M(eie)I a.e.

We denote by {g(f) the set of all infinite dimensional matrix

valued functions, each row of which being in L2(f). We then give

84

{T(f) the row-wise convergent t0pology, i.e. if Q“, Q E I&(f)
then 6“ a Q in I%(f) sense if and only if Q: a 61 for all
i 6 2+, where Q1 denotes the i-th row of Q. When f = I, then
we write L2 and I; instead of L2(f) and Lé(f) reapectively.
7.2.10 Definition. Let S. be the inflation of S, defined on
on Hé(¢0 into Ié(f) by the relation (8(9)1 = 3(51), where
E 6 ITEM) and i 6 2+.
The following lemma gives some properties of SI

7.2.11 Lemma, With the above notation we have
(i) (§,n) - (§§,§fl)f, where for each Q and Y in {g(f)

we let (§.Y)f = [(Q1,Yj)]:,j=1
(ii) S. is one-to-one
(111) §k§ + n) = 51s) +-§XTD
(iv) S, is a continuous transformation

(v) mg) - A(s'g), whenever Ag is defined.

EEQQE- (1) ' (iv) 18 0bV10US. To see (v) consider

§kA§> §k[A§‘]:=1> = [S((A§>i>]:=1

a: Joana jab:
[S<j§1a,,§ >11=1 [jglaijscg >11=1 A(S§>.

Now we digress to discuss some Fourier analysis of infinite
dimensional matrix valued functions. For a matrix valued func-
+m
tion Q - [th]m,L8l whose elements are summable, we define its

n
n-th Fourier coefficient An (amL) by

n is -in9

1 2"
8% II Egg (Pm(e )e d9 .

We first prove the Parseval identity.

85

7.2.12 Lemma (Parseval identity). Let Q = [¢1,¢2,¢5,...] and

m m
Y = [$1,¢2,¢3,...] belong;to L2. Let Ak and Bk be the k-th

Fourier coefficients of ¢h and 7m reapectively. Then

21'? * +00 no
(a) (M) = 31;; M819)? (Jame = 2 2 A: 13‘1“,
k=-co m=l
211 +1» (3
2 1 ie 2 _
an M -—- a; use ”we - 2 21W
=.Q m=1
Proof. Let a: = [e ikeé M] Then 3:, 'w‘< k1< w,‘1 S‘m < m,

becomes a complete orthonormal system. We also observe that
(Y,§:) = A: and (Q,§:) = Bi. Now standard Hilbert space argu-
ments can be used to complete the proof of the theorem.
7.2.13 Remark, The space L2 consists of all weakly measurable
Lz-valued functions Y = [11:12....], for which “Y“zz = 121111‘2
is integrable. L

We now prove the Riesz-Fisher theorem for infinite
dimensional case.

7.2.14 Lemma. Let be a sequence of infinite dimensional

{ “1““,

has
matrices. Then {An}n=-m is the Fourier coefficients of a func-

_ a +1» n 2
tion Y in L2 if and only if 2 E ‘a l < m» for all
L=1 n=-oo ml, +00

m 6 2+, Furthermore in this case we have Y = 2 Aneine.

n=na

Proof. If Y 6 LE, then for each m 6 2+, we have

n 16 2
' 8 2 a \ = \w ) d9
na—Q L31 M L31 {In-a‘ In"; LE 1 2" _j‘ m(e ‘

2n a .
192
fingleme )\d9<oo.

The last inequality follows because each row of Y belongs to L2.
4a m
For the other way assume that Z 2 ‘3; ‘2 < a, for each m 6 2+
nB-co L=1

86

41:

Then for each mac 6 2+, nE-Jamd

there exists a square integrable function *mL whose n-th Fourier

< m. Hence for each m,L 6 2+

n on
coefficient is amL' Let Y [me]m,L=1' Then Y has An as

its n-th Fourier coefficient, and Y is in ii, because

211 m . on 211' . 00 +6)

2 2 2
g 2 \¢M(e1°)\ d9 = z (E ”M(elen de = z 2 MM < co .
{1:1 L=1 L=1 “3"“,
+w .
Now standard arguments show that Y = 2 Anelne.
n=-m

An important consequence of the last lemma is that if An

is the n-th Fourier coefficient of a function in L. and {B 1+”
2 n n=—m

is composed from. A s and zeros, then, {B }+m is also the
n n n'B-cn

Fourier coefficients of a function in Lé. This allows us to give

the following definition.

7.2.15 Definition. (8) If Y 6 L2 and has Fourier coefficients

d ._
0+, Y_ an YO_ will denote the function in L2

whose n-th Fourier coefficient is An for all n >'0; n 2 0,

An’ then Y , Y

n < 0 and n s 0 and zero for the remaining n's respectively.

YO will be the constant function Y0 = A0.

(b) L:, Lg+. L3, L2. will denote the subset of all functions

in L5 whose n-th Fourier coefficient vanishes for all n s O,

n‘< 0, n 2 O, n >>O respectively. Note that Y
-O+ -—+ -- .- —

belongs to L2 , L2, L2, L2 whenever Y E‘Lz.

The proof of the following lemma is obvious and hence is

o_|_’ Y+9 Y_, Yo-

omitted.

7.2.16 Lemma. (a) The sets LO+

-fl-
2, 2, L2 are closed sub-

Spaces of Li with L; J-EE.

(b) Let Y 6 L2 and let Yi denote the i-th row of Y. Then

87

- = = + = +
(1) w Y+_+-Y0 + Y_ Yo_ Y+ Y_ Yo+

an M2 = MHZ + \wguz + M2 = \w3.n2+ “vhf. 1e 2.

(m) uvin. Mu, Mn, win s M. i

* * 'k *
(1V) (Y+) = (Y )_) (Y_) = (Y )+ '

7.2.17 Remark. Similar definitions and properties can be given

for L instead of £5.

2

We now prove a convolution rule for the functions in L2.

7.2.18 Lemma (Convolution rule). Let An and Bn be the n-th

Fourier coefficients of Y and Q E {é respectively. Then the

* on
n-th Fourier coefficient of YQ is 2 A 3*
k=-m k n- -k
Proof. The n-th Fourier coefficients of the (j,L)-th element of

me

2
* *
Y0 is given by %;'£(Y(eie)§ (eie)) d9, which is

LLe

211m

211 .
1 ie * ie -ine _ l 19) - 19 -ine
_an (we )9 (e ))e de - “2" “12-: M >cpw<e )e de .

Consider

211' m _ 211 m . .
z \vm(e1°)cpm(eie>e “ewe =3 2 Mjm(ele)Hchm(ele)\de
m=l

m=l

211 a 1::
e19 2 35
guy“! >\><m§1\¢m<ee>\2>de

211m £21100

$( 2 W>de)( 2
& m'l‘wjm(e \ gm 1|Qbm

_ 2n w ._
Now since Y and 6 6 L2 we get 3 mgl\¢jmqkm\de < m. Hence

(eie % .

