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6/01 cJCIRC/DatoDue.p65-p.15

INVARIANT VECTOR SUBSPACES OF L10 WITH APPLICATIONS

By

David Allen Redett

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2003

ABSTRACT

INVARIANT VECTOR SUBSPACES OF L19 WITH APPLICATIONS

By

David Allen Redett

For a majority of this dissertation, we study invariant vector subspaces of the
Hardy and Lebesgue spaces for both the circle and torus. When studying invariant
vector subspaces, some kind of completeness is required of our invariant vector sub—
space. Classically, one only considered those invariant vector subspaces that were
closed. In this dissertation, we take a different approach. Rather than studying
those invariant vector subspaces that are complete in the metric induced from the
norm on the larger space, we consider those invariant vector subspaces that can sup-
port any norm that makes them complete and a Hilbert space. This idea was first
introduced by de Branges in his proof of the famous Bieberbach conjecture. He con-
sidered those invariant vector subspaces of H 2(T) that could support a norm that
would make them Hilbert spaces. Since then, these spaces, affectionately called de
Branges spaces, have been studied by Dinesh Singh, U. N. Singh, Vern Paulsen and
Sanjeev Agrawal. These gentlemen made many nice contributions to this area. We
also begin making connections to Strongly Harmonizable Stable Fields. We hope in
the future some of the work from this dissertation may be used to help completely

understand the “prediction” of these random fields.

To my family for their continuing love, support and understanding.

iii

ACKNOWLEDGMENTS

Several individuals deserve my gratitude. First and foremost, I wish to express
my deep appreciation to my advisor, Professor V. Mandrekar, for his guidance and
encouragement throughout the preparation of this dissertation. For his efforts, I am
extremely indebted.

I would also like to thank the other members of my Dissertation Committee:
Professors Michael Frazier, Joel Shapiro, William Sledd and Clifford Wei]. Each of
them has contributed significantly to my graduate school experience.

Next, I would like to thank Professor Alexei Alexandrov for a helpful conversation
during the preparation of this dissertation.

Finally, I would like to acknowledge the financial support of the Department
of Mathematics at Michigan State University, and in particular the granting of the
Graham Research Fellowship for the summer of 2003 which gave me the much needed

time to complete this dissertation.

iv

Contents

1 Background 1
1.1 Preliminaries ............................... 1
1.2 Terminology and Notation ........................ 6
1.3 Hilbert Spaces Contained in Banach Spaces .............. 9
2 Invariant Vector Subspaces of L2(T) 12
2.1 Known Results .............................. 12
2.2 Hilbert Spaces Boundedly Contained in L2 ............... 17
2.3 Hilbert Spaces having L2-Closures that are Simply Invariant ..... 19
3 Invariant Vector Subspaces of L”(T) 21
3.1 Known Results .............................. 21
3.2 An Extension of a Result of Singh and Agrawal ............ 23
3.3 Hilbert Spaces Contractively Contained in L" for 1 s q < 2 ...... 24
4 Invariant Vector Subspaces of LP(T2) 33
5 Random Fields 50
5.1 Introduction ................................ 50
5.2 Some Probability Background ...................... 52

5.3 Strongly Harmonizable Stable Fields ..................

A Orthogonal Decomposition of Isometries in a Banach Space
B A “Beurling Type” Theorem in HP(T)

BIBLIOGRAPHY

vi

64

67

Chapter 1

Background

1.1 Preliminaries

To keep this thesis some what self contained we include some basis definitions and
theorems used throughout. The information in this section can be found in any stan-

dard text on real or functional analysis. For instance, see [19].

Let C denote the complex plane.

Definition 1 A complex vector space is a set V, whose elements are called vec-
tors and in which two operations, called addition and scalar multiplication, are de-

fined, with the following algebraic properties:
1. x + y E V whenever :12, y E V. Further, a: + y = y +113.
2. :r+(y+z) = (:1:+y)+z whenx,y,z€ V.
3. 06 V such thatcr+0=xfor alleV.

4. To each a: E V there corresponds a unique vector -:r E V such that :r+(—a:) 2 0.

5. as: E V whenever a E C and a: E V. Further, 1:1: 2 1:.
6. 0461:) = (own: for a,fl E C and a: E V.

7. a(x+y)=a:r+ozy, (a+fl)x=a:r+fizrfora,fi€C andx,y€V.

A subset M of a complex vector space V is called a vector subspace of V if

.M is itself a complex vector space, relative to the addition and scalar multiplication

which are defined on V.

Definition 2 A complex vector space X is said to be a normed linear space if to
each :1: in X there is associated a nonnegative real number “51:“, called the norm of :r,

such that
1. “:1: + yll S llscll + “y“ for all a: and y in X,
.2. Ilaxll = |oz|||5r|| ifa: is in X and a E C,
3. “as“ = 0 implies a: = 0.

If we define d(:c,y) 2 Ha: — y”, then it is easy to check that d is a metric on
X. A Banach space is a normed linear space which is complete in the metric d.

Completeness, then, means if {:rn}n20 is a sequence in X such that
“in - Emll -t 0

as n, m —> 00, then there exists an :r in X such that
”$7: _ 37“ _) 0

as n —> 00.
The simplest example of a Banach space is C with norm |z| for z E C. The
Banach spaces that are of interest in this dissertation are the LP-spaces, which we

define now.

Definition 3 Let Q be an arbitrary measure space with a positive measure u. If

1 S p S 00 and if f is a complex measurable function on 52, define
(inlflrdu)g 13p<oo
inf {a : u(f‘1((oz, oo])) = 0} p 2 00

and let LP(Q,u) consist of all f for which

llfllp =

||f||p < 00.

We call ||f||p the LP-norm off.
If a Banach space X has a norm that satisfies the parallelogram law
Ill? + yll2 + “:5 - 31H2 = 2(llirll2 + llyllz),

we call X a Hilbert space. In such a case, we may define a sesquilinear functional on

X by
1 2 2 -1 - 2 - 2
(2:, y) = Z(|l$+yll — Ilir-yll )+2;,-(||$+zy|l - llx-zyll )-

This sesquilinear functional is called the inner product for the Hilbert space X and

”Hill = Wat’s?)-

C is also the simplest example of a Hilbert space with inner product given by
(21, 22) = 212—2 for 21, 22 6 C. We point out that of all the Lp(§2,u) spaces, L262, u)
is the only Hilbert space with inner product given by (f, g) = In fg du.

Hereafter, ’H will always denote a Hilbert space.

If (x, y) = 0 for some 11:, y E 7i, we say that x is orthogonal to y, and sometimes
write a: .L g. If M is a vector subspace of 71, let .Mi be the set of all y E ”H which

are orthogonal to every 2: E M.

A set of vectors ua in ”H, where or runs through some index set A, is called
orthonormal (o.n.) if it satisfies the orthogonality relations (”(10,113) = 0 for all
a yé fl, oz 6 A and fl 6 A, and if it is normalized so that ||ua|| = 1 for each or E A.

A vector subspace M of a Banach space X is called a subspace of X if M is
itself a Banach space, relative to the norm which is defined on X. The following

theorem gives us some very important properties of subspaces of a Hilbert space.
Theorem 1 Let M be a subspace of H.

1. Every x E ’H has a unique decomposition

x = Px + Qx
into a sum of Px E M and Qx 6 Mi.

2. Px and Qx are the nearest points to x in M and Mi, respectively.

3. The mapping P : ’H ——> M and Q : H —> M‘L are linear.

4- llilfll2 = IIPIIIII2 + llellz-
Corollary 1 IfM 75 7-1, then there exists y E ’H, y # 0, such that y J. M.

Definition 4 A linear transformation of a complex vector space V into a complex

vector space W is a mapping A of V into W such that

A(ax + fly) = 0M1?) + flMy)

for all x,y E V and for all a,fi E C. In the special case that V = W we call A a

linear operator.

Consider a linear transformation A from a normed linear space X into a normed

linear space 32, and define the norm of A by
IIAII = sup {llArrll :2: e x, Hill 31}.

If “All < 00, then A is called a bounded linear transformation. We denote by
[3(X, y) the collection of all bounded linear transformations from X to y. If X = y,
we simply write B(X).

Recall that H denotes a Hilbert space. We call A E B(H) an isometry if
||A(x)|| = “as“ for all x E H.
An isometry A E B(H) is called a shift or pure isometry if

A"(’H) = {0}

:08

and unitary if the range of A is H.
Definition 5 A vector subspace M of H is invariant under A E B(H) if
A(M) Q M.

We say it is simply invariant if the above containment is strict. (i.e., A(M) C M,

but A(M) 7e M.)

A subspace M of H reduces A E 8(H) if both M and M i are invariant under

A.

Definition 6 IfA E 3(H), then the adjoint of A, denoted A", is the unique operator
on H satisfying

(Minty) = (x,A‘(y))-

for all x and y in H.

For A1, A2 6 3(H) we say A1 and A2 are doubly commuting if A1 commutes
with A2 (i.e., A1A2 = A2A1) and A1 commutes with A; (Note: A1 commuting with

A; is equivalent to A2 commuting with A‘f).

1.2 Terminology and Notation

We let U and T denote the unit disc and unit circle in the complex plane, respectively.
The Hardy space H”(U), (1 _<_ p < 00) is the Banach space of holomorphic

functions over U which satisfy the inequality

sup Twerdma) <oo

OSr<1

where m denotes normalized Lebesgue measure on T. The norm [l f ll,D of a function

f in HP(U) is defined by

1/p
”pr 1:33] ([1, |f(r€)l”dm(€)) .

When p = 2, we have a Hilbert space and the norm can be simplified. For f in
H 2(U) with Taylor series 22°20 f (n)z” the norm simplifies to
00 . 1/2
llfll2 = (2 Iran?) .
n=0
The Hardy space H °°(U) is the Banach space of holomorphic functions over U

which satisfy the inequality

sup|f(z)| <oo.
zElI

The norm ||f||00 of a function f in H°°(U) is defined by

”filoo = sup|f(z)|-
zEIJ

It is well known (see [19]) that every function in HP(U), (1 g p S 00) has a
nontangential limit at [m] almost every point of T. Let f* denote the boundary

function of an f in HP(U); then
f" E HP(T) E spanLP(T’m){E" : n 2 0}.

It is also know (see [19]) that f can be reconstructed by the Poisson integral as well

as the Cauchy integral of f *. Phrther,

||f||p = llf‘llp

where the second norm is the LP (T, m) norm. For this reason, we identify HP(U)

and HP(T) and no longer distinguish between f and f". Therefore, these Banach

spaces of holomorphic functions H ”(U) may be viewed as a subspace of LP (T, m).
For f in LP(T) = LP(T, m), S will denote the operator of multiplication by the

coordinate function. That is,

A good part of this dissertation is spent studying vector subspaces of L" (T) invariant
under S.

We let C2 denote the cartesian product of two copies of C. The unit bidisc in C2
is denoted by U2 and the distinguished boundary T2, where U and T are the unit
disc and unit circle in the complex plane, respectively.

