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Homological algebra of Hilbert spaces endowed with a complete Nevanlinna-Pick kernel

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Homological algebra of Hilbert spaces endowed with a complete Nevanlinna-Pick kernel
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Thesis (Ph.D.)--University of Florida, 1998.
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Includes bibliographical references (leaves 46-48).
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HOMOLOGICAL ALGEBRA OF HILBERT SPACES
ENDOWED WITH A COMPLETE NEVANLINNA-PICK KERNEL











By

ROBERT S. CLANCY


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1998




























Copyright 1998


by


Robert S. Clancy














This work is dedicated to the memory of my father Robert James Clancy,

who provided me an unassailable example of what a man should strive to be.














ACKNOWLEDGEMENTS


I would like to thank my advisor for his patience and generosity. More than

a mentor, he has inspired me. I am in his debt. I would also like to thank the

members of my committee for their contribution to this work and to my education.

I would like to thank the staff of the Math Department, especially Sandy, who has

made my time at Florida much easier. I am especially grateful to my close friends

who have lent me their support throughout this project. Finally, this work could

not have been completed had it not been for my mother and Jennifer. I will never

be able to thank you both enough.














PREFACE


Homological algebra has long been established as a field of research separate

from the topological problems of the late nineteenth and early twentieth century
which spawned the subject. Indeed, by the middle of this century a rich body of

knowledge had been developed and cast in the abstract setting of category theory.

Concurrent with the abstraction of these results from the topological setting was

a diversification in their application. Although sometimes ridiculed for the level of

abstraction, category theory thrived as new and exciting homological applications

appeared in group theory, lie algebras, and logic, among other areas as well. Still,

the geometric insight afforded by the original toplogical problems remains a powerful

influence. Each of the areas mentioned has been cross fertilized by interactions with

the other areas by the ability of category theory to give a precision to "analogous"
results in different areas of research.

In the following work operator theoretic results which we establish are given

homological meaning. The homological framework that has been developed serves
not only to provide what we believe to be the proper perspective from which to

view these operator theoretic results, but enriches the operator theory by suggest-

ing directions for further research. It is our opinion that when the results which

homological algebra seek are established, they will provide fruitful insight into op-
erator theory itself.

It would be unfair however to represent the development of the following

results as a strict application of homological algebra to operator theory. As each








area of mathematics develops, there arises an organization of the subject providing

some implicit valuation to certain results above others. The organization of the

material takes place both by the logical ordering of the work, and also subjectively

by the community of mathematicians working in the area. One result in operator

theory which has achieved some status in the latter regard is the commmutant lifting

theorem. Operator theorists have been very successful in employing the commutant

lifting theorem to solve many problems within their discipline. Hence the proof of a

generalization of the commutant lifting theorem in our setting provides an operator

theoretic rationale for this approach. Additionally, specialists working in closely

related areas, e.g. control theory, are taking more and more abstract approaches to

difficult problems in their disciplines.

In short, our opinion is that both operator theoretical and homological per-

spectives are necessary. The interplay between the two is rich and similar relation-

ships have proven to be very powerful in other areas of mathematics.

Our notation is standard for the most part. The field of complex numbers

is denoted by C. We use the math fraktur font Qt, 93, T,... to denote categories.

Gilbert spaces are always complex and separable and usually written in math script

X, or calligraphy R-. The set of bounded linear maps between Hilbert spaces X- and

'C is written L(I-, X) or (-) if 9 = C. Elements of (H-, X) are referred to as

operators; in particular, operators are bounded. Roman majuscules T, V, W,... will

typically be used to denote operators. An important exception to this convention is

the model operator Sk defined in Section 3.6. The definition of a Nevanlinna-Pick

reproducing kernel is given in Chapter 3, after which we reserve k to denote a (fixed)

Nevanlinna-Pick kernel and refer to k as an NP kernel. Given a Hilbert space H-

and an element h E 9, the function ph(T) = IJThJJ defines a seminorm on L(9C).

The topology induced upon L2(9-C) by the family of seminorms {Ph I h E -} is called

the strong operator topology.








We assume the reader is familiar with the standard results of functional

analysis, such as is covered in Conway [9]. Specifically such results as the Banach-

Steinhaus theorem, the principle of uniform boundedness, and various convergence

criteria in the strong operator topology are assumed. Perhaps less well known, but

of great importance in the sequel is the Parrott theorem.

The Parrott Theorem Let 7- and C) be Hilbert spaces with decompositions

Wo 3)-i, respectively )Co E IC1 and let Mx be the bounded transformation from R
into IC with operator matrix




with respect to the above decompositions. Then

infHjMxl = max{ I 0 A B } (2)

This result first appeared in a paper of S. Parrott [25], in which it is used to obtain a

generalization of the Nagy-Foias dilation theorem and interpolations theorems. As

such, the expert will not be surprised at the utility this theorem has afforded us.