)lzde)

we can change the order of integration and summation,

2n . . . 2n m .
_l_ 19 * ie -ine =.l_ ie‘- 19 -in9
2" g (V(e )e (e )>jme 2n m§1w1m(e )cznfle >e de
an 211 on a n-k
1 . 19 - 19 -ine k -
= z: — t (e )(p (e )e = '2: z a b
“1.1 211$ Jm {m m=1 k=-m jmw

The last equality is by the usual convolution rule. If we change

the order in the last term we get

%;-in<w(eie>¢*(eie>)jte““9de
= ; Eakfimk= :XAK') =(':AB*) .
k'_m “=1 jm Lm k=-m k n-k j; k=:m k n-k jL
We also need the following definition.
7.2.19 Definition. Let Q be in Lg+. Then 9 is called the optimal

 

factor of f if the following three conditions hold
i i * '

<1) f(e 9) = @(e °>¢ (e19) a.c.

(ii) i 2 O

0
' * - *
(iii) For any Y E Lg+, f(ele) = Y(eie)Y (e19) = Q(eie)§ (€19),

*
we have Q0 z/YOYO .

The following uniqueness theorem can be proved exactly as
in the finite dimensional case (c.f. [15]), and hence we omit its
proof.

7.2.20 Theorem (Uniqueness theorem). Let §,Y be bounded linear

operator valued functions such that Q-l, Y.1 exist and are bounded.

LE‘: 9, ¢-1) Y: Y-1 be in 1:24..

If
a(eie)¢*(e19> = w<e19)w*(eie),

then there exists a unitary operator U0 such that

89

@(eie) = Y(eie)Uo .

Furthermore U0 3 I, if either Q0 and Y0 are equal, or they

are positive definite.

7.3 Further Analysis of Time and Spectral Doma_i_n_. In this section
we develop some results which will be needed in the next sections.

Let gn, -oo < n < an, be a SSP. We denote by gin“,
-oo < n < co the L-dimensional subprocess of it, i.e. for each

i

n E Z, we have (gL n)1 = 5“ if 0«< i s L and zero if i >'L.
9

The following theorem gives some relation between E and
EL, the predictor error matrix of gn, -m < n1< m and gL,n’
—o < n < co respectively.
7.3.1 Theorem. Let gn, -m < n < m be a SSP such that (g0,§0)
is a bounded operator. Let E and 2L be the two sided predictor
error matrices of §n, ~o‘< n < o and gL’n, -m < n < m respec-
tively. Then 2 2 K21 if and only if EL 2 XZIL for all L > 0.

Proof. If 2 > 121 then clearly EL > AZIL for all L > 0. To

 

prove the other way, let us assume EL > xZIL for all L > O,

and suppose 2‘< 121, i.e. suppose there exists a sequence cm,
0

-co< n < do with Z \cilz = 1 such that
i=1
O
'—- 2 2
2 c C = 1' < k . (1)
1,1,1 1 11 .1
Let c = A - 1' and take ‘N1 >’0 such that
on 2 2 _ , 2
2 ‘cj\ (LA—#2 l2 , for all n >N1. (2)
1‘“ X

We have

90

H 2 c E H = ( E c g , 2 c E ) = 2 c (E .E )C. < m.
i=1 i 0 i=1 i 0 i=1 i 0 k,j=1 i 0 0 J

because (§O,§O) is bounded. Hence there exists N2 > 0 such

that
Q .
H z ci§3H < 6/4, for all n >.N2 . (3)
i=n

If go is the two sided linear predictor of gm, -m < n < m

then (1) means

A2) '2: g z E—=z C( ) _—
x i.j=1Ci iJ j ij i QO’QO 1101

7 i m i a i 2

= H 31° 1 Co“ = “iglci g0 ' (iflci go‘K(o))H '

0
Hence we get WQ( 2 c1 g3)“ = K. < k: where Q is the projection
i=1
on K(O)J-FIH(m). Let QL be the projection on H(ao [‘IKL(O)‘L
Then since KL(O) 1 H(O) we obtain QL 1 Q. So there exists

N3 > 0 such that

\\Qn( Sleigé)“ < l" + 3/4, for all n > N (4)
1:

3 .

Let N = max(N1,N then by (4) we obtain that

.. 2.13)
‘pN(1z1c1§3)“ < 7" + 3/4. Hence we get

HQN1< z c1§O)+QN<2113'c§)u<x +e/4.

So we get
N i
\pN(i§1ci§0)H ' WIN (1 2+ 1615.0)“

( 2 C151) +.QN(121cf§ s 1' +'e/4 .
“Q“ i=N+l ° 0 >“

91

Hence
\\Q(:c§i)\\sx'+e/4+\\Q( ;c§1)n<i'+e/2

N
Thus HQN(izlci§3-)“2 < 0" + e/Z)2 and hence by (2) we get

N m
2
\\QN( z c153)“ < 0.’ + .2/2)2 + [0.2 - (i' + e:/2)2 - x2 2 \ci\2]
i=1 i=N+1
2 2 w 2 2 m 2 m 2
<>.->. 2\ci\=x(z\ci\ - z\ci\).
i=N+1 i=1 i=N+l

N 1 2 2 N 2
Hence ‘pN(i}31ci§o)u < ), 121\ci\ and thus we get

N N N
i i 2 2 2
12c§-<2c§ (0))st2c\,

01'

N N
H131c1<§3 - <§3\KN<0>)H2 < xzi§1\ci\2 .

So we get H 1:: c g: ”2 < 12 I; \c ‘2 which implies
i=1 i ’0 i=1 i ’

N N ___ 2 N 2
2 cit e.<). 8 \c1\
i,j=1 13 3 i=1

Hence we get ZN < )(le, which is a contradiction.

A similar theorem for the one sided predictor errors,
G and GL was proved by Gangolli [8].

We will need the follow ing theorem due to Masani [14].
7.3.2 Theorem. Let En, ~00 < n < 0°, be a finite dimensional
SSP with spectral distribution F and two sided predictor error
matrix 2. Then g“, -m < n < an is minimal full rank if and

1

only if F'.1 exists a.e. and F'- 6 L1. In this case we have

To progress further in this section, we suppose our
stationary stochastic process satisfies the following condition.
7.3.3 Assumption. Let g“, -m‘< n1< a be a SSP with a spectral
density f such that m(eie)I s f(eie) s‘M(eie)I a.e. with
1/m(eie) and M(eie) being summable.

Let fL be the spectral density of the L-dimensional sub-

process of go, -m1< n < n. Then m(e19)I s f(eie) $'M(eie)l

10319) s (l/m(eie))l.

and hence for all L >'O, (1/m(ele))I s f;
Hence by theorem 7.3.2, the subprocess §L n’ -m‘< n < m is

’
minimal full rank for each L >’O. Now applying theorem 7.3.2

to these subprocesses we get

211
a ,l_ -l 19 -l
2L [2n 5 EL (e )de] , for all L > 0.

Thus by theorem 7.3.1, 2 > E%-
-1 2" 19 n -1
have G >’z > Lifi'g (l/m(e ))d9] I.

2" 19 -1
g (l/m(e ))de] 1]: We also
Summarizing we get the following lemma.
7.3.4 ngmg. If 5“, -m‘< n <iw is a SSP with density f
satisfying the assumption 7.3.3, then G >'2 > A1 for some
7. > 0. Now since 6‘1 exists we let 9k = /c—:T “k’ and we call
it the one sided normalized innovation process of the processes
in, -w‘< n <,m.
Now using the last lemma in conjunction with several
results in [12], we prove the following theorem.