The Hardy space HP(U2), (1 _<_ p < 00) is the Banach space of holomorphic

functions over U2 which satisfy the inequality

SUP T2 |f(r§1,r§2)|pdm2(§1,§2) < 00

0§r<1
where m; denotes normalized Lebesgue measure on T2. Note, holomorphic here

means holomorphic in each variable. The norm N f Hp of a function f in HP(U2) is

defined by
1/p
“flip : SUP (/ lf(7'€1,7‘€2)lpdm2(€1,€2)) -
03r<l T2

When p = 2, we have a Hilbert space and the norm can be simplified. For f in
H 2(U2) with multiple Taylor series Emmzo f (n, m)z]‘z§" the norm simplifies to
- 1/2
llfllz = ( 2: limit?) .
n,m20
The Hardy space H °°(U2) is the Banach space of holomorphic functions over U2

which satisfy the inequality

sup |f(z1,22)|< oo.
(21,22)€U2

The norm N f [[00 of a function f in H °°(U2) is defined by

“film: SUP |f(zi,22)|-

(21,22)EU2
It is well known (see [18]) that every function in HP(U2), (1 g p 3 00) has a

nontangential limit at [m2] almost every point of T2. Let f * denote the boundary

function of an f in HP(U2), then

 

fi 6 Hp(T2) E spanLP(T2’m2){€f§§" : n,m Z 0}.

It is also know (see [18]) that f can be reconstructed by the Poisson integral as well

as the Cauchy integral of f ’. Further,

“flip = “fillp

where the second norm is the LP(T2, m2) norm. For this reason, we identify H P (U2)
and HP(T2) and no longer distinguish between f and f“. Therefore, these Banach

spaces of holomorphic functions H ”(UZ) may be viewed as a subspace of U’(T2, m).

For f in LP(T2) : LP(T2, m2), SI and 52 will denote the operators of multiplica-

tion by the first and second coordinate functions respectively. That is,

S1(f)(z1,22) = Zif(31, 32)

and
32(f)(21, 22) = 22f(21, 22)-

A good part of the remaining portion of this dissertation is spent studying vector

subspaces of LP(T2) invariant under 31 and 32.

1.3 Hilbert Spaces Contained in Banach Spaces

We say a Hilbert space H is contained in a Banach space X if H is a vector subspace

of X. If further, there exists a 0 < C < 00 such that
lellx S Cllrlrllu

for all x in H, we call H boundedly contained in X. If C = 1, we say that H is
contractively contained in X.

We give some examples to illustrate the concept.

Example 1 L2(T) is contractively contained in LP (T) for 1 S p g 2. Similarly, for

the Lebesgue spaces on T2. A

Example 2 If X is a Hilbert space, then any subspace of X is contractively contained

inX. A

Example 3 For perhaps a more interesting example take X = H2(T) and H =

gH2(T) where g is in H°°(T) with norm on H given by

llgfllu = ||f||2

for all f in H2(T). By the definition of the norm on H, H is clearly a Hilbert space.
Since 9 is in H°°(T), H is clearly a vector subspace of H2(T). Further we point out

that

”Mb 5 ”gllooufll2 = HglloollngH-

So, H is always boundedly contained in H 2(T) and is actually contractively contained

in H2(T) if ”all... s 1. A

Alternatively, we can describe this idea using operator theoretic terminology.
Let A be a bounded operator from a Hilbert space IC into a Banach space X.

Define H A to be the range of A in X and equip H A with the inner product given by

(AL/1y)?“ Z (1‘, yllc

with at least one of x, y orthogonal to the K er(A). Then H A is boundedly contained

in X since

||A$||x S llAllllxllic = llAllllAirlluA-

Fhrther, H A is contractively contained in X, if A is a contraction. Every Hilbert
space H boundedly contained in X is such an operator range; it is the range of the
inclusion map of H into X. We note that if X is a Hilbert space and A is a partial
isometry, then H A is an ordinary closed subspace of X since for all x orthogonal to
the Ker(A) we have

llAivllx = llxllzc = IIAIBIIHA-

Conversely, if H A is an ordinary subspace of X, then A is a partial isometry.

In our above examples, for
Example 1 X = LP(T), H = L2(T) and A = I. A
Example 2 X is any Hilbert space, H is any subspace and A = I . A

10

Example 3 X = H2(T), H = H2(T) and A = My, where M9 is the operator of

multiplication by g. A

11

Chapter 2

Invariant Vector Subspaces of

L2(T)

2.1 Known Results

We begin by discussing vector subspaces of H 2(T) that are invariant under S. The
first result is due to Beurling. He characterized all invariant subspaces of H 2(T).

Before we give his characterization we need a definition.
Definition 7 A function (15 in H°°(T) is called inner if |¢| = 1 a.e. on T.

We now give Beurling’s characterization of subspaces of H 2(T) invariant under

Theorem 2 (Beurling [1]) A subspace M of H 2(T) is invariant under S if and

only ifM = ¢H2(T) where q) is an inner function.

We won’t go into any details, but we point out that Beurling’s Theorem can be

proved using the following decomposition.

12

Theorem 3 (Halmos-Wold Decomposition, see [8]) Let V E 8(H) be an isom-

etry.

1. There is a unique decomposition
H = 9 G3 [I

such that g and .C are reducing subspaces for V, S = Vlg is a shift operator on

Q, and U = V]; is unitary on .C.

2. Define IC = H e V(H). Then {V"(IC) 2:0 is an orthogonal family of subspaces

of H satisfying
g = Z 69V"(IC)

n=0

and

We call 5 and U the shift and unitary parts of V, respectively.

Many of the remaining theorems from this section can also be proved using the
Halmos—Wold decomposition. Some of our results also rely heavily on this decompo-
sition.

An important corollary of Theorem 2 (p. 12) is used later. The proof can be

found in [9]. Before we state it, we need another definition.

Definition 8 A function g in HP(T) is called outer if the linear combination of

functions

9(5), 69“), 629(6),

are dense in HP(T).

With this terminology, we get

13

Corollary 2 Each function f in H 2(T) has a factorization

f=<i>g

where d is inner or constant and g is outer. This factorization is unique up to a

constant factor of modulus 1.

We point out that this corollary is true for HP(T), 1 S p S 00, not just H 2(T),
see Rudin, [19] for details.
We now turn our attention to a generalization of Theorem 2 (p. 12) due to de

Branges. He proved the following theorem.

Theorem 4 (de Branges, see [20]) M 9:9 {0} is a Hilbert space contractively con-
tained in H 2(T) invariant under S and S acts as an isometry on M if and only if
M = gH2(T) for some 9 in the unit ball of H °°(T) unique up to a constant multiple

of modulus 1 with ||gf||M = ||f||2 for all f E H2(T).

We recall that every subspace of H 2(T) is contractively contained in H 2(T). We
also point out that S acts as an isometry on every subspace of H 2(T). So, this is a
nice and reasonable generalization of Theorem 2 (p. 12).

It was pointed out later by U. N. Singh and Dinesh Singh that de Branges’

contractively contained condition could be relaxed. Their result is stated here.

Theorem 5 (U. N. Singh & Dinesh Singh [23]) M 7Q {0} is a Hilbert space

that is a vector subspace of H 2(T) invariant under S and S acts as an isometry on
M if and only ifM = gH2(T) for some 9 E H°°(T) unique up to a constant multiple

of modulus 1 with ||gf||M = ||fl|2 for all f E H2(T).

We now turn our attention to the results known in L2(T). We start with a result

due to Helson and Lowdenslager. They were able to characterize the subspaces of

14

L2(T) that are simply invariant under S. We note that every subspace of H 2(T) that
is invariant under S is in fact simply invariant under S. Their work not only extended
but also generalized the work of Beurling. We point out that Beurling’s original proof
of Theorem 2 (p. 12) weighed heavily on analytic function theory. Recall, H 2(T) is
just H 2(U) in disguise. So Beurling’s techniques could not be applied to characterize
the simply invariant subspaces of L2(T). Helson and Lowdenslager used a Hilbert

space approach to solve the problem in L2(T).

Theorem 6 (Helson & Lowdenslager, see [9]) A subspace M of L2(T) is sim-
ply invariant under S if and only ifM = gH2(T) where g is in L°°(T) and lg] = 1

a.e. on T.

In contrast to the situation in H 2(T), in L2(T) there are subspaces invariant
under S that are not simply invariant. Weiner was able to characterize these. In the
following theorem, we use the term doubly invariant to mean invariant, but not

simply invariant.

Theorem 7 (Weiner, see [9]) A subspace M of L2(T) is doubly invariant under

S if and only ifM = 13L2(T) where E is a measurable subset of T.

In recent work [15], contractively contained Hilbert spaces in L2(T) were studied.
There were examples given to show that there are contractively contained Hilbert
spaces in L2(T) satisfying the conditions of Theorem 4 (p. 14), but not of the form
gH2(T) with g in L°°(T) nonzero a.e. These examples show that a direct general-
ization to L2(T) is not possible. Additional conditions are required for these Hilbert
spaces to have the form gH2(T) with g in L°°(T) nonzero a.e. In [15], Paulsen and

Singh gave an additional condition, namely a continuity condition on the norm of

15

M in addition to its contractive containment. Their motivation was to generalize

Theorem 6 (p. 15) along the lines of de Branges’ generalization of Theorem 2 (p. 12).

Theorem 8 (Paulsen 86 Singh [15]) Let M # {0} be a Hilbert space contrac-
tively contained in L2(T) simply invariant under 5' and on which S acts as an isom-

etry. Further, suppose there are p, 2 g p g 00 and 6 > 0 such that
“an s awn. for all f in M n LP(T)- (2.1)

Then there exists a unique b (up to a scalar multiple of modulus I) in the unit ball

of L°°(T), which is non-zero a.e. such that
1. M = bH2(T) with ||bf||M = ||f||2 for all f in H2(T),

2. b’1 E L3(T) and Ilb‘llls g 6, where

00.19:?

= 2

8 £3 2<p<oo
2, p=oo.

(Note: We do not assume in (21) that M D LP(T) sé {0} forp > 2.)

Paulson and Singh also had a doubly invariant result. Before we state that we
need a little terminology. If T E BUC) where [C is a Hilbert space, we denote by
R(T) the range of T which is a vector subspace of IC. If we endow R(T) with the

norm
thlnm = inf{|lk|| 1T,“ = h},

then ’R.(T) is a Hilbert space in this norm called the range space of T and it is
boundedly contained in IC. If T is a contraction, then the range space of T is
contractively contained in IC.

We use M¢ to denote the operator of multiplication by (b on L2(T).

16

Theorem 9 (Paulsen & Singh [15]) LetH be boundedly contained in L2(T). Then
S acts unitarily on H if and only if there exists a function 45 in L°°(T) such that
H = R(M¢) isometrically; i.e., “filly = ”h”72(M¢) for all h in H. When H is con-

tractively contained in L2(T), we have ||gb||oo g 1.

2.2 Hilbert Spaces Boundedly Contained in L2

In this section, rather than finding conditions so that our Hilbert space is of the form
gH2(T) with g in L°° (T) nonzero a.e., we describe all Hilbert spaces boundedly

contained in L2(T) which are invariant under S and for which S acts as an isometry.