TABLE OF CONTENTS




ACKNOWLEDGEMENTS ............................ iv

PR EFA CE . . . . v

ABSTRACT .. .. .. .. .. .. .. .. ... .. .. .. .. ix

CHAPTERS

1 INTRODUCTION ............................... 1

2 HOMOLOGICAL ALGEBRA ......................... 6
2.1 Foundations .. .. .. .. .. .. .. .. .. ... 6
2.2 Resolutions . . . 10
2.3 The Ext Functor ........................... 12

3 OPERATOR THEORY ............................ 14
3.1 Introductory Remarks ........................ 14
3.2 Classical Hardy Spaces ....................... 15
3.3 D ilations . . . 17
3.4 The Commutant Lifting Theorem ................. 19
3.5 Reproducing Kernels ......................... 20
3.6 Tensor Products and The Model Operator ................. 24
3.7 Fundamental Inequalities ............................ 25
3.8 Constructions ....... ............................ 29
3.9 Kernels ....... ................................ 32

4 HOMOLOGICAL MEANING ............................... 38
4.1 Introduction ....... ............................. 38
4.2 The Category b2 .................................. 38
4.3 Projective Modules ...... ......................... 39
4.4 The Commutant Lifting Theorem ..................... 40
4.5 The Existence of Resolutions ......................... 41

5 CONCLUSION ....... ................................. 43
5.1 Summary ...................................... 43
5.2 The Horizon ....... ............................. 44

REFERENCES ........ ................................... 46

BIOGRAPHICAL SKETCH .................................. 49














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



HOMOLOGICAL ALGEBRA OF HILBERT SPACES
ENDOWED WITH A COMPLETE NEVANLINNA-PICK KERNEL

By

Robert S. Clancy

August 1998

Chairman: Dr. Scott McCullough
Major Department: Mathematics

In this work, the representation of operators upon a Hilbert space endowed

with a Nevanlinna-Pick reproducing kernel. A generalization of the commutant

lifting theorem in this context. It is shown that in an appropriate category there

are projective objects. Furthermore it is demonstrated that objects in this category
have projective resolutions.














CHAPTER 1
INTRODUCTION

Our results concern Hilbert spaces endowed with a reproducing kernel k(z, ) =

Zn=0 anz such that ao = 1, an > 1, (z, I) 1 E=- n and b, > 0 for
n > 1. Following Agler [2] we call such a kernel k a Nevanlinna-Pick (NP) kernel.'

A classical example of a Hilbert space endowed with an NP kernel is the Hardy

space H2 with the Szeg6 kernel k(z, () = 1- In this case an = 1 for all n > 0,

while b, = 1 and b, = 0 for all n > 2. Classical results of complex analysis teach us

that the Szeg6 kernel is a reproducing kernel for 1112. The multiplication M, defined

via f(z) -+ z. f(z) defines an isometric operator on the space 12. It is also true that

the set of analytic polynomials is dense in 12. In fact {zn}o_0 form an orthonormal

basis for 12. Then relative to this basis, the effect of the multiplication operator M.

is to shift the Fourier coefficients "forward," hence the name shift operator. Now

given a function f E H2, we may define the map Mf 1112 -+1 2 via g i-4 fg. If Mf

is bounded, then we speak of the (multiplication) operator with symbol f. While

it is clear that the operators Mf commute with the shift operator M, Beurling [5]

showed that this essentially characterized the commutant of the shift operator M,.

Another classical Hilbert space of interest is the Dirichlet space. Here our

kernel is given by

~(1.1)
k~z Z + 1"

'Actually, we also require k to have a positive radius of convergence about (0, 0), and there
exists a constant C such that -' < C'. The rationale for these conditions is found in Section
3.6.








The Dirichlet space is the completion of the pre-Hilbert space consisting of analytic

polynomials endowed with the bilinear form
= '(s + ).(1.2)


Agler [2] has shown that the Dirichlet kernel defined in 1.1 is in fact an NP kernel

for the Dirichlet space. Again, multiplication by z is a bounded operator on the

Dirichlet space.

One is tempted then to establish mutatis mutandis much of the same body

operatory theory for the Dirichlet kernel that exists for the Szeg6 kernel. Indeed, we

believe this idea visible in much of what follows, although the following (counter)

example cautions us to choose the generalizations of the Szeg6 kernel judiciously.

Specifically, if we consider the Bergmann kernel where

k(z,() = E(n + 1)z' Q, (1.3)

then we see that k is not an NP kernel. In fact one can explicitly calculate that

b, = 2 while b2 = -1. In the sequel, we will see that the non-negativity of the b,

for n > 1 is crucial in establishing our results.