7.3.5 Theorem. Let En, -m < n < an be a SSP with density f

satisfying assumption 7.3.3. Then 5“, -m‘< n'< a is purely

93

nondeterministic and f is factorable as

f(eie) = 9(e19)¢*(eie),

where Q(ei 6) = 2 Cke and 2 Cijlz < m, for each
k=0 k=0 j= —l

i 6 2+, Furthermore 6 is the optimal factor with CO ==/C
and CR = (§O,e_k ), for each R E Z+ .

Proof. Let d(ei 9) = (l A m(ei 9))1, then f(ei 9) 2 d(ei 6) a.e.

and we have

2n . 2n .
g \log(1 A m(e19)\de s & log(l/(l A m(e16)))d9

5 log En(l/(1 A m(eie)))de s log[g(l/m(eie))de + I d6].
211 EC
where E = {9, o s e s 2", m(e 6) < 1}. Hence g \log(l A m(e e)de s
log[in(l/m(e19 ))de + l] < m. 'Now by [17], p. 165, we see that f
is factorable and hence by [17], p. 163, the SSP g“, -m < n < a

is purely nondeterministic. Thus by [17], pp. 155-156, g“,

on

-ml< ne< a, has a one sided moving average, g = 2.C “a k’ such
m m k k=0k

that for each i 6 2+. 2 2 \C ‘ < m and e , -m‘< n1< m
j=l k=1 13 n

is the one sided normalized innovation process of g“, -m‘< n < w.

19 “ 1ke .
Now take Q(e ) = 8 Cke . It is clear that 6 is an analytic

k!0
factor of f, and we have

fl
(5 .9 ) ‘ ( E C 9 .9 ) = C (e ,e ) = C .
0 O k=0 k -k 0 0 O 0 0
Also we have
a

(go.e_k) = ( 2 Ck9_k.9_k) = (Cke_k,e_k) = ck(e_k.e_k) = Ck -
k=0

Finally, in order to show 6 is an optimal factor, assume that

94

Y is an analytic factor, then we have f = 66 =
N
H('N:-1) = 6<§k. -N s k s -1). (§O\H(-N,-1)) = kg Ak§_k. Then

G = lim GN’ where

N-Om
N N
=(§-2A§,,§-2A§)
GN o k=1 k k 0 k=1 k -k
2n N . * . N .
= fi- I (I - zAkelkg) f(e‘gm - 2: Ake‘keme
" o k=1 k=1
1 2" N we * * N ike
=E'I(I' ZAke )Y‘HI- ZAke )de.
0 k=1 k=1
Hence
N N * N N * N
k=0
N °° ik
where Ek. is the k-th Fourier coefficient of Y(I - 2 Ake 6).
k=l
It is easy to see that E? = Y0 for all N. Thus
GN 2 YOYO.

Now let N a m to get

* *
QOQO = G 2 YOYO .
The proof of optimality given above is adapted from the proof
of the similar result due to Masani, for finite dimensional case.
We will need the following lemma.
7.3.6 Lemma, Let En, -m1< n1< a be a SSP with spectral density
f satisfying assumption 7.3.3. Let en, ~m4< n < m and Q be
its normalized innovation and generating function respectively. Then

(a) e-nie -1(eie

Q ) belongs to 'Lé(f) and corresponds to en in

g(m) .

95

(b) For Y in {g(f), Y6 belongs to L2

(c) For any Y in Lé(f), if we let Ak be the k-th Fourier

n
coefficient of Y0, then ( Z Ake1k9)§-1

k=-n_,
Proof. (a) Since an belongs to H(ao, there exists a correspond-

v Y in Lé(f) sense.

ing element of Lé(f), say Y. Now consider

so

me ___ °° i(n+j)e 3 ije
(e ”um (j§0(§0’ej)e )wi jgn(go’en-j)c,me .(1)
On the other hand, we have
211 211‘ a .
~ike 19 * 19 -ike ie 19
f \y d = d
g e < (e > (e >>L m e g e (jE1f41(e >¢jm<e >) e
_ -ike

The last equality is by definition, since for arbitrary Y, Q

- -Lme‘” -
in L2(f) we have (§.Y)L’m (9 ,Y ) Ii,§=1mcifijwjmde. Now
by (2) we get

2n

& e‘1k9<£(eie)w*<eie)>L’mde = (e‘ik91.v), m = <§O,en_k> (3)

L.m°

Now (1) and (3) imply that for each L,m 6 2+, (enie§(eie))L m

*
and (f(eie)Y (619))L m have the same Fourier coefficients.

nie ie . 19 * 19
Hence for each L,m 6 2+. we have (e Q(e ))L,m (f(e )Y (e ))L,m

which means

e"‘°¢<e‘°) = f(e1°>v*(eie) = Q(e19>¢*(eie>v*(e1°).

ine *-1
e

**
Now since Q(eie) is invertible we get Y (e19) = Q (e16).

Thus

eie) = e-nie -l(eie)

Y( 9 a.e.

96

-nie -1

so e Q (eie

) corresponds to en'

(b) Suppose Y 6 Lé(f). It suffices to show that (Y§)k'€ L2,
k

for all k 6 2+. We observe that (Yé) - [¢k,1’¢k,2’°°']§'

Now we observe

2n 2n
5 “(mknzde = g \\[¢k,1:‘¥k,2,...]q\\2de
gg ([wk 1"“, 2,...JQ, [Wk,1,‘¥k,2,...]§)de

=t§ (Uk,1’wk,2’°”]f’ [¢k,1,¢k’2,...])de .

2n 2n m _
Hence {E H(W) k2“ d9 '3 1 figlwk’jfidwkdde < co. The last in-

equality follows from wk 6 L2(f).

(c) Using the Parseval identity of lemma 7.2.12 twice, we get

n m m m

)Luzde= 2 2:\ak

ike
\ z
k=..n j=1 Lj k=—m j=1‘ Lj‘

1 2n n
"' ( 2: e
2. g u 1.--,fk

= i; flmfinzde = uom‘u < «>-

n —
The last inequality follows by (b). So ZlAkeike is in L2-
“ ike -1 - ='“
Hence 2(Ak e )6 belongs to L2(f) for each n 2 0.
k--n
(This follows by similar argument provided below‘.) Now to show
n
8(A keike)Q-EY converges to zero in L2(f), it is SUffiCient to show
kh“ “ me -1 L
that (( 2 Aka )Q - Y ) ,q 0 for each L 6 2+. Consider
ks-n
eike 1 2" “ ik e -1 4, 2
“(zAkew'Wn11E“((mzAke )é -Y)\\de
k--n

ike

2n
75E \\<<< 22 A e )e' - w>/£)*’u2 de

k--nk

%;;i \\<( z Ae “‘9 - mil/o‘nzde =

k--n

 

Illillfl‘llku

97

2n n _ 2
= L \\<< 2 A eike - W‘s Vin de
2n k=~n k
2n n n
= %;£ ((k z likeike - W’s 1/f ff (1*, (k z Akaike - whde .
gun =-n
Hence
n . _ 2 2n n . 2
U( E Akelken 1 - “L“ = :79; “(k 2 AkeIke - we)!“ de.
=-n =-n
Hence
“ ike -1 L 2 “ ike L 2
\\(( 2 Ake )@ - Y) H = \\( z Ake - M) H de. (4)
k=-n k=-n

Now since Ak's are the Fourier coefficients of Y6, by the
Parseval identity the right hand side of (4) converges to zero

n
and hence “( Z Akeikeyb-1 - Y)L“2 converges to zero.

k8-n
Now we prove the following corollary which is important.
7.3.7 Corollary. Let Pv(eie) = [e-ive§(eie)]o+§-1(eie), v > 0.