Theorem 10 If M is a Hilbert space which is boundedly contained in L2(T), invari-

ant under S and S acts as an isometry on M, then there exists (t, 91, g2, . . . E L°°(T)
M = (inf{nqni : ¢q =

1 2
(1519] + 2,- ||f,-||§) / . Note, we do not assume that ii, 91,92, . .. E L°°(T) are nonzero.

such that M = ¢L2(T)+E, giH2(T) with norm |]¢p+zigifi

 

 

Proof: By Theorem 3 (p. 13), we get

M = u e f: eS"(N) (2.2)

n=0
where S is unitary on H and N = M e S (M) If H 75 {0}, then by Theorem 9

(p. 17) we get that there exists (b E L°°(T) such that
H = ¢L2(T)

with norm
”2le = inf{l|qll2 = «sq = 451)}-
IfN # {0}, let gl 6 N with ||g1|]M = 1. Then, {glei"9}n>0 is an orthonormal se-

quence in M. Let f E H2(T). Then f(e“’) = 2:10 f(k)e“‘9. Let fn = 2220 f(k)e"‘9.

17

Then fn converges to f in L2(T) and a.e. Consider the following computation.

”fullg IfA(/€)l2

II
M:

a-
II
o

|f(k)l2|l916ik9||i4

M: Em.

||f(k)gle‘k”||ia

2

a.
II
o

 

 

i angler”
k=0

 

 

M
Since (fn)n is Cauchy in L2(T), (22:0 f(k)gle‘k9)n is Cauchy in M. Since M is a

Hilbert space, there exits a h in M such that

 

 

Z f(k)gleik0 — h” ——> 0 as n —> oo.
k:0 M

Since M is boundedly contained in L2(T), we get that

——>0 asn—>oo.
2

 

 

Z ffk)91€ik0 — h
k=0

 

 

So a subsequence converges almost everywhere. But
n A .
Z f(k)e'k9 —> f a.e. as n —+ 00.

So

So h = 91 f. Therefore,
”glfllM = ||f||2-

Since M is boundedly contained in L2(T), we get

“91f“2 S CllglfHM = Cllfllz

Since f was an arbitrary element of H 2(T), we get that 91 multiplies H2(T) into

L2(T). Since L2(T) = H2(T) EB ei9H2(T), we conclude that 91 multiplies L2(T)

18

into L2(T). Since 91 was an arbitrary element of N, we conclude that N must be

contained in L°°(T). Now fix an orthonormal basis {9,} in N. 80, we have

= Z [If/cili-

2
M k

 

 

ngfk
I:

Now putting this altogether we get for 45p + 2k gkfk E M that

 

 

 

 

 

 

 

 

 

 

2 2
”4519 + 29s,. = Input. + 29m. = inf{nqn§ = ¢q = ip} + 2: llfklli-
k M k M k
Therefore,
1/2
[lip + 29.12. = (inauqng = ¢q = 4520} + z: llfklli)
k M k

as desired. A

The above result explains the examples given in [15].

2.3 Hilbert Spaces having LQ-Closures that are Sim-

ply Invariant

In this section, we characterize all Hilbert spaces contained in L2(T) which are in-
variant under S, for which S acts as an isometry and whose L2(T)-closure is a simply
invariant subspace of L2(T).

Our motivation for this section comes from Theorem 5 (p. 14), Corollary 2 (p. 14)
and the fact that all subspaces of H 2(T) are simply invariant.

Let g be any element of L°°(T) having the modulus of an outer function a.e.

Consider M = gH2(T) with [IgfIM = ||f||2 for all f in H2(T). Then it easily

 

follows that M is a Hilbert space invariant under S and S acts as an isometry on
_ 2
M. Further note that ML (T) is a simply invariant subspace of L2(T), (see [2],

p. 142). Our result gives the converse.

19

Theorem 11 If M 76 {0} is a Hilbert space that is a vector subspace of L2(T) such

__ 2
that M is invariant under S, S acts as an isometry on M and ML (T)

is a simply
invariant subspace of L2(T), then M = gH2(T) with g E L°°(T) which has the

modulus of an outer function a.e. and is unique up to a constant multiple of modulus

1 With ”gfllM = llf|l2 for all f E H2(T)-

Originally, we formulated and proved this result directly. The proof below was
suggested to us by an unknown reviewer.

Proof: By Theorem 6 (p. 15) we have that My”) = ¢H2(T) for some unimod-
ular function ()5. Then M’ = EM is contained in H 2(T) and with norm “Eplljw =
lelM for all p in M is a Hilbert space invariant under S and S acts as an isometry
on M’. So by Theorem 5 (p. 14), we get that M’ = bH2(T) with b 6 H°°(T) with
norm ||bf||Mr = ||f||2 for all f in H2(T). So M = ¢M’ = ¢bH2(T) with norm

||¢bf||M = “bf“M’ = ||f||2 for a11f in H2(T)- A

20

Chapter 3

Invariant Vector Subspaces of

LP(T)

3.1 Known Results

We begin by discussing vector subspaces of HP(T) that are invariant under S. For
the first result we give credit to de Leeuw and Rudin, who first proved this result for

H1(T). Here we give the characterization of all subspaces of HP(T) invariant under

S.

Theorem 12 (de Leeuw & Rudin, see [9]) A subspaceM of HP(T) is invariant

under S if and only ifM = ¢HP(T) where 45 is an inner function.

Remark 1 We point out that even though this theorem is a strict Banach space
result for p 75 2, a proof found in [.9] shows that this is really a corollary of Beurling’s

H 2(T) result. See Appendix A and B for an alternative approach.

At this point one might expect a generalization as done in the H 2(T) case. Per-

haps one would expect a vector subspace X of H P (T) invariant under S which is not

21

closed in the H1p (T)-norm but is able to support a new norm that makes it complete.
Considering what we know in the H 2(T) case, we may think of X as being of the

form gH”(T) where g is in H °°(T) with norm

||9f||x = llfllp

for all f in H P (T) X is a Banach Space in this norm and invariant under S. Further,
S acts as an isometry on X. However, given the complicated structure of a Banach
space, that is not a Hilbert space, this seems like quite a task. We will slightly
modify this idea to get a better handle on the problem. Rather then allowing vector
subspaces of H 1"(T) that support any norm, we will only allow those vectors subspaces
of HP(T) that can support a norm, that will make them a Hilbert space. Such
ideas have already been studied by Dinesh Singh and Sanjeev Agrawal [22] who
characterized certain Hilbert spaces contained in some Banach spaces of analytic

functions. In particular, they proved the following H ”(T) result.

Theorem 13 (Dinesh Singh and Sanjeev Agrawal [22]) If M is a Hilbert space

contained in HP(T), invariant under S and S acts as an isometry on M, then
M = bH2(T)
for a unique b:
1. [fl 3 p g 2, b E H%(T). When p z 2, we mean H°°(T).
2. pr> 2, b=0.
Further. ”bfllM = “fllz for all f in Him (1 s p s 2)-

We now turn our attention to LP (T) We start with a generalization of Theorem
12 (p. 21). This is due to Forelli, who characterized the subspaces of L” (T) that are

simply invariant under S.

22

Theorem 14 (Forelli, see [9]) A subspace M of LP (T) is simply invariant under

S if and only ifM = ng(T) where g is in L°°(T) and |g| = 1 a.e. on T.

As one might expect, there is also a doubly invariant result. We give credit for it

to Weiner.

Theorem 15 (Weiner, see [9]) A subspace M of LP(T) is doubly invariant under

S if and only ifM = lELP(T) where E is a measurable subset of T.

The same complications arise in LP (T), as did in HP(T), so here too, we only
consider vector subspaces of LP(T) that are Hilbert spaces. Paulsen and Singh [15],

in addition to their aforementioned L2(T) result, showed the following.

Theorem 16 (Paulsen & Singh [15]) Let M be a simply invariant Hilbert space
contractively contained in L’ (T) for some r > 2 and on which S acts as an isometry.

Further, suppose there are p, 2 S p S 00 and 6 > 0 such that
||f||M S 5||f||p for all f in M 0 INT). (3.1)

Then M = {0}. (Note: We do not assume in (3.1) that M F) LP(T) 79 {O} for

10>?)

Later, we investigate the situation in U (T) for 1 S q < 2 and give conditions so

that the Hilbert space is of “Beurling type”.

3.2 An Extension of a Result of Singh and Agrawal

Here, we give an extension of Theorem 13 (p. 22). It is easy to see from Theorem 11
(p. 20) that this result can be extended to certain vector subspaces of LP(T) in the

following way.

23

Theorem 17 If M is a Hilbert space contained in LP(T), invariant under S, S acts

as an isometry on M and MM”) is a simply invariant subspace of LP(T), then
M = bH2(T)
for a unique b:

1. If1 S p S 2, b E L723?(T) and has the modulus of an outer function a.e. When

p = 2, we mean L°°(T).

2. pr>2,b=0.

Further, [lbfllM = ||f||2 for all f in H2(T) (1 S p S 2).

Proof: By Theorem 14 (p. 23) we have that MINT)

= ¢HP(T) for some uni-
modular function (15. Then, M’ = 3M is contained in HP(T) and with norm
”PPM/W = ||p||M for all p in M is a Hilbert space invariant under S and S acts
as an isometry on M’. So by Theorem 13 (p. 22), we get for p > 2 that M' = {0}
and hence M = {0} and for 1 g p g 2, M' = gH2(T) with g e H7233(T) with
norm IlgfllM: = ||f||2 for all f in H2(T). So M = ¢M’ = qbgH2(T) with norm
ll¢9fllM = ”QfHM’ = ||f||2 for all f in H2(T). A

The converse of the theorem is clear except for the fact that the closure is simply

invariant. This follows from a straight forward variation of a theorem in [2], page

142.

3.3 Hilbert Spaces Contractively Contained in Lq
for 1 S q < 2

In this section, we give conditions so a contractively contained Hilbert space in Lq(T)

for 1 S q < 2 is of “Beurling Type”.

24

Before we give our result, we consider the following situation. Fix 1 S q < 2 and
let b be an element of the unit ball of Lzl-QHT) with b 75 0 a.e. Then b multiplies
H 2(T) into L“ (T). Let’s call M the range of such a multiplication; i.e., M = bH2(T)
and endow M with the norm

llbeIM = llfll2.

Then M becomes a Hilbert space contained in L" (T). In fact, M is contractively

contained in Lq(T) since

llbfllq s llbllgazllfllz by Halder’s Inequality

S llbfllM-

The last inequality follows from the definition of the norm on M and the fact that
b is in the unit ball in LZ—ZEWT). We further point out that M is simply invariant
and that S acts as an isometry on M. Unfortunately, this is not enough to hope
for a characterization as pointed out by Paulsen and Singh in [15]. We consider
the following extra condition, which is similar to the condition found in Theorem 8

(p. 16), but slightly modified to fit the L4 (T) setting. Suppose that b"1 is in L3 (T)

where

00 if p = 2

S :

2 - 2

F33 1f 2 < p S 535.
Let 6 = Hb'llls which we point out is strictly greater then zero. We now make the
following calculation.

llbfllM = ||f||2
= llb‘lbfllz

< ||b‘1||_2%||bf||p by Hblder’s Inequality
= (lllbfllp-

25

So we get
llbfllM S 5|lbfllp-
Our theorem gives the converse.
Theorem 18 Let M ¢ {0} be a simply invariant Hilbert space contractively con-

tained in L‘l (T) (1 S q < 2) and on which S acts as an isometry. Further, suppose

there are p, 2 S p S 23}; and 6 > 0 such that
”fHM S 5||f||p for all f in M 0 INT) (3.2)

and if the unitary part of M is nonzero then so is its intersection with LP(T). Then
there exists a unique b (up to a scalar multiple of modulus 1) in the unit ball of

_22_ . .
L2-q (T), which 28 non-zero a.e. such that
1. M = bH2(T) with IlbfllM = ||f||2 for all f in H2(T),

2. b‘1 E L‘(T) and [lb—1|], S (5, where
00 ifp = 2
55, if 2 < p g 2335.

To prove our theorem we need a lemma.

Let Mg denote the linear transformation of multiplication by 9.

Lemma 1 If Mg : L2(T) —-+ L‘l(T) (1 S q < 2) is bounded, then “Mg“ 2 ||g||2_2_g_.

So, in particular, 9 E LTi-qHT).

Proof: Let f be any element of L2(T). Then by Holder’s Inequality we get that

llgfllq s llgllglmh. Therefore, we get

HM.“ = sap{||gfllq 2 “Ah 3 1}
s sup{|l9ll,_thllf||2 : |lf||2 _<_ 1}

_<_ llgllgu
"q

26

For the other inequality we note that there exists a measurable function a with
la] 2 lsuch that cry 2 lg]. Now let E, = {x : |g(x)| < n} and define f = xEnlgIEi-Ta.

Then [fl2 = XEnlgla—z-gi. So, f E L°°(T). Also, fg = xEnIgIYE—a. So we have

_29_ 1/0 l/q
(/ Iglz-qdm) = (/ lgflqdm)
En T

= “Ms(f)”q

s IIMgllllfllz
29. 1/2
= IlMgII(/E viz-cam) .

Dividing through by ( f E" [5]]52'-q?dm)1/2 which is finite we get ( fT X En | g|§2'-g?dm) ”(I—U2

”Mg“. Noticing that 1 / q — 1 / 2 = 23’5“ and applying the monotone convergence theo-
rem we get ||g|| 2%,; S ”Mg”. Putting these two inequalities together gives our desired
result. A
Proof of Theorem 18 (p. 26): By Theorem 3 (p. 13) we may write M as
00
M = ’H e Z esnw),

n=0
where S is unitary on H and N = M e S(M). Let M1 = M e H. We point out
that M1 # {0}, otherwise, we contradict the fact that M is simply invariant. So we
have that N 75 {0}. Therefore, we may choose an arbitrary element b of N with unit

norm in M. Then, {being} 0 forms an orthonormal sequence in M. Let f E H 2(T).

n2

Then f(ei9) = 2,210 f (k)e"‘". Let fn 2 22:0 f (k)e"‘9. Then fn converges to f in

L2(T) and a.e. We also have
llfnllg = Z lfA(/’€)|2
= Z |f(k)|2||b6ik6||ivl

= Z llf(k)be”“’llit

n

2 f(k)beik0

k=0

 

 

 

 

M
Since (fn)n is Cauchy in L2(T), (22:0 f(k)be"“9)n is Cauchy in M. Since M is a
Hilbert space, there exists an h in M such that

Z f(k)be”"9 — h” ———> 0 as n ——> oo.
[:20 M

 

 

Since M is contractively contained in Lq(T), we get that

n

2 fans” — h

k=0

—>0 asn——)oo.
q

 

 

 

 

So a subsequence converges almost everywhere. But

it
f(k)e”°o ——> f a.e. as n ——> oo.
k=0

So

2 f(k)be”°(’ —> bf a.e. as n —> oo.
k:0

80 h = bf. Therefore,
llbfllM = ||f|l2.

Since M is contractively contained in L"(T), we get

llbfllq S llbfllM = llfll2- (33)

Since f was an arbitrary element of H2(T), we get that b multiplies H 2(T) into
L" (T) Since L2(T) = H 2(T) EB m, we conclude that b multiplies L2(T) into
L" (T) Using inequality (3.3) (p. 28), we see that Mb is a bounded transformation
from L2(T) to Lq(T); that is a contraction. So by Lemma 1 (p. 26) we conclude that
b is in the unit ball of LTEQHT). Since b was an arbitrary normalized element of N,
we conclude that N must be contained in L72-93(T).

Now we show that no element of N can vanish on a set of positive measure

28

unless it is identically zero. Choose any nonzero element d in N. Suppose that d is

identically zero on a set of positive measure; call the set E. Let

n onE
kn:

1 on EC.

Then kn is in L°°(T) for all n. Let
h,, = exp(k,, + iii”)

where k1, denotes the harmonic conjugate of kn. So hn is in H °°(T). Replacing b by

d in the above computations gives
“hnHSZ : “dhnHM S dildhnllp-

First note that dhn E LiziHT) g LP(T). By the construction of hn, the right
hand side of the above inequality is bounded by a fixed constant independent of n
whereas the left hand side goes to infinity as n —> 00. This contradiction shows our
supposition must be incorrect. So no element of N can vanish on a set on positive
measure unless it is identically zero.

Next we show that N is one dimensional. To do this, we suppose not. So we can
find a b1 in N with unit norm orthogonal to b in M. By our decomposition, we get
that bH2(T) is orthogonal to b1H2(T) in M. Since b and b1 are in L%(T) and in
M1, we see that {b, b1} C M1 0 LP(T). Further, M1 0 LP(T) is invariant under S.
Let A(M1 fl LP(T)) denote the annihilator of M1 (1 LP(T), which is a subspace of
L35 (T). By the definition of the annihilator we get that the annihilator is invariant
under S since M1 (1 L”(T) is invariant under S. We now show that A(M1 fl LP(T))
contains an element that multiplies b and b1 in to L°°(T). Let f be any non-zero

element of A(Ml fl LP(T)). Let {pn(z)} be a sequence of analytic polynomials that

29

converges boundedly and pointwise to

expl—(lfl + z'|f|~)l expl-(Ibl + Z'lbl~)laXPl-(|b1| + z'll)1|~)lo

Clearly, pn f converges to

expl-(Ifl + Z'|f|")l expl—(Ibl + z'lbl")lexpl-(|b1| + z'll11|~)lf

in L391; SO expl—(lfl + z'lfl‘)]exp[-(Ib| + ilbl~)]exPl”(lbll + z°|l91|")lf is in A(M1 fl
LP(T)) and clearly multiplies b and b1 into L°°(T). So now let g be any non-zero

element of A(M1 fl LP(T)) that multiplies b and b1 into L°°(T). Note then that
/1r 9(ei9)b(ei9)ein9d0:0 n:0,1,2,.”

and

/ g(e‘”)b1(e‘”)e‘"" d0 = o n = 0, 1, 2, . .. .

Let’s write k = gb and k1 = gbl. Then k and k1 are in H °°(T). Since b and k do not
vanish on a set of positive measure, we conclude that 9 does not vanish on a set of

positive measure. Now consider the function

kki kbl E b1H2(T)

_
_—

bkl e bH2(T).

Since bH2(T) fl b1H2(T) = {0}, we get that k—Sl = 0, but this is a contradiction
since k—Sl does not vanish on a set of positive measure. From this contradiction we
conclude that our supposition must be incorrect. So N must be one dimensional.
Note that ifwe use exp[—-(If|+i|f|”)] exp[—(|b|+i|bl“)] exp[—(|b1|+i|b1|~)]f instead
of g in the above calculations, then we get that f does not vanish on a set of positive
measure. This follows from the reasoning employed above concerning the k and k1

and the observation that exp[—(|f| + ilfl")] exp[—(|b| + ilb]")] exp[—(|b1| + ilbll")] is

30

a bounded analytic function. So every nonzero member of A(M1 fl LP(T)) does not
vanish on a set of positive measure.

Now we show that H = {0}. By hypothesis, we need only show that HflL” (T) =
{0}. To do this, we suppose not. Let gt be an element of H n LP(T). Let f be a
nonzero element of A(M flLP(T)). Let {pn(z)} be a sequence of analytic polynomials

that converges boundedly and pointwise to

eXpl—(lfl + z'|f|~)l expl—(Mll + z'|<z5|~)l-

Clearly. m converges to expl-(Ifl + ilfl”)] expl-(Iel + il¢|”)lf in Lea; so

u E eXpl‘(|f| + z'|f|~)l eXPl—(W + Z'|<i>|")lf

is in A(M fl LP(T)). Since A(M fl LP(T)) C A(M] fl LP(T)), the above observation
shows us that u does not vanish on a set of positive measure and by the construction
of u, u multiplies Q5 in to L°°(T). Now (152" is in H (1 LP(T) for all integers n.
Therefore,

/ 1r u(e’0)q5(ei0)ei"9d6 = 0 for all integers n.

Therefore, ugb = 0, but u does not vanish on a set of positive measure. Therefore,
a) = 0. So HflL”(T) = {0}.
To finish the proof we need to establish our conditions on b‘l. To do this, we

first consider the case when 2 < p S 233;. Let

. 1 .
E, = {e”9 : it < |b(e'9)| < n}.

Now define
25—1; log |b| on E,

0 on Ef,

31

and

hn : exp(kn + ilin).

Note that hn is H °°(T) for all n in 2+. We make the following computation.
1 .2; 1/2
(— / lblz-P d9) 3 Henne = thnnM

27r E
1 1/P
_ P P
6(27r/rlbl |h,,| do) .

The last inequality holds because bhn is in M and b is in Liz-EMT) and ha is in

|/\

H°°(T); so bhn is in L’Tz-gHT) which is contained in LP(T). So bhn is in M n I)"(T).
Further the above inequality holds for all n. So the quotient is bounded by 6 for all

n. Letting it go to infinity, we get that
Ilb‘1II. s 6

as desired. Now, for the case where p = 2, we note that for h a trigonometric
polynomial

llhllz = thHM S (lllbhllp-

Therefore,

1
g [Tater -1)lh|2d0 2 o

for all trigonometric polynomials h, from which it follows that

”VIII... 3 6. A

32

Chapter 4

Invariant Vector Subspaces of

LP(T2)

In this chapter, we begin the second part of this dissertation. Here, we consider vec-
tor subspaces of LP(T2) that are invariant under 51 and 52. We start by considering
vector subspaces of HP(T2). In fact, let’s start with the case p = 2. Naturally, one
might start with the subspace M = ¢H2(T2) and hope, as in the H 2(T) case, that
these are all of the subspaces of H 2(T2) invariant under both SI and 52. Unfortu-
nately, that is not the case. Rudin [18] showed that, unlike H 2(T) where all subspaces
invariant under S are generated by a single inner function, there are subspaces of
H 2(T2) invariant under S1 and S2 that are not generated by a single function. In
fact, there are subspaces of H 2(T2) invariant under SI and S2 that are not even
finitely generated. Further, he showed that there are subspaces of H 2(T2) invariant
under 81 and 32 that contain no bounded elements, again in contrast with the H 2(T)
case where every subspace invariant under S contains a bounded function, in fact
an inner function. To my knowledge, the description of all the subspaces of H 2(T2)

invariant under 51 and 32 is still unknown. However, some work has been done to

33

that end. The first result is due to Mandrekar.

Theorem 19 (Mandrekar [13]) Let M 76 {0} be a subspace of H2(T2) invariant
under 51 and S2. Then, M = qH2(T2) with q inner if and only if SI and $2 are

doubly commuting on M.

We want to extend Mandrekar’s result to HP(T2), 1 S p S 00. Before we do this
we give a result of Ghatage and Mandrekar in L2(T2) to prevent proving a similar

result twice.