Then Fv(ele) belongs to Lé(f) and correSponds to the linear
predictor iv = (§V\H(0)) under the isomorphism S. given in
definition 7.2.10.

Proof. Consider

2n
\\(1‘v)”\\2 i7; \\(rv<eie))"/f(e19)“2de

2n
i; g n<rv(ei°>/£<eie))”u2de

2n
1 -1 e 1 -1 1 2
'27; \\([e v Q(e e)‘jmfi /f(e e))”\\ de

2n _
i;- {E “([e ”eueienmfi 1(e19)/f<e‘°)nzde

211 _
51-; J‘ “(Le ”eueienop‘nzde .
0

98

The last equality follows as in the proof of part (c) of the last
i 2 -i 2
lemma. Hence “(Pv(e 9))Lu - “([e veééeie)]0+)LH s “Q(eie))LH2.
L 2 1.. n * 3 1.. n
Thus “(Fv) “ s 2" g (eLQQ ,eL)de 2" £ (eLf,eL)d6‘< m, because
f is weakly summable. Hence Fv(eie) = [e-iveQ(eie)]0+§-1(eie)

belongs to L2 Now let C be the k-th Fourier coefficient of

k
the generating function Q; Fv(eie)§(eie) = [e-iv9§(eie)10+ 3
CD
2 C eike belongs to L and by lemma 7.3.6 part (c), we see
v+k 2
k=0
that
N ike -1 19 ie —
(EC 8 )Q (8)-o1"(e)inL(f)-
k=0 v+k v 2

Now let Yv be the random function in H(m) corresponded to
TV in Lé(f). Then by part (a) of lemma 7.3.6, we have
N _

2 C 9_ v v , in H sense.
k=0 v+k k v

N A
On the other hand, 2 Cv+ke-k converges to gv, because

.. k=o . .

§v = kgocv+kek. Hence Yv = gv, i.e. 5v corresponds to Fv,
under the isomorphism S;

Let gn, -m¢< n < m be a minimal full rank process with
two sided innovation process gm, -m‘< n < m. We write ah =
8-1Cn and call an, -w‘< n < w the normalized two sided
innovation process of 5“, -w‘< n < a.

Now we prove the following lemma which is crucial in
getting our algorithm.

7.3.8 Lemma. Let g“, «m‘< nI< m be a SSP with density f
satisfying assumption 7.3.3 and let an, -m < n < m be its two
sided normalized innovation process. Then

(a) The sequences ah, -a‘< n < m and gn, -m < ni< a are bi-

orthogonal.

99

co

*
(b) an 3 Z Dken-l-k’ where 9“, -co < n < an 18 the one sided

k=0
normalized innovation process of gn, -w < n < co, and Dk is
the k-th Fourier coefficient of the inverse 9-1 of the generat-
0
ing function Q(eie) I z Ckeike.
k=0
(c) {I e 1.3+ .
Proof. (a) We have
(a g > = (2‘1; 5 > = 2'1<; § )
0’ O 0’ O 0’ O

2'1(§0 - <§0|EKO)),§O)
= 2'1(§O - <§0\E10>). g0 - (soliko>>.
So we get
(ao.§0) = 2'1<co.co) = 2'12 = I . (1)
For n aé 0 we have go 6 f(n) J. O’k’ Hence

(Gk: g0) = 0 ° (2)

Using (1), (2) and the stationarity of our process, we get (a).

(b) and (c). Since ark 136(0) :2 H(n) for n < O, we get

*nim)=§0@- (a
n

“0
Now since (10 E 5(a) and H(oo) - H(wo) G) g(ek, ~00 < k < co) , we
get 010 6 g(ek’ -co < k < an). In fact a0 6 g(ek, k 2 0), because
for each k < 0, 9R C H(-l) : {(0) .L 00' Hence
°° °° °° k 2
a0 - kfoAkek, with 133.0 j2.31mi” < co, (4)

for each i 6 2+. Now by Wold's decomposition we have

100

a)
En = 2 Ckpn-ke+ V“, where vn E H(-m). Hence by (3) and (4) we
k=0
obtain
a: on n *
(a .E ) = ( z A e . z c e ) = z A,C ‘ .
0n j=ojjk-okn-k J‘OJn-j

Hence by part (a) we have

n *
z A c _. = 5 I . (5)
j=0 j n J n0

Then by the convolution rule we get

—ike

( 2 AR? )§*(eie) = I a.e. (6)
k=0

*
(6), together with the invertibility of Q , implies that

 

*- m -'
e 1 = z Ake Ike a.e. (7)
k=0
Taking adjoint, we get
- m -' *
Q 1 = ( Z Ake Ike) a.e. (8)
k=0
-1 1 a ik
So the (L,m)-th entry of Q is (Q- ) = z a e 9.
Lam k=o “I’L—
Hence the k-th Fourier coefficients of (Q'l) is ak , for
L,m m,L
*
each m,L, 1 s m,L'< m. Hence if we let Dk I Ak then we have
m 0
Q'1 g 2 Dkeike
k=0
in the following sense
-1 1 m k -ike
(e <e°>>M= “we . <9)
’ kgo :
-1 19 2 -1 19 i9 -1 -
Now from “6 (e )u - “f (e )u s 1/m(e ) we see that Q 6 L2,
Q Q
2
which means 2 2 \d: \ < o, for all L, l s Ll< an This and
m=l k=0 'm

(9) implies that

101

) = 2 D e
k=0 k

-1 ie

6 (e ike

, in {g(f>.
This completes the proof of (b) and (c).

7.4 Determination of the generating function and the linear predictor.
In this section we shall express the generating function of a SSP,
gm, -a.< n1< m satisfying some boundedness conditions (to be made
precise later) in terms of the Spectral density f by an iterative
procedure as in the finite dimensional case [14]. We shall then
derive a computable expression for the linear predictor error
matrix. We mention here that because of infinite dimensionality
our convergents here would be in a weaker sense than the convergents
of the corresponding results in [14].

Here we suppose that our process in, -m < n < m has a
Spectral density f satisfying the following assumption.
7.4.1 Assumption. Let f(eie), the spectral density of our pro-

cess satisfy
m(eie)I s f(eie) s M(eie)I,

with M(eie), 1/m(eie) and M(eie)/m(eie) being summable.

We need the following lemma.
7.4.2 Lemma. Let gn, -m‘< n1< a be a SSP whose spectral density
f, satisfies the assumption 7.4.1. Then there exists a nonnegative
real valued function f1(eie) and a nonnegative infinite dimensional

matrix valued function f2(e16

) such that
(a) f<e19> - £1<e1°>£2<e‘°)
(b) f2(e19) - I +-N(eie), where “N(eie)“B < l a.e.

(c) £20219) satisfies assumption 7.3.3

102

19
(d) £1<e

Proof. (a) Let f1(eie) =35(M(eie) +m(eie)) and f2(eie) =

) and l/f1(ele) being summable.

f(eie)/f1(eie) which are defined a.e. and satisfies (a).
(b) Let N(eie) = f2(eie) - I = (f1(eie)/f2(eie)) - I. Since

m(ele)l s f(eie) s M(eie)l, for all e's we get

19 2 19 M(eie) - m(eie)
\N(e ) = f(e ) — I s .
\ “B ‘m(eie) + M(eie) \ M(ele) +,m(e18)

Now since m(eie) > 0 a.e. we get HN(eie)\\B < l a.e.

(c) Now from f2(eie) = I + N(eie) = f(eie)/f1(eie), we get

19 19
2m(e ) 19 I S f2(eie) ‘ 2M(e ) I .

M(eie) + m(e ) M(eie) + m(eie)
Hence (m(eie)/M(eie))l s £2(e19) s 2(M(e19)/M(eie)) = 21, which
completes the proof of (c).
(d) f1(eie) and 1/f1(eie) are summable, because we have
0 s g(e”) s M(eie) and o s 1/£1(e19) s 2/m(eie).

The following theorem gives the relation between the gen-
erating functions of the spectral densities f, f1 and f2.
7.4.3 Theorem” Let f be the spectral density of a SSP which
satisfies the assumption 7.4.1. Let f1(eie), f2(eie) and
N(eie) be as in theorem 7.4.2. Let 6, ml, 62 be the generat-
ing functions and G, g1, G2 be the predictor error matrices
of the spectral densities f, f1, f2 respectively. Then
(a) §-1, (l/qh)I and 651 are in {3+}

(b) Q B qifiz.
(c) G - 3162.
2522;. (a) is clear from theorem 7.4.2 and lemma 7.3.8. For

(b) consider

103
'k _ — _ §* '3: - *
m - f — flfz slsléz 2 = (cplizxcpléz) - Y‘i’ .

where Y = qd62. By (a) and convolution rule we see that Y

and Y-1 belong to Lg+.

of 6 and Y are positive matrices, we can apply the uniqueness

Since the O-th Fourier coefficients

theorem 7.2.21 to conclude 6 = Y. Hence 6 3 mléz-

Now (c) follows because

* * __. __. __. __.*
=/§I/é:/CTZ/§; .__ 8162'

Since f1 is a real valued spectral density one can find
its generating function by the usual method. So in order to find
Q we just have to get an algorithm to find the Optimal factor of

f Hence in view of the last theorem, we can assune that our

2.

spectral f satisfies the following condition.

7.4.4 Assumption. Let f be a spectral density of a SSP such

that f(eie) = I + N(eie), where N(eie) is a Hermiation valued

function with the following two properties.

(i) \\N(eie)uB < 1 a.e.

(ii) m(eie)l s I + N(eie) i MI, where M. is a positive

constant and l/m(eie) is summable.
From now on we will be working under the set up of assump-

tion 7.4.4.

7.4.5 Definition. For any Y 6 L2 define 0(Y) = (YN)+, this

makes sense because “N(eie)“B s 1. Now for each Y 6 L; we de-

fine 5 by (5101 a"'001'1), for all 1 62+.

We omit the easy proof of the following lemma.

104

7.4.6 Leanna. (a) 9 is abounded operator on L2 into L2

with the Banach norm less than or equal to one.
(b) 9(1) = 11+, 92(1) = (N+N)+,...
Now we prove the following lemma.
7.4.7 Lemma. Let 6 and G be the generating function and the
predictor error matrix of the spectral density f(eie) =

I + N(eie) satisfying condition 7.4.4. Then

— - -l
swam/G 6 )= 1,
where :0 is the identity operator on 172.

Proof. Let Y = /G 6-1. Then by theorem 7.3.8 part (c), Y be-

longs to L2 and Y0 = I. Hence Y = I + Y+. Next, since

I + N(eie) = 6(eie)6*(eie) we get

- * -
w+1rN fG6I(I+N)=/G6 gig.

Hence Y++(YN)+ (Y+YN)+=O. Thus Y-I+(YN)+=O. Hence

d+5>m =1.

We next state the following theorem.
7.4.8 Theorem. Let a and 5 be as in the definition 7.4.5.
Then
(a) 0 is a strict contraction on L2H, i.e. 0 9‘ Y E LCZH. implies
s... new < M-
(b) 3+; is one-to-one on L2 into itself.
(c) (9Y3) " ($891!), for all Y,X 6 L;
M. (a) By assumption 7.4.4 there exists an e > 0 and a

set Cs with positive measure such that

105

11N(eie)11B <,/1 - e, for all e 6 Ce.

0+

Let 0 1‘ Y 6 L2 Since 1191111 = 11(YN)+11 < 11YN11 we have

11 . 211 .
19112 s i1 11v<e19>u<e191112de e)§;1s 11Y(e19)11211N(e19)11§de

<1 - e>%-1111<12de + 171/0 1111112., -

Hence we get 11911112 S11Y112 'g-‘1811Y112L2de. Now since 0 9‘ Y € L34-

one can see that £611Y(e19)11L2de > 0, which means 1191/11 < 1111111.

(b) Let Y 6 L2 and suppose that (.11 + 9) (Y)= . Then
(J+9)(Y 1) = Y i4-001'1) = 0 for all i 6 2+. Hence Y1 =
Q(Yi). So Y1 6 L3..- and consequently 11Yi11 = 110(Yi)11. Hence
by part (a) we get Y1 = 0 for all i 6 2+, which means Y = 0.
This completes the proof of (b).

.1.
(c) For Y and X in L2 we have

(9”) = (<11N)+,x1 = (11”,) .
(The last equality follows from Parseval identity.) Hence

(91%)!) = (TEX ) = (YNJO = (‘1’,XN)

= (mm) = <11.<m>+) = (11.91:).

We now show that for the operator (:6 + 53-1, the usual
geometric series converges; the convergence, as one expects, is
strongly and in L2(f) sense.

7.4.9 Theorem. Let 0 and 5 be as in definition 7.4.5. Then

(a) 9n _. 0 strongly in L2, as n ... ao, i.e. for each Y 6 L2,

lim119n 1111- o.

106

_.n _. ._
(b) 9 _. 0 strongly in L2, i.e. for each Y ELZ, i 62

lim 110nY111 = o.

n-co:

(c) If Y is in the range of :a +5 then

+9

n —
G + 9) -1(Y) = lim 2 (-1)k5k(Y) , in L2.
n—m k=0

Proof. Let Y 6 L2. Then using theorem 7.4.8 (a) and an argu-
ment similar to the one used by Masani in theorem 4.8 of [14],
one can show that 119nY11 -* 0, as n _. co. This completes (a).
Now (b) and (c) immediately follow from (a).