Theorem 20 (Ghatage & Mandrekar [5]) LetM 95 {0} be a subspace of L2(T2)
invariant under SI and S2. Then, M = qH2(T2) with q unimodular if and only if

51 and 32 are doubly commuting shifts on M.

Here the extra condition shifts is very important.

The above two theorems can be shown by exploiting the following decomposition.

Theorem 21 (Halmos-Wold Four-Fold Decomposition, [11], [24]) Let V1, V2 6

B(H) be isometries with V1 and V2 doubly commuting on H.
1. There is a unique decomposition
H = HS, EB Hsu EB Hus EB Huu
such that

(a) Vial”) C H33 and VilH” is a shift fori = 1,2.

(b) V1(H3u) C Hsu and V1

 

H,“ is a Shift.
(c) V2(H,u) = Hsu and Vglng is unitary.

(d) VIA/Hus) C ”as and V2|Hus is a Shift

34

(e) V1(Hus) = Hus and Vlqu“ is unitary.

(f) V,(Huu) = Hm, and Vilma is unitary fori = 1, 2.

2. Define 1C1: H@V1(H), 1C2 = HGV2(H) andlC = (HGV1(H))fl(HeV2(H)).

Then we have

71,, = iguana/2mm),

H... = goevlminrzovsucnt

it... = :0@lémlflfzoV1"(lC2)l,

u... = 0 WWW)-
n,m20

Notice, this is just Theorem 3 (p. 13) applied to two operators. The double
commuting condition is required to make everything work out as we expect.

Now, we extend this result to LP(T2), 1 S p S 00. As a corollary we will
get the HP(T2) extension. Before we do this, we need some terminology. We let
RP (U2) denote the class of all functions in U2 which are the real parts of holomorphic
functions. We point out here, in contrast with functions analytic in U, not all real
harmonic functions in U2 are the real parts of holomorphic functions in U2, see [18]

for more details. We also recall that,
f : T2 ——) (—oo, 00]
is called lower semicontinuous, (l.s.c.) if
{(691,602) :f(ei01,ei02) > a}

is open for all real 0.

The proof of the following lemma is found in [18] (p. 34).

35

Lemma 2 Suppose f is a l.s.c. positive function on T2 and f E L1(T2). Then there
exists a singular (complex Borel) measure 0 on T2, 0 Z 0, such that P[f — do] E

RP(U2).

Before we give our next lemma we make some observations. First of all, if f
is continuous, then f is l.s.c.. If f and g are two continuous functions, then so is
(f V g) (x) = max { f (x), g(x)}. This follow from a straight forward 6 — 6 argument
once you note that (f Vg) (x) = %[f(x) +g(x) + |f(x) -— g(x)|]. Finally, if 0")le is
a sequence of l.s.c. functions, then f (x) = sup" fn(x) is also l.s.c. This follows from

the definition of l so. and the fact that

{x:f(x) >0} = U {xzfn(x) >01}.

n=l
Lemma 3 Suppose f is real-valued on T2 and f E LP(T2) for 1 S p < 00. Then
there exists two positive l.s.c. functions gl and 92 in L”(T2) such that f = g1 — 92

a.e. on T2.

Presently, we only need this result for p = 1, but later we will need this result for
other values of p.
Proof: Since f is real-valued on T2, f E LP(T2) and continuous functions are

dense in LP(T2) there exists gbl continuous such that

”f — ¢1Hp < 2-1

and by the reverse triangle inequality we get

llahllp < (1+ 2||fl|p) 2“.

Now we can find (152 continuous such that
[[(f — (151) _ ¢2llp < T2

36

and by the reverse triangle inequality we get
||¢>2||p < 2‘2 + llf - ¢1llp < 3 ° 2’2-

Continuing in the manner we get the existence of a sequence of real-valued continuous

functions (rbn),, such that
f = Z ¢n
n=1
in LP(T2) and

”in“, < C . 2-" for all n, where o = max {1 + 2||f||,,, 3}.

Now, for e > 0, define
If): = ((25,,V0)+6-2"”
and
it); z (—¢,, V 0) + c - 2’".

Then it: and 1,1); are positive continuous functions with (bu = if): — 1,1); . So
f = [(223 —l/J;) = 222:: — 221»; in LP(T2).
":1 n=1 n=l
Since

(llth0llp+e-2‘”) S Z(I|¢n”p+€°2—n)

1 n=1

(C-2—"+c-2‘”) < 00

M8

1'1

«3
Z llwillp S
n:l

[V18

<

H
p—a

n

we get that there exists a gl in LP(T2) such that
gl 2 i If): in LP(T2).
n=1
Similarly, we get that there exists a 92 in LP(T2) such that
92 = 2312/); in LP(T2).

37

So we have that

f =91—92 in ”(T2)-

It is left to show that 91 and g2 are equal to positive l.s.c. functions a.e. Let

it
8n = 2110?-
k=l

Since 3,, converges to 91 in LP(T2), there exists a subsequence that converges to g1
a.e. But since 3,, is monotone increasing, we get that 3,, converges to 91 a.e. and
further that sup 3,, = lim 3". By our above observation, we conclude that sup 3,, is
l.s.c. It is clear that sup 3,, is positive. Therefore, gl is equal to a positive l.s.c.
function a.e. Similarly, we get that 92 is equal to a positive l.s.c. function a.e. So f

is equal a.e. to the difference of two positive l.s.c. functions. A

Lemma 4 Suppose f is real-valued on T2 and f E L1(T2). Then there exists a

singular (complex Borel) measure 0 on T2, such that P[ f - do] 6 RP(U2).

Proof: If f is real-valued on T2 and f E L1(T2), then Lemma 3 (p. 36) asserts
the existence of two positive l.s.c. functions 91 and 92 in L1(T2) such that f = gl — 92
a.e. By Lemma 2 (p. 36) there exists nonnegative singular measures 01 and 02 such
that Plgl "do1] and P[gz --dog] are in RP(U2). Letting o = o1—02 we get a singular

measure such that

P[f - d0] = P[(gl —92) " d(01— 02)]
: Pl(91 — d01) — (92 — d02ll

= P[91 -’ d01] — P[92 — (102].

So, P[f — do] is in RP(U2). A

38

Theorem 22 Let M 75 {0} be a'subspace of LP(T2), 1 S p < 2, invariant under 51
and 52. Then M = qHP(T2) where q is a unimodular function if and only if 81 and

52 are doubly commuting shifts on M O L2(T2).

Proof: Let N denote M F) L2(T2). Then N is a closed invariant subspace of
L2(T2) and by hypothesis Sl and 52 are doubly commuting shifts on N. Therefore,
by Theorem 20 (p. 34), N = qH2(T2) where q is a unimodular function. Now since
N is contained in M and M is closed, the closure of N in LP(T2), which is qHP(T2),
is contained in M. So we need to show that N is dense in M. To do this, let f E M,

f not identically zero. Then define

0, Ifl S n,

log Ifl“, Ifl > n-

U":

Note that an E LP(T2) for all n since

flunrdm = / lloglfl“‘|”dm=/ Iloglfllpdm
|f|>n |fl>n

</ pdm< ”<00.
_ Wm _quI.

So in particular, un E L1(T2) and real valued for all n. So by Lemma 4 (p. 38), there
exists a sequence {on},,20 of singular measures such that P[un — don] E RP(U2)
for all n. So there exists a sequence of analytic functions (R). such that Re(F,,) =
P[un —do,,]. By the M. Riesz theorem, which holds on the polydisc (see [17]), we have
||Fn||p S Cpllunllp for all n. Now since un E LP(T2) and un converges to 0 in LP(T2),
we get Fn converges to 0 in LP(T2) and hence at least a subsequence converges to

zero a.e. Let (1),, = exp{F,,}. Then

1’ lflSn)

Ifl", |f|>n

l¢nl :

39

and (Pa tends to the constant function 1. By construction, (1),, f is a bounded function
dominated by f for all n. Also, ¢n f E M because 0% is bounded analytic and hence
is boundedly the limit of analytic trigonometric polynomials. Since (15,, f is bounded,
it is in N. As n goes to infinity 051: f converges to f in LP(T2) by the dominated
convergence theorem. So each f in M is the limit of functions from N. So N is
dense in M as desired.

Conversely, if M = qHP(T2) with q unimodular, then MflL2(T2) = qH2(T2). So

SI and S2 are doubly commuting shifts on M n L2(T2) by Theorem 20 (p. 34). A

Corollary 3 Let M 75 {0} be a subspace of HP(T2), 1 S p < 2, invariant under 51
and S2. Then M = qH"(T2) where q is an inner function if and only if 51 and 52

are doubly commuting on M (l H 2(T2).

Proof: H ”(T2) is a subspace of LP(T2); so M is a subspace of LP(T2). Note that
M 0 H2(T2) = M n L2(T2) since M C HP(T2). Since S1 and 82 are shifts on all
subspaces of HP(T2), we get M = qHP(T2) where q is unimodular by the previous
theorem. Since q E qHP(T2) C HP(T2), we see that q is holomorphic, and hence
inner. The converse is just a special case of the above theorem. A

We use the notation H§(T2) = {f E HP(T2) : f(O, 0) = 0} in the next theorem.

Theorem 23 Let M 7E {0} be a subspace1 of L”(T2), 2 < p S 00, invariant under
31 and Sg. Then M = qH§(T2) where q is a unimodular function if and only if S1

and S2 are doubly commuting shifts on A(M) fl L2(T2).

Proof: If M = qH{,’(T2) where q is a unimodular function, then A(M) =
qHF¥L1(T2). Therefore, A(M) fl L2(T2) = qH2(T2). It then follows from Theo-

rem 20 (p. 34) that 51 and 52 are doubly commuting shifts on A(M) fl L2(T2).

 

lAssume further star-closed when p 2 oo.

40

Conversely, if 51 and Sg are doubly commuting shifts on A(M) fl L2(T2), then by
Theorem 22 (p. 39) we get that A(M) = qH35(T2) where q is a unimodular func-
tion. Therefore, M = qu,’(T2) where q is a unimodular function. When p z 00 we

need that M is star-closed to make our final conclusion. A

Corollary 4 Let M ¢ {0} be a subspace2 of HP(T2), 2 < p S 00, invariant under
S1 and 32. Then M = qu(T2) where q is an inner function if and only if SI and

52 are doubly commuting shifts on A(M) fl L2(T2).

Proof: A similar argument as used in the above corollary gives the result. A
We now consider the ideas from the first part of this dissertation; namely, the
idea of our vector subspaces being Hilbert spaces. Our first result is due to Dinesh

Singh. He proved a generalization of Theorem 19 (p. 34).

Theorem 24 (Singh) N is a Hilbert space which is a vector subspace of H 2(T2)
such that N is invariant under SI and S2 and for which S1 and S2 are doubly com-
muting isometries on N if and only if there exists g in H°°(T2) unique up to a
factor of modulus one such that N = gH2(T2) with norm HgfIIN = ||f|]2 for all f in

H2(T2).

We now slightly modify the proof of this theorem to prove a general HP(T2)

, result. The following theorem is just a two-variable analogue of Theorem 13 (p. 22).

Theorem 25 If M is a Hilbert space contained in HP(T2), invariant under 31 and

S2 and if 51 and Sg are doubly commuting isometries on M, then
M = bH2(T2)

for a unique b:

 

2Assume further star—closed when p z oo.