We know that the range of :5 + 5 is a subset of L:,
containing I. Let us write

Y = (5 +5)'1(1) =1 -N++ (N+N)++... 61:2.
The function Y is thus available from the Spectral density by
an iterative method. We shall now show that the generating func-
tion 6 of our SSP and its predictor error matrix G are easily
obtainable from Y.
7.4.10 Theorem. Let f, the Spectral density of our SSP, satisfy
the assumption 7.4.4. Then (a) Y =,flG 6-1, (b) YfY* = G a.e.

Proof. (a) Since by theorem 7.4.9 (c) and lemma 7.4.7 we have
(:7 +5301) = I = G +5)(/G {1).

On the other hand, by theorem 7.4.8 (b), 3 + 5 is one-to-one.

Hence Y = ./G 6-1 which gives (a).

(b) By (a) we get /G = Y6 and hence we get

G =/G f6 = (1m (Y6)* = we"? = 1191*.

107

Since 6 and 6.1 E {3+ it follows that Y and Y.1 E Lg+.
m - a) i

Let Y(eie) = 2 Akeike, Y 1(em) =- 2 Bke k9.
k=0 k=0

' = - ‘+ -...
From the series Y I N+_ (N+'N)+ we see that

A0 = I, and for each n > 0
moo

A =-I"'+ EP'I" '2 EF'F' 1'" +... (1)
n _ m

n “_1 n -n n=lp=1 p n-p m-n
where Pi iS the k-th Fourier coefficient of N. Thus the co-
efficient Ak is determinable. The coefficient Bk can be found

from the relations

AOBO = I BOAO

A031 +1A1B0 = 0 = BOA1 +'BIA0 (2)

Since A0 = I matrix inversion will not be encountered in finding

B . Now for C and D the k-th Fourier coefficients of 6

k k k’
and 6.1 respectively, we have Ak = /G Dk’ Bk = Ck fG-l. But
Q
*
G can be evaluated from G = YfY , so we can get 6 = Z Ckeike.

k=0
The last thing we are going to do in this section is, given

the SSP fin, -m1< n < a, with Spectral density f, to find a Scheme
for computing Ev, the linear predictor of log v.

Let gn, -m1< n < a be a SSP with Spectral density f.
Let M be a constant and m(eie) be a scalar valued function
with l/m(e19) being summable. Suppose m(ei9)I s f(eie) 5 MI.
It is clear that under this condition, f satisfies assumption 7.4.1,
and hence one is able to find C and D the k-th Fourier co-

k k’

efficients of 6 and 6"1 respectively. So one can compute

108

- v9§<eie 6-1

. . . 1
Evk’ the k-thkFourier coeff1c1ents of [e ”0+ , in
fact Evk = “200 Dk-n' Also in this case, we have
n
- - -0+ ik -i ‘ -
[e ive6(eie)] 6 1 6 L . Hence 2 Ev ke 9 -o [e ve6(ele)"1 6 1
0+ 2 k==0v 0+

as n -+ co, in :2 sense. From f(eie) 5 MI it follows that,
as n -* on,

n
ZEe

ike _, E -ive ie --1
vk e
k=0

6(e ”0+6 (em), in 172(f).

Applying the isomorphism S- we see that

n
lim 2E €- =§ , in H(oo) sense.
n~m k=0 Vk k v

So we have a scheme for finding Ev.

Meanwhile we proved the following theorem.
7.4.11 Theorem. Let En, -oo < n < on be a SSP with spectral
density f satisfying m(eiefl 5 He”) 5 MI, where M is a

18

constant and m(e ) is a scalar valued function with a summable

reciprocal. Then with the above notation we have

n —
lim 2, E vkeike = [e We Mum 19), in 12(5).
nam k=0
n .. _
Hence we get lim 2 E vkg-k =§v, in H(oo).

nan k=0

9

CHAPTER VIII

MINIMALITY AND INTERPOLATION OF INFINITE
DIMENSIONAL STATIONARY STOCHASTIC PROCESSES

8.1 Introduction. In this chapter we investigate the problems

of interpolation and minimality for infinite dimensional stationary
stochastic processes. We will continue with the notation of the
last chapter. We will first extend to the infinite dimensional
case, most of the results of H. Salehi [21], [22], [23].

Using his technique in [233 we prove two infinite dimensional
extensions for a result due to P. Masani [143 on the minimal full

rank stationary stochastic processes (c.f. theorem 7.3.2).

8.2 Minimality and interpolation. We assume that our stationary
stochastic process satisfies the following assumption.

8.2.1 Assumption. Our SSP has a spectral density f, with

o < m(eiefl :2 He”) s M(eie)l a.e.

8.2.2 Defipition. Let M and N be weakly measurable 1 X m
matrix valued functions. We say <M;N> is Hellinger integrable

with respect to f if

2n
.. *
<M,N>f g é—fl-g M(eiefi 1(319)1‘1 (819)69 < 0°-

We denote by H2(f) the class of all 1 X m matrix valued func-
tions M, for which <M,M> is Hellinger integrable with respect

to f.

109

110

The following lemma gives some properties of H2(f)
functions.

8.2.3 Lemma, (a) M 6 H2(f) if and only if Mf-k 6 L2.
(b) M. and N in H2(f) implies that <M;N> is Hellinger
integrable with respect to f.

(c) M. and N in H2(f) implies that M +-N E_H2(f).
EEO—f; Since d‘l,N>f = (M,N)f, the proof follows from the
corresponding properties of L2(f).

The following lemma is needed later to establish the
isomorphism between H2(f) and the space L2(f) introduced in
§7.2.

8.2.4 Lemma, Let 'M(eie) = Q(eie)f(eie) and N(eie) =
Y(eie)f(eie), where Q and Y 6 L2(f). Then <M;N>» is Hellinger

integrable with respect to f and
01,N>f = (Qav) f.
The proof is clear because

2n
L 19 -1 is * 19
<M,N>f a 21W]; M(e )f (e )N (e Ne

2w

'3 f” (Q(eieuuienf‘lmie)(V(ei°)f<e1°))*de
0

2n
-= -:--£ a(e‘°)f<eie)v*<e19>de = woof.
TI

Let T be the linear transformation defined on L2(f)
into H2(f) by TQ 8 9f.
Some important prOperties of T are stated in the next

theorem.

111

8.2.5 Theorem. (a) T is a linear Operator on L2(f) into
H2(f), i.e. for any a,b E k and any Y,Q G L2(f), we have
T(aQ + bY) = aTQ + bTY.

(b) T is an isomorphism, in fact <T§,T‘Y>f = (Q,Y)f for all
Q and Y 6 L2(f).

(c) T is onto. In fact, if M E H2(f) then T(Mf-l) = M.
‘ggggf. (a) is obvious. (b) follows from lemma 8.2.4. To see

1

(c) we just have to show that Mf- is in L2(f), which is the

case because

_ 2n . _ . . _ .
(Mf'1,Mf 1) i3); (M(eleu 1(elenfueie)(M(e‘ew 1(e19))*de

2n . _ *
-:-;£ M(ele)f 1(eie)M (eieme < co.