41

1. [fl S p S 2, b E H%(T2). Whenp = 2, we mean H°°(T2).
2. pr>2,b=0.

Further, “beM = ||f||2 for all f in H2(T2) (1 S p S 2)-

Note that the converse of this theorem is also true. Before we prove this theorem,

we give several lemmas. The first two lemmas are due to Slocinski [24].

Lemma 5 (Slocinski [24]) Suppose that V1 and V2 are commuting isometries on
a Hilbert space H and write Rf = H e V,(H) (i = 1,2.). Then the following are

equivalent:

1. There is a wandering subspace L for the semigroup {I/lnl/2m} I )0 such that

H = f: it elem/M).

n=0 m=0

2. V1 and V2 are doubly commuting shifts.

3. Rf n R} is a wandering subspace for the semi-group {KW/2m} and

n,m20

it = Z Z an‘Vemwt a Rt)-

n=0 m=0

Lemma 6 (Slocinski [24]) Suppose V1 and V2 are commuting isometries on the

Hilbert space H at {0}. If Rf“ fl 1sz = {0} where R} = H 9 Vi(H) (i = 1, 2.), then

’H a Z Z avrvemmt 0 R2).
n=0m=0

Lemma 7 [ft/2 is positive and l.s.c. on T2 and 2/2 E LP(T2), then if) = |f| a.e for

some f E H”(T2).

42

Proof: Since l.s.c. functions attain their minimum on compact sets, (for a proof
see [16]), we may assume without loss of generality that w > 1. Applying Lemma 2
(p. 36) to logr/J asserts the existence of a singular measure 0 2 0 and a holomorphic
function g in U2 such that Re(g) = P[logw — do]. Put f = exp(g). Then f is

holomorphic in U2 and
|f| = leXD(g) = exnflog y) = w
on T2. Since it) E LP(T2), f E HP(T2) as desired. A

Lemma 8 For all h E LP(T2) with 1 S p < 00, there exists a positive, l.s.c. (b E

LP(T2) such that ¢ 2 Ih] a.e. on T2.

Proof: If h E LP(T2), then [h] E LP(T2) and real-valued. So, by Lemma 3 (p.

36), there exists two positive l.s.c. functions d9 and w in LP(T2) such that
|h| = q) - if) a.e. on T2.

So

|h| S q) a.e. on T2. A

Lemma 9 Let f be an element of HP(T2) that multiplies H2(T2) into HP(T2). Then

f multiplies L2(T2) into LP(T2).

Proof: Let g be an element of L2(T2). Then by Lemma 8 (p. 43) there exists a
positive l.s.c. function (15 in L2(T2) such that [g] S (f) a.e. on T2. Then by Lemma 7

(p. 42) there exists an h in H 2(T2) such that |h| = (b a.e. on T2. Now consider

/ lfglpdmz / Iflplglpdmaz

T2 T2

< [2 |f|p|h|p dmg since |g| S (t = |h| a.e. on T2
'r

= / |fh|pdm2 < 00 by hypothesis. A
T2

43

This next lemma is a straight forward calculation found in [22]. We include it for

completeness.

Lemma 10 If g is a measurable on T2 that multiplies L2(T2) into Lq(T2) where

1S q S 2 then 9 E LFQEHTZ’). When q = 2, we mean L°°(T2).

Proof: So
flfgl" dm < 00 for all f E L2(T2).
That is,
/]fIq|g|”dm < 00 for all |f|q e L2/‘I(T2).
Hence,

/ Iglqh dm < 00 for all h e L2/‘1(T2) with h 2 0.
Since every h in L2/‘l(T2) is equal to (hl — hz) + i(h3 — h4) where hj is in L2/‘l(T2)
and h,- 2 0 forj = 1,2,3,4, we have

|g|qh E L1(T2) for all h E L2/9(T2).

Now by an inverse of Hblder’s Inequality found in [26], we may conclude that lglq is

in the dual of Lz/‘1(T2); that is,
lgl" e Liam).

Hence,

9 E Lil-15(T2).

So the set of multipliers of L2(T2) into Lq(T2) (1 S q S 2) is the space LT2-qi(T2). A
Proof of Theorem 25 (p. 41): We first consider the case 1 S p S 2. Observe

that flfzoSflM) = {0} (i = 1,2.). This observation and our doubly commuting

44

hypothesis give us that
= Z Z @V1"V2’"(Rt0 Rt) by Lemma 5 (p. 42)
n 0711 0

and that

Bi 0 Ri # 0 by Lemma 6 (p. 42)
1 2

where Rf = HeV;(H) (i = 1,2.). So we may take 9 from 12% flRéL, with ||g||M =1.
Then {gem1 e‘m92}n m>0 is an orthonormal sequence in M. Let f be an arbi-
trary element of H2(T2). Then f(e‘91,e‘92) = gozwo (n, m)e‘"91e‘"‘92. Let

fnm(e‘91,e‘92) 2 22:0 [[10 f (k, l)e"‘91e"”'~’. Then fnm converges to f in L2(T2) and

a.e. along rectangles. We make the following computation.

lf‘(lc,l)l2

[V]:

llfnmlli =

a-
II
o

Iflk, l)|2llge”°"‘e“’”llit (4-1)

a-
II
o

H
M:
Ma this 2M3

llflk, 096"”1 6”"? “in

II
M:

7!"
II
o

l

2":

k=0 l

H
o

2
get/€01 61102

MB

 

 

 

 

II
o

M

Since (fnm)(n m) is Cauchy in L2(T2), (22:0 2E0 f (k, l) ge"N91 €192)“I m) is Cauchy in

9

M. Since M is a Hilbert space, there exits a h in M such that

n m
E Z: f(k, l)ge"°(’lei’92 — h” ——+ 0 as (n, m) ——+ 00 along rectangles.

 

 

k=0 l=0 M
Thus,
00 00 A . ’
h z Z Z f(k, l)getk9162l02
k=0 1:0
and since

9(k l)eik018i192

ii
M8
M8

a.
II
0
~
II
o

45

we have for fixed m and n,

h : f(030)9 +f(021)gei02 +f(170)gei91+ ° ° ' +

+f(m, n)ge’"’ole’"”'~’ + hle’(”’+1)”l + hge”(”+1)’92 (4.2)

where

h1=flm+1,0)g+f(m+ 1,1)ge’”2 +f(m+2,0)ge”’l +

and

h2 =f(0,n+1)g+f’(0,n+2)ge“92 +f(1,n+1)ge’91+-~-

It’s clear that hl and h2 are in M and hence in HP(T2). Thus from equation (4.2)
(p. 46), we see that the (m, n)-th Fourier coefficients of h are the same as the (m, n)-
th Fourier coefficients of the formal product of the series of g and f. This means
that h = g f in HP(T2) and hence in M. This observation along with equation (4.1)

(p. 45) gives us that
llgfllM = Ilfllz.

Since f was an arbitrary element of H 2(T2), we see that g multiplies H 2(T2) into
M g HP(T2). By Lemma 9 (p. 43) we conclude that g multiplies L2(T2) into LP(T2).
Lemma 10 (p. 44) shows us that any d that multiplies L2(T2) into LP(T2) must be
a member of Lag-$(T2). Thus g must be in H 2'29Lr’(T2).

Note that 233’; 2 2 when 1 S p S 2, so g is in H2(T2).

It’s left to show that Rf n R; is one dimensional. Suppose there is a 91 in
Rf fl RiL with unit norm and g _L gl in M. Then by the same computations above
we get that ng2(T2) is also contained in M and by our decomposition we get that
gH2(T2) _L ng2(T2) in M. Further, ggl = 919 is in gH2(T2) as well as ng2(T2).
So, 991 = 0. As 9 and g1 do not vanish on a set of positive Lebesgue measure unless

they are identically zero we get a contradiction. Hence Rf fl RQL is one dimensional

46

as desired.
Now we consider the case p > 2. Suppose M 74 {0}. Proceeding as in the
previous case we get that g multiples L2(T2) into LP(T2) C L2(T2) and hence g is

in H °°(T2). Choosing an appropriate 6 > 0 such that
E = {(601,602) I |g(ei018i02)| > 6}

has positive measure, let b be a function that vanishes on the complement of E which
is in L2(T2) but not LP(T2). But then, gb is in LP(T2) and so b will lie in I}’(T2) since
g is invertible on E. Hence a contradiction. So our supposition must be incorrect.
So, M = {0} A

Before we give a corollary of this result, we need a definition.

Definition 9 A function g in HP(T2) is called outer if the linear combination of

functions

101,662) 1‘01 (8 1'01 H8292) eidgg(ei61 6’02) i01 i02 (8 i01’ei02),u.

g(e ege eege

are dense in H”(T2).

Before we proceed, we make an observation. In HP (T), a function f being outer

is equivalent to

loglf(0) =2 —/_: logm (etude

In HP(T2), this is not the case. Let’s call the functions in HP(T2) that satisfy

log|f(0) fiefl f:logIIe*"lei92)Id61dot

weakly outer. It is known that outer implies weakly outer but weakly outer does

not imply outer. See Rudin, [18] for more details.

47

Corollary 5 Suppose M is a Hilbert space contained in L”(T2), invariant under 31

and 52, 51 and S2 are doubly commuting isometries on M.

Case 1: For 1 S p S 2, if 31 and SQ are doubly commuting shifts on M” flL2(T2),
then
M = bH2(T2)
for a unique b E L725(T2) having the modulus of an outer function a.e. When

p = 2, we mean L°°(T2).

Case 2: Forp > 2, if S1 and 52 are doubly commuting shifts on Ann(—M-Lp)flL2(T2),

then M = {0}.
Further, ||bf||M = ||f||2 for all f in H2(T2) (1 S p S 2).

Note that the converse of this theorem is also true.

Proof: Case 1: By Theorem 22 (p. 39), we have that M” = ¢HP(T2) for
some unimodular function d). Then, M’ 2 EM is contained in HP(T2) and with
norm llapllM: = ||p||M is a Hilbert space invariant under S1 and 32, SI and 52 act
as isometries on M’. We also see that 51 and S2 doubly commute on M’. So then
by the above result we get that M’ = gH2(T2) with g E H 5255(T2) with norm
llgfllM' = ”fllz. So M = ¢M’ = ¢9H2(T’) with Ham I|¢gf||M = llgfllM' = ||f||2-
Further, since the closure of M in LP(T2) is 6H2(T2), we have that 9 must have the
modulus of an outer function a.e.

Case 2: By Theorem 23 (p. 40), we get that M” = ¢H§(T2) for some unimodular
function qt. Then, M’ = 5M is contained in HP(T2) and with norm IlagllM: = ||g||M
is a Hilbert space invariant under S1 and $2, 31 and SQ act as isometries on M’. We
also see that 51 and 32 doubly commute on M’. So then by the above result we get

that M’ = {0}. Therefore, M = {0}. A

48

Before we give another corollary we recall some definitions. Let BM 0(T2) be

the class of all L1(T2) functions f such that

1 1
an.=sup|-,—|/I|f—m/In<oo

where the supremum is taken over all squares of T2 and II | denotes the normalized
Lebesgue measure of I.