Now since L2(f) is a Hilbert Space, the following
corollary whose proof is omitted, is an immediate consequence
of the last theorem.
8.2.6 Corollary. H2(f) is a Hilbert space over complex numbers.
8.2.7 Definition. We denote by Hé(f) the Space of all m X m
matrix valued functions, each row of which being in H2(f). For

M and N in §2(£), we define <M,N>f by

= 1 J
(d~i,N>f)i,j <M ,N >f .
Let the transformation T. on Lé(f) into Hé(f) be the
inflation of T. The results of theorem 8.2.5 and the usual
technique can be used to show that T. is a one-to-one transforma-
tion on {g(f) onto H9(f) which is an isometry, in fact for

all T and Y 6 I--'2(f) 9

112

4%. TY>f = (Q, ‘1’)f -

1 e i§(£) and T(Mf'l) = M.

Furthermore for any M E H2(f) , Mf-
Now let us give the following definition.

8.2.8 Definition. Let g“, ~oo< n < as be a SSP. Let J be a

 

subset of integers Z. We write mJ = 5(§j, j E J), SJ = 51;, fl H(oo),
where J' = Z\J. We say that

(a) J is interpolable with respect to 5“, -oo < n < on if

fiJ = {0}.

(b) gn, -co < n < an is interpolable if each bounded subset, J, of
integers is interpolable with respect to En, -co < n < ea.

(c) The process gn, -co < n < on is minimal if for each integer

j, J = {j} is not interpolable with respect to En, -co < n < on.

8.2.9 Definition. (a) For each element § Elf-1‘] we write

1 -i

P§(e 6) = 2 (§.§J)e 19.
JEJ

(b) We define the operator Q on 6 into H2(f) to be

J

Q: g Pg'

Part (a) of the next theorem shows that Qg E H2(f) , for
each g 6 6J.
8.2.10 Theorem. (a) Let g 6 SJ and ‘1' 6 1:2(f) such that
gY-g. Then P IYf.

E

(b) Q is an isometry on 5 into H2(f), in fact

J

(gmf = 42mm,

where g and “n are any two elements of H(oo).

Proof. (a) Let Y e {5(f) and g - g». Then

113

(§ gk)= Ufeieh= 721E302 191>f<e‘3)~1:r*<eie)de
On the other hand we have
2n 2n .
‘1_ Mike 1 1kg
{8 p§<e 9)de= 2" {c z (5 g )e jee) }de

2" jEJ

(21(1<"J')ede = (La) .

= jEJ Zn in (§,§je

Hence Yf and P have the same Fourier coefficients, which

E
hmplies

P = Yf .

(b) Let g and nefiJ and §,Yef2(f) such that §Y=§

and so = n. Then we have

42%.wa = <9 ,P >f = <I'Y,T§>f

(M)f = (m).

The following theorem gives a characterization for inter-
polability of an infinite dimensional SSP.
8.2.11 Theorem. Let 5“, -o‘< n < m be a SSP. Then it is
interpolable if and only if for any trigonometric polynomial P
with matrix coefficients, either Piszero in H2(f) or P é H2(f) .
2322;. Sufficiency. Suppose fiJ # [0} for some bounded subset
J of 2. Then there exists 0 i § 6 SJ. Hence we get 0 # (§,§) =
<P§,P§>f. Hence Pg
a,a).

is a non zero trigonometric polynomial in

Necessity. Suppose there exists a non zero trigonometric

polynomial in Hé(f). Then é = Pf.1 E {g(f). Hence there exists

114

g, o 15 g 6 £11..) such that a} = :5. We have.

2n
1
(ask) = 21710. 9<e19)f(eie>e keds
211
_1__ 1k is -1 is ie
=2ޣ 9m )f (e )f(e )d9
= 1 in ikep( le)de ._ 2 L A ei(k-j)ed9
2 J‘EJ 2 Jo ‘3

Hence

A_k k EJ
(g)gk) ={ ,
o k c! J

where P(ele) = g A_je-ije. So we see that (§,§k) = 0 if
jEJ _ ._

j 6 J and hence g E m;]. But obviously E E H(m). Therefore

g 6 SJ and furthermore

Pg = z (5.5,)e‘1j9 = 2 A_J.e‘ije = P.
jEJ jEJ

Hence P = Pg. Now since 0 # (§,§) = (T§,T§)f = <P§,P§>f =
<P,P>f. Therefore 53 i {0}. Hence J is not interpolable with
respect to g“, -m < n < n. Thus gn, -m < n < m is not inter-
polable.

In the next two theorems we give generalizations of theorem
7.3.2.
8.2.12 Theorem. Let gn, -o < nl< m be a SSP whose density, f
satisfies m(eie)l s He”) 5 M(eie)I with M(eie) and 1/m(eie)
being summable. Then the process g“, -m < n < m is minimal

full rank and we have

2n
1 -l i -1
E a [Egg f (e 9)de] .

115

Proof. By lemma 7.3.4, 2 >'xI for some positive number 1 > 0.
Hence the process is minimal full rank. By theorem 8.2.10 (b)

we have
2 = (QO’QO) = dlgoa ng>f '
But ng = Pgo = (go,go) = 2- So we get
211
1 -1 ’
2=<z.z>f=§;ng (e19): dg-
Now since 1/m(eie) is summable one can see that
211
l_ -1 i
z = mug f (e ewe]: .
Now because 2 2 [I we get
2n
1 -l i -1
z=tgjf (e ewe] .
0

Theorem 8.2.11 and lemma 7.3.4 give sufficient conditions
for minimal full rank processes. The next theorem provides a
necessary and sufficient condition for a process to be minimal
full rank. The next theorem also gives a natural extension of
theorem 7.3.2.
8.2.13 Theorem. Let gn, -m‘< n < m be a SSP with spectral
density f satisfying 0 < m(eie)I s f(eie) sZM(eie)I a.e.,
where M(eie) is a summable scalar valued function. Let fL
be the top left L x L submatrix of f. Then

(a) the process g“, -o¢< n < a is minimal full rank if and

only if there exists a constant p such that

211’ _
g (g(eien 1as s ML,

116

uniformly for all L, 1 S L < as.

(b) We have
211 .
2..- inf L-I (fL(e19))'1de]‘1
15L<au 0

 

Proof. (a) The process gn, -m'< ni< m is minimal full rank if
and only if I: 2 ),I for some 1. > 0. By lemma 7.3.1 we see that
2 2 11 if and only if 2L 2 [IL uniformly in L, 1 s‘L < m.
Hence the process En, -9 < n < m is minimal if and only if
2L 2 11L uniformly in L, l s L«< m. But by lemma 7.3.2 we know

211
that 2L1=21ng (fL (eie))1de. So g“ , -m<< n‘< a is minimal

2n 19

full rank if and only if g (fL (e uniformly in L,

I
)) Ide<uL
15L<co.

(b) We know that EL 4 2 strongly, as n a m. 'Now taking limit

on both sides of
2n
g _1_ 19 -1 -1
2L 2M1 <£L<e )) de]
we get

2n
1 19 "1 -l
2 a linf” [EX (fL(e )) 1

APPENDIX

 

APPENDIX

*
Here we consider an example of a weakly summable B(I,I )

valued function f = f(eie) on the unit circle. Using the

technique of lemma 4.3.1 we construct a quasi square root of f.

We adhere to the notations of lemma 4.3.1.

as

*
Let I = L1, X'= L2. Hence I = L . Take the partition
n n
E =[L2-—-—gm,-££—:-—1-m,l$n<oo of [0,211), foreach
n 2n-1 2n-l
x = (x1,x2,...) 6 L1 we define

f(eie)x - (x1,x2,...,xn,0,0,...) if e 6 En.

n
2
Since (f(eie)x)(x) - 2 \xj\ for e 6 En we see that f is
1:1
a nonnegative operator valued function. It is clear that
f = f(eie) is measurable and \£(e19)\B s 1 for each 9. Now

we give the construction of a quasi square root of f as out-

lined in lemma 4.3.1. We first obtain the operator T. Since

2n
(Txxrx) = (T*Tx><x> = fi— 3 (f(e‘°)x><x>de
TT

Q

2n
d6 ‘ “’ 2 Z
"n-i £n\xj\2 2"n=1 j= 1 2n xj\2

= z z -- x z -' x
n-l j-l 2n ‘ 1‘2: 1:1 “=1 2“ ‘ J‘
” 1 2

a Z x
j=l 21'1 \ 1‘

117

 

118

2
one can see that T: L1 a L can be taken as

2
Tx 8 (x1, XZA/Z, xBA/Z ,...).
Hence the function g in the proof of lemma 4.3.1 is given by

n
o -1 _-
g(e‘9)(a.b) = 2 2j a,b., e e E ;
j=1 J J n

2

a = (al,az,a3,...) and b = (b1,b2,b3,...) in L .

Consider the countable set {xi}:=1 in the proof of lemma 4.3.1

to be given by xi = (611):.1. Then Tx = (6 / 2j-1)m

1 31 j=1°
the Gram-Schmidt orthogonalization of {Txi}:=1 becomes

Now

{e1}:;1, where ei = (531):;1. Hence the matrix valued function

0
[gij]i,j=l is given by

21‘1 if i = j
g .(eie) = g(e19)<e..e ) =
1’3 1 3 o if 1.# j

Now it is easy to see that the functions Fj 6 L2(X) of the proof

of lemma 4.3.1, can be taken to be the constant functions

Fj(e16) g Q/éj'l bji):;1. The operator A is given by

A(e19)(al,a2,a3.-~) = (81, 82/2. 83/22...” an/Zn-1.0.0.-~) e e E -

Now for each 9 E En’ we have

New»: = (Mac) (.19) = m1. x2//2.x3//22,...>
a (x1,x2,...,xn,0,0,...)

Hence we have

19 .
Q(e )x (x1,x2,...,xn,0,0,...), e 6 En'

119

Obviously Q(ele) is measurable and

(f(e1°)x) (y) = (Q(eie)x, Q(e‘em, x,y e x .

Clearly Q(eie) is bounded. So we have obtained an explicit
form for this quasi square root.

In this example since f' is countably'valued function,
following [1 ] one could factor each value of f separately to
determine Q, whose measurability is automatic. However in
general when f is not countably valued this procedure may not

yield a measurable quasi square root.

REFERENCE S

10.

REFERENCES_

S.A. Cobanjan, The class of correlation functions of stationary
stochastic processes with values in a Banach space, Sakharth.
SSR Mecn. Akad. Moambe, 55 (1969) 21-24. (Russian) MR 42 #6929.

, Certain properties of positive operator measure in
Banach spaces, Sakharth. SSR Mecn. Akad. Moambe, 57 (1970)
273-276. (Russian) MR 42 #6930.

, Regularity of Banach space valued stationary pro-
cesses and factorization of operator valued functions, Sakharth.
SSR Mecn. Akad. Moambe, 61 (1971) 29-32. (Russian) MR 44 #7631.

R.G. Douglas, 0n factoring positive operator functions. J.
Math. Mech., 16 (1966) 119-126.

, 0n majorization, factorization and range inclusion
of operators on Hilbert space. Proc. Amer. Math. Soc., 17
(1966) 413-415.

K. Fan, On a theorem of Weyl concerning eigenvalues of linear
transformations 1. Proc. Nat. Acad. Sci. 0.8., 35 (1949)
652-655.

W. Fieger, Die anwendung eigiger mass-und integration theoretisch
theoretischer sfitze auf matrizelle Riemann Steiltjes-integrale.
Math. Ann., 150 (1963) 387-410.

R. Gangolli, Wide sense stationary sequences of distributions
on Hilbert space and the factorization of operator valued
functions. J. Math. Mech., 12 (1963) 893-910.

P.R. Halmos, Introduction to Hilbert Space. Chelsea Publishing,
New York, 1957.

H. Helson, Lecture on Invariant Subspaces. Academic Press, New
York, 1964.
120

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

121

E. Hille and R.S. Phillips. Functional analysis and semi-
groups. Amer. Math. Soc. 0011., 1957.

V. Mandrekar and H. Salehi. The square-integrability of
operator-valued functions with respect to a non-negative

operator-valued measure and Kolmogorov isomorphism theorem.
Indiana Univ. Math. J., 20 (1970) 545-563.

P. Masani. Recent Trends in Multivariate Prediction Theory,
Multivariate Analysis. Academic Press, New York, 1966, 351-382.

, The prediction theory of multivariate stochastic

processes, III. Acta Math., 104 (1960) 142-162.

, Cramér's theorem on monotone matrix valued functions
and the Wold decomposition (in "Probability and Statistics"
The Harald Cramér's volume, Ed. by U. Grenander, Stockholm
(1959) 175-189.)

M.G. Nadkarni. Prediction theory of infinite variate weakly
stationary stochastic processes. Sankhya, Series A, Vol. 32,
Part 2 (1970) 145-172.

, On a paper of Ramesh Gangolli. J. Math. Mech., 17
(1967) 403-405.

R. Payen. Fonctions aléatoires du second ordre a valeurs
dans un espace de Hilbert. Ann. Inst. Henri Poincare, 3
(1967) 323-396.

Yu. A. Rozanov. Some approximation problems in the theory of
stationary processes. J. Multivariate Anal., 2 (1972) 135-144.

H. Salehi. 0n determination of the optimal factor of a non-
negative matrix valued functions. Proc. Amer. Math. Soc.,
29 (1971) 383-389.

, The Hellinger square integrability of matrix valued
measures with respect to a nonnegative measure. Ark. Mat., 7,
No. 21 (1967) 299-303.

22.

23.

24.

25.

26.

27.

122

H. Salehi. 0n the Hellinger integrability and interpolation
of q-variate stationary stochastic processes. Ark. Mat., 8,
No. l (1968) 1-6.

, Applications of the Hellinger integrals to q-variate
stationary stochastic processes. Arka Mat., 7, No. 21 (1967)
305-311.

N.N. Vahanija and S.A. Cobanjan. Processes stationary in the
wide sense with values in a Banach space, Sakharth. SSR Mech.
Akad. Moambe, 57 (1970) 545-548. (Russian) MR 42 #6935.

N. Weiner and P. Masani. The prediction theory of multivariate

stochastic processes, 1. Acta Math., 98 (1957) 111-150.

, The prediction theory of multivariate stochastic

processes, 11. Acts Math., 99 (1958) 93-137.

, 0n bivariate stationary processes and the factoriza-
tion of matrix valued functions. Theory of Probability and

Its Applications (Moscow), English Edition, 4 (1959) 300-308.

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