BM 0(T2) is a Banach space under the norm

llfll = Ilfll. + |f(0)l-

VM0(T2) is the closure of the continuous functions in BM 0(T2). BM 0A(T2) =
BM0(T2)flH1(T2) and VMOA(T2) = VMO(T2)flH1(T2). By the John-Nirenberg
theorem [25], we get that BM OA(T2) C HP(T2) for p < 00. We are now ready to

state our corollary.

Corollary 6 If M is a Hilbert space contained in BM 0A(T2) (VM0A(T2)), in-
variant under $1 and 5'2 and if SI and 5'2 are doubly commuting isometries on M,

then M = {0}.

Proof: By the J ohn—N irenberg Theorem mentioned above we get that BM OA(T2) C
HP(T2) for p < 00. So in particular, BMOA(T2) C HP(T2) for p > 2. So by Theo-

rem 25 (p. 41), M = {0}. A

49

Chapter 5

Random Fields

5.1 Introduction

Let (9,]: , P) be a probability space (measure space with P(S2) = 1) and {Xmm :
(m, n) E Z2} be a family of random variables (complex measurable functions) on
(9,319) such that E|X,,.,,.|2 < 00 (f0 |Xm,,,|2dP < 00) for all (m, n) e 22. We
assume that EXmm :2 0 for all (m, n) E Z2. The {Xmm : (m, n) 6 Z2} is called a

second order random field. For (m, n), (m’, n’) E Z2, we define

 

C((m, n), (m’, ”Il) :- EXm,nXm’,n’

the covariance function of {Xmm : (m, n) E Z2}. We call a second order random

field (weakly) stationary if
C((m, n), (m’, n')) = r(m — m’, n — n’).
One can prove using the Bochner Theorem that

7-(m, Tl) = [r2 e-(imA+in0)F(dA, d0).

50

The measure F on the torus is called the spectral distribution of the weakly
stationary random field {Xm‘n : (m, n) E Z2}. One can use the extension of Stone’s

Theorem, to Show that
Xmm = / ze-“mvnolzwxdm (5.1)
'1‘

where

2:3(1‘2) ——> L(X)

is an (orthogonal scattered) measure with
L(X) = mL2<“>{Xm,n : (m, n) e 22}

and B(T2) denotes the collection of all Borel sets of the torus. Here EZ(A)Z(A’) =

F (A n A’). It is easy to check that the map
Xm,n __) eim-+in-

is an isometry from L(X) onto L2(T2, F). This isometry can be used to study ana-
lytic properties of F corresponding to prediction questions related to the stationary
random field. The “prediction” of the “future” from the “past” observations is de-
fined by giving an order on Z2. For an ordering induced by a semi-group, this problem
was studied by Helson and Lowdenslager. In their context the semigroup S satisfied
8 U (—S) = Z2 and 8 fl (—8) = {(0,0)}. The “analyticity” was defined by using
functions of the form

f(A, 6) : Z: a(m,n)eim/\+in9

(m,n)ES

where Z |a(,,,,,,)|2 < 00. Using the above isometry one can show that the stationary

random field satisfies n(m.n)€Z2 29—poin{Xk,, : (k, l) <<s (m, n)} = {0} if and only if

Xm,n: Z am—k,n—-l€k,l (52)
(k,l)€$

51

where {5“} are orthogonal random variables and Z |a(,,,,,,)|2 < 00. The represen-
tation (5.2) (p. 51) is called the moving average (MA) representation. For all
different orderings, the problem is studied in [14], where also the MA representation
for the semigroup given by the quarter plane (m 2 0, n _>_ 0) is studied.

Recently, based on data from finance, insurance and hydrology, it is found that
one needs to study the stationary random fields which are not second order (i.e.,
E |X,,,,,,|2 = 00). For this, one needs to study the models given by so called stable
random fields. In this case, we show that the class of random fields of type (5.1) (p.
51) and (5.2) (p. 51) are disjoint following the work of [12]. We then generalize some
other results of [12] for some half-space ordering for stable random fields contained
in random fields of the form (5.1) (p. 51). Presently, we have not completed the
project. However, we indicate at the end the open problems for our future research
which connect with the invariant subspaces of weighted spaces L”(T2, w) (1 S p < 2)

and an analogue of a result of Zygmund [27].

5.2 Some Probability Background

Let ((2, f) be a measurable space. A positive measure P on (Q, .77) is called a prob-
ability measure if P(§2) = 1. In what follows, P will always denote a probability
measure, the term event is used to mean a member of .7: and random variable is

used to mean a complex measurable function on (9, f).

Definition 10 Two events A and B are said to be independent if PM 0 B] =

P[A]P[B].

Definition 11 Two random variables X and Y are independent if the events X E

A and Y E B are independent for any two Borel sets A and B on the line; i.e.,

52

P[[X e A] n [Y e B]] : P[X e A]P[Y e B].

Definition 12 A finite collection {Xj : 1 S j S n} of random variables is said to
be independent iffor any n Borel sets A1, A2, . . . , An on the line
P] n [X,- e A,]] = H P[X, e A,].
1335" 131$”
Definition 13 A collection of random variables indexed by a parameter is called a

random process; e. g., {Xn : n E Z}. When the index set is a collection of ordered

pairs one calls the random process a random field; e.g., {Xmm : (n, m) E Z2}.

Definition 14 For a k-dimensional random vector X = (X1, . . . ,Xk), the distri-
bution u (a probability measure on R") and the distribution function F (a real

function on R") are defined by
,u(A) = P[(X1,...,Xk) e A], A e B

and

F($li'°'7xk) : P[Xl S x1)"',Xk S 113k] 2 ”(53)
where 3,, = [y : y, S x,,i=1,...,k].
Definition 15 By a field of i.i.d. random variables we mean a field of random
variables that are independent and identically distributed. Thus, if {Xmm : (n, m) E

Z2} is a field of ii d. random variables, then the Xmm ’s are independent and all have

the same distribution function F (say):
P(Xm,n S x) = F(x) for all (m, n) E Z2 and for all x.

Definition 16 A distribution function F is stable if for each n there exists con-
stants an > O and b", such that, if X1, . . . , Xn are independent and have distribution

function F, then Lia—sin + bn also has distribution function F.

53

Remark 2 If 0 < a S 2, then exp(—|t|°‘) is the characteristic function of a sym-
metric stable distribution; it is called the symmetric stable law of exponent a.

The case a = 2 is the normal law, and a = 1 is the Cauchy law.

5.3 Strongly Harmonizable Stable Fields

By an S (15 field we will mean a family of complex random variables {Xmm : (n, m) E
Z2}, such that for every (n1,m1), . . . , (nk,mk) E Z2, the joint distribution of the
2k-dimensional random vector ReXnmnl, Ianwm, , Reannk, Iankm. is
symmetric stable with parameter a. For each real SaS random variable X there

exists a number |X la 2 0 such that
Eexp<z°tX) = exp{-|X|2|tl°‘}

for all t E R. It is well know that an SaS process is a pt" order process for any
p satisfying 1 < p < a. For a linear space of $015 random variables, the function
X -—> IX [a defines a norm for 1 < oz < 2. The norm is related to the usual 0(9)

norm by

llelp = C(p,a)|Xla

where C(p, a) is the following constant depending on a and p, 1 S p < a S 2:

 

2p-l fooo s—p/(a—l)(1 _ 6—3) (18 1/P
C(p,a) _ [ af0°° v—P—lsinzv dv '

If {Xmm : (n, m) E Z2} is a complex SaS field, then L(X) we will denote the
closure in probability of the set of all linear combinations of {Xmm : (n, m) E Z2}.

The Schilder norm of a complex S (23 random variable Z = X + iY, is defined by

(FX,Y(T))1’°‘, 1 g a g 2

Pkg/(T), 0<a< 1

llZIIa =

54

where F xy is the unique symmetric measure on the circle T such that
Eexp (itX + isY) = exp ( —/ ltx + sylaFX,y(dx, dy)). (5.3)
'r

If o = 2, then 2||Z||§ = EIZIZ. If 1 < a < 2, then [I - Ila is a norm on L(X) which is
equivalent to any [JD-norm for 1 < p < a. (If 0 < oz < 1, then H ”a gives merely a

metric.)

Definition 17 A complex SaS field {Xmm : (n, m) E Z2} is said to be harmo-
nizable if there exists an L(X)-valued Borel measure Z on T2 such that for every
(n, m) E Z2,

Xmm = [[26‘(‘m*+’"9)2(d/\,d6). (5.4)
If, in addition, the measure Z takes independent values on disjoint Borel sets, then

{Xmm : (n, m) E Z2} is called strongly harmonizable.
If {Xmm : (n, m) E Z2} is strongly harmonizable 305 field, then the mapping

falflfdz

is an isometry from L"(T2,,a) onto L(AZ) :2 span{Z(B) : B E 8(T2)}. Here u is

a nonnegative measure related to Z by the formula

“3(3)”; 13 a s 2
MB) =
||Z(B)lla, 0 < a <1

for B E B(T2) where B(T2) denotes the collection of all Borel sets of the torus.

Definition 18 A complex 305 field {Xmm : (n, m) E Z2} is said to have a moving
average (MA) representation if there exists a sequence (amm) E €°(Z2) and a field

of i.i.d. SaS random variables {Ymm : (n, m) E Z2} such that

Xm,n : Z am—k,n-—lYk,l (man) 6 Z2-
(k,l)EZ2

55

It is know that mapping
{ak,l} E 60(Z2) —+ Z am-k,n—1Yk,z E L(Y)
(k,l)EZ2
is an isometry.
In what follows, [1 means the Fourier transform of the measure u. We write f
to mean fdmg. Also, BF means the collection of all functions which are the Fourier
transform of a member of LP. Recall that the Fourier transform is defined as follows.

For a measure on T2,

[1(771, Tl) : [r2 e—(imA+in9)d#(A,0)

and for a measure on Z2,

fl(/\, 0) = Z e“(im»\+in9)#(m’ n)

(m,n)EZ2

If u is a measure on T2 which takes values in 80(Z2), then for B E B(T2)

i<B><A.0> = : e-<'“+“">u<B)<k,z).
(k,l)EZ2

where ,u(B)(k,l) is the (k, l)-th coordinate of u(B).

Theorem 26 ([4], Theorem 2.7) If F is a function on T2 with the property that

A

fF E U15p<2€P(Z2) for each f E C(T2), then F = 0 a.e. [m2].

Theorem 27 Let 1 < a < 2 and let Xmm = ZUCJKZ? am—k,n—1Yk,z, where (amm) E
(“(Z2) and {Yum : (n,m) E Z2} is a field of i.i.d. SaS random variables. Suppose

there exists an L(Y)-valued measure Z such that for each (m, n) E Z2,
Xmm = / ze‘(‘m"+i”9)Z(d/\,d0). (5.5)
'1‘
Then Xmm = O for all (m,n) E Z2.

56

Proof: One can assume that ||Yk,,||a = 1, (15,1) E Z2. Since L(Y) is isometric to
€°(Z2), the formula (5.5) (p. 56) implies that there exists an €“(Z2)-valued measure
u such that for all (m, n), (k, l) E Z2

an-.. = [T2 e~<im*+‘"9>u<d/\.d9>(k,z)
where u(B)(k,l) is the (k,l)-th coordinate of u(B). Note that (aw) E €“(Z2) C
82(Z2). Thus, a E L2(T2) and
A2 ei<k-m>*+i<’-">95(A, 0) dm2(/\,6) = ak_m,,_,. (m, n), (k,l) e 22.

Thus p(dA,d0)(k, l) = exp(ikA + il6)a()\, 6) dm2(A, 9), (k, l) E Z2 and for each Borel
set
[1,,errome/te)amazes) = u<B>(k, l)

= [B u(dA,d0)(k,l)

= [T2 e‘k*+i'913(.\,6)5(,\,6) dm2()\,6) (5,1) e 22.

Hence [i(B) = 1361 for each Borel B E B(T2). Since u is a measure and the Fourier

transform is continuous from ”(22) into L"'(T2) with 51; + i = 1, we conclude

that (f gdu)‘ 2 get for each continuous 9. Therefore, by Theorem 26 (p. 56),
(amm) E 0. A

Definition 19 Let
My“) 2 span{e""‘91+’l""92 : n S k,m E Z}

and

Mia,” ___ W{emol+imog : n E Z,m S k}
where the closure is in L"(T2,u). A strongly harmonizable 505 field is ((1,1)-
regular if M8213) := (1,, Méa’l) = {0} and (a,2)-regular if M823) := 0,. ME”) =

{0}.

57

In what follows, p]- j = 1, 2, is the marginal of u. That is, for all B E B(T),
MB) = MB X T)

and

712(3) = MT X B)-

This is in contrast with terminology used before. Recall that m2 meant normalized

Lebesgue measure on T2.

Theorem 28 M82) = {0} if and only if each nonzero f E Méa’ll is different from
zero a.e. [m <8) #2]. If M9231) = {0}, then u << m (8) W and there exists a B E B(T)

with u2(B) = 0 such that fT log (M W (01,62))dm(91) > ——00 for 92 93 B.

Proof: (4: ) Let’s suppose that MEX,” 7E {0}. Let f E M_ a 001’) with f 75 0. Fix

6’2 E T, write f92(-) for f(-, 62) and define
Mfg— ._ spanLa <Td#>{e' ”f9,(-) : (n,m) e Z2}

and

JV— — spanLoVr lfozlad"){e"" ' .(n,m) E 22}.

Finally, defining Tm, : N —> M L92 by gb i—) ¢f02' TM, is an onto isometry. Since
continuous functions are dense in L“(T, |fg2|°du), we get that N :2 L"(T, Ifgzladu).
Therefore, lgfg, E M £9, C M8231) C Méa’ll for all B E B(T). Thus contradicting
our hypothesis.

(=>) Note that Mg? Q M121). Therefore, M12533 2 {0}. So by Theorem 2.6, p. 18
of [14] u << m 8) M and there exists a set B E 8(T) such that u2(B) = O and for
02 E B, fr log (W(61a02))dm(01) > —00. By Theorem 7.33 of [27], one can find

a $92 E H“ such that |¢Q,I°— - %’::i-), for all 62 6! B. The mapping e” —> ein'oigz,

58

(n 2 0) extends to an isometry from Méa’ll to H 0. Since each non-zero function in
H a is different from zero a.e. [m] and in particular (by, 76 0 a.e. [m] for all 62 g! B,
every function in Méa’l) has the same property. So, every nonzero member of M (a I)

is different from zero a.e. [m <8) M]. A

Theorem 29 An 3615 field is (a,1)-regular (0 < a S 2) if and only ifu << m®u2
and there exists a B E B(T) with u2(B) = 0 such that IT log (firm(01’02)) dm(01) >

—00 for 02 ¢ B.

Proof: It suffices to prove sufficiency. As proved in Theorem 28 (p. 58), the
existence of a set B E B(T) such that u2(B) = 0 and [1. log (d——t‘—-(m®m) (01, 02)) dm(l91) >
—00 for 02 E B implies that no member of M ( ’ 1') is different from zero a.e. [m (8) [12].

Now using Theorem 28 (p. 58), we get M10351) 2 {0}. A

Theorem 30 IffT log (“mm2 )(01,02))dm(01) = —00 a.e. [ug], then {Xmm : (n, m) E

Z2} is (a,1)-singular (i.e., M5,“) = L°(T2,u) for all n) (0 < a S 2).

Proof: M5,“) Q M,(,"’1) for all n. From the assumption and Lemma 2.7, p. 19
of [14], it follows that M5,”) = L2(T2, u). Now M5,“) is the closure of M5,“) in
L“(T2,u) giving M5,“) = L°‘(T2, p). A

We now state open problems which are needed to be solved in order to obtain

complete generalizations of the work in [12].

Problem 1 Extension of the result of Guadalupe [7] to the case of the Torus under

appropriate assumptions.

Problem 2 Extension of the result of Zygmund [Theorem 7.33, [27]] to the case of

the torus.

59

Appendix A

Orthogonal Decomposition of

Isometries in a Banach Space

The following results are found in [3]. Let X be a Banach space. For x, y E X, we
write x J. y if for all a E C,

III?“ S llx + ayllo (A-l)

Remark A.1 This is a nonsymmetric notion of orthogonality but it is equivalent to

the usual concept of orthogonality in Hilbert space.
WewriteM iNforM,N§ X ifxE M andyENimpliesxiy.

Definition A.l A semi-inner-product (s.i.p.) on X is a function [-, ] from X x

X into C with the following properties:
1. [, y] is linear for each y E X
2. llxhyll S ||$||l|y|| for 10,21 E X

3. [x,x] = “x“2 for all x E X

60

4. [x,ay] = 6[x,y] for all x,y E X and a E C.

Remark A.2 A particular Banach space may have many s.i.p. ’3 consistent with the

norm and the notion of orthogonality will be dependent on the s.i.p.
For M,N Q X, we write [M,.N] to mean {[x,y] : xM,y E N}.

Theorem A.1 Let X be a normed linear space and M and N be subspaces of X

with M _L N. Then there exists a sip [~, ] such that [N,M] = {0}.

Lemma A.1 Let X = M GEN where M and N are subspaces ofX with M J. N.

ThenN = {x E X: [x,M] = 0} for some s.i.p. [-, -].

Lemma A.2 Let M and N be closed subspaces of X with M _L N. Then M ED N

is closed.

By a smooth Banach space, we will mean a Banach space that is uniformly F réchet

differentiable. That is, for all x and y in the unit sphere S of X and A real,

Hm llrr + Ayn — urn

exists
A—iO /\

 

and this limit is approached uniformly for (x, y) E 8 x 8.

Remark A.3 In a smooth Banach space, the s.i.p. is unique, so we may write MJ‘

for {x E X: [x,M] = 0}.

Lemma A.3 Let X be a smooth, reflexive Banach space and suppose that {Mk}

and {Nk} are a sequence of closed subspaces such that
1. X = M]; EB N};
2. Nk _L Mk

61

3. N]; 9 Nk_1 and Mk_1§ Mk

Let M =[U,‘:°:1Mk] andN = (72°:le then X = M EBN andN _L M.

Definition A.2 Let V be an isometry on the normed linear space X. V is said to
be orthogonally complemented (a.e.) provided there exists a closed subspace M

ofX such thatX=M€BV(X) and V(X) J_M.

Remark AA 1. V is ac. if and only if there exists a projection P : X —-> V(X)

of norm 1.
2. In a Hilbert space, each isometry is orthogonally complemented.

3. Every isometry of LP (1 S p < 00) is orthogonally complemented.

Definition A.3 An isometry V : X —> X will be called a unilateral shift if there

exists a subspace E Q X such that
1. V"(£) _L V’"(£) for n > m
2. X : $zozo Vn(£).

Theorem A.2 (Generalized Wold Decomposition) Let V be an isometry on a
smooth, reflexive Banach space X. If V is ac, then there exist closed subspaces X1

and X2 such that
1. X1 and X2 are invariant under V,
2. V] .13 is unitary (surjective),
3. VIM, is a unilateral shift,
4. X = X1 EB X2.

62

Corollary A.l If V is an isometry on a smooth, reflexive Banach space which sat—

isfies:
1. V is ac.
2- mini/"(M = {0}.

then V is a unilateral shift.

63

Appendix B

A “Beurling Type” Theorem in

Hp(T)

In this section, we turn our attention back to S—invariant subspaces of H P (T) Recall
that the problem was first solved by Beurling for the case p = 2. Later, de Leeuw
and Rudin solved the problem for p = 1. A duality argument gives p = 00. We solve
a weaker version of the problem for p E ’P := {p : 1 < p < oo,p 7‘- 2}, using the
tools just developed. Before we give our theorem, we point out that all subspaces
of LP(T) are smooth, reflexive Banach spaces for p E ’P. So, the s.i.p. on LP(T) is
unique. It is given by

[9. f] = Ilflli‘” fr gimp-2 dm.

We now give our result.

Theorem B.1 M is an S-invariant subspace of HP(T), p E ’P with M 5' ac. if

and only ifM = ¢HP(T) with (b inner.

64

Proof: (<2) If M = ¢HP(T) with (b inner, then clearly, M is an S-invariant
subspace of HP(T) and S is o.c. on M since

M=ceés"(c)=cesw)

n=l
with S(M) 1 c where c = {art : a e o}.

(:) We start by noting that M satisfies all the conditions of Corollary A.1 (p. 63).
Therefore, S is a unilateral shift. That is, there exists a subspace .C 9 M namely,
£ = S(M)J‘ such that

M = é S"(£)
n=0
with S"(£) _L S "‘(L) for all n > m. We need only show that .C is one dimensional and

spanned by an inner function. Let (t E .C with ”all,D = 1. Then by our decomposition

of M we get that oz" .1. (b. That is,
] ¢z”$|¢|”‘2 dm = o for all n 2 1.
T

That is,

[rznlrtlpdmzo foralanl.

Taking complex conjugates of both sides we get
] znlrtlpdm = 0 for all n 9e 0.
T

Therefore, lab]? is constant and hence |q§| is constant. Since “45“,, = 1, we get that
|¢| 2: 1 a.e. So, (15 is inner. It is left to show that .C is one dimensional. To do this,
let’s suppose not. Then from Lemma 4, p. 440 of [6] there exists w 75 0 in .C of norm
one with w _L (b. By our decomposition we get that wz" J. 45 and (1)2" J. 11). Also
doing that same calculations above with w in place of (f) we see that w is also inner.

That is, both (23 and it have constant modulus one a.e. on T. Writing our above

65

orthogonal relations in term of the s.i.p. we get
Ail/WWI“? dm = 0 for all n 2 0

and

[r (bZn-IZII/le_2 dm 2 O for all n 2 1.

Since d5 and 2]) are unimodular we get
Luznadm = o for all n 2 0 (B.1)

and

[raznwdm = 0 for all n 2 1.

Taking complex conjugates of (B.1) (p. 66), we get
[r a.e-"Edm = 0 for all n 2 0.

Therefore, 45E = O a.e., but that is a contradiction since 43 and 1]) are unimodular. So
our supposition must be incorrect. That is, I, is one dimensional as desired. A
Although the above result is weaker than Theorem 12 (p. 21), it shows a property
of the subspaces of the form ¢HP(T). in the future, we want to examine in the
question of a Generalized Wold Decomposition for two isometries in this context and
get the analogue of Corollary 3 (p. 40) with conditions on the isometries acting on

the subspaces of HP(T2) directly.

66

BIBLIOGRAPHY

67

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