We arrive at our results by first fixing an NP kernel k(z, C). We define a

pre-Hilbert space on the space of analytic polynomials via the bilinear form

< t>= (1.4)
a,
We denote by H2(k) the completion of this pre-Hilbert space. Upon the Hilbert

space H2(k) we define the operator Sk(f) = zf to be multiplication by the polyno-

mial z. We see that Sk is bounded (see footnote 1). Given X, a complex separable

Hilbert space, we then define the operator cSk =I 0 Sk on the space { 0 H2(k).

Of particular interest to us are those Hilbert spaces )C and operators T E (]C) for

which there is an intertwining W : 0 H2(k) -+ )C of HSk and T. By intertwining

of operators, say T on '7- and V on IC, we mean a bounded linear map W: 7- -+ V

such that WT = VW. One of our fundamental results is








Theorem Let N, X- and M be complex separable Hilbert spaces and C c Z(N).

If there is a partial isometry P : X 0 H2(k) -+ N which intertwines the operators

j-cSk and C, then for every bounded intertwining f : 3M H2(k) -+ N there exists a

bounded intertwining F : M 0 H2(k) -+ X 0 H2(k) of the operators .Sk and RcSk

such that 11F1 11fl1 and PF = f.

Diagrammatically this is represented as

MOH2 (k)

F f (1.5)

2 P
X H H(k) N

This leads directly to Corollary 3.8.4, and Corollary 3.8.5 which we see as a gener-

alization of the commutant lifting theorem. We then establish

Theorem Let M/[ be a Hilbert space endowed with an operator C from L(Vf). Sup-

pose that there is a surjective partial isometry PAr : 1"12(k) -* M which intertwines

the operators C and WSk. Let C denote the kernel of the partial isometry PAr. Then

there is a partial isometry PKc : H2(k) OKC --+ IC which intertwines the operators KcSk

and Kj-kC.

The homological importance of this last result is that we will be able to show

that objects will have projective resolutions. In the classical case of the forward shift

on the Hilbert space H2, this result reduces to the observation that upon restricting

an isometry to an invariant subspace, the restricted operator is again an isometry.

Thus when our kernel is the Szeg6 kernel, we obtain a proof that every object will

have a two step projective resolution. In other words we obtain a proof that the

Ext' for n > 2 functors are trivial. Achieving the same level of knowledge when

the Szeg6 kernel is replaced by the Dirichlet kernel has proven to be a challenging

problem which at the present remains open.








In what follows we briefly describe the order of presentation. In Chapter 2

we introduce the necessary Homological Algebra we will require. The treatment is

very specific to our needs and we only establish that part of the theory which we

will later employ. In particular, in Section 2.1 we define a category, and products.

No discussion is made of more general limits. We then define an additive category,

in order to describe chain complexes and homotopy. A significant development in

this material is the treatment of exact sequences, which we briefly explain. We show

that one may decree a class of sequences to be exact. Once done, we can define a

projective object, and establish what is meant by an acyclic chain complex. We then

establish the solvability of two mapping problems which arise in Section 2.2. It is in

Section 2.2 where we establish that projective resolutions (relative to our definiton

of an exact sequence) are essentially unique. More precisely, projective resolutions

are unique up to a homotopy equivalence. This uniqueness then allows the definiton

of the Ext functor in Section 2.3.

Chapter 3 contains the operator theoretic results described above. Following
some introductory remarks which place the results in context, we define a reproduc-

ing kernel Hilbert space in Section 3.5, and then define an NP kernel. Given an NP

kernel k, we define the Hilbert space H2(k), which by its construction will be en-

dowed with the given NP kernel as a reproducing kernel. In Section 3.6 we define our

model operator u{Sk and show that it is a bounded operator on the Hilbert space

X- 0 H2(k), where Jf is a complex separable Hilbert space. Section 3.7 contains

technical results necessary for the proof of the theorems in Section 3.8.

In Sections 3.8 and 3.9 are found the statements and proofs of the results to

which we give homological meaning in Chapter 4. We use the Parrott Theorem in

the proof of Theorem 3.8.3, from which we are able to establish Corollary 3.8.4. We

show in Chapter 4 that Corollary 3.8.4 implies the objects of the form X- 0 H2(k)

are projective in the category b52 defined in Section 4.2. Theorem 3.8.3 also allows








us to establish Corollary 3.8.5, which is our generalization of the commutant lifting

theorem. In Section 3.9 we establish the existence of several limits of sequences of

operators in the strong operator topology. The operators so defined are then used

in the proof of Theorem 3.9.9.

In Chapter 4 we establish the category in which we work and then give the

homological meaning of some of our results. Notably, we show that Theorem 3.9.9

demonstrates that projective resolutions exist for every object in the category f)2 in

which we work. These results show that an Ext functor can then be defined. Chap-

ter 5 addresses specific questions that remain to be answered, as well as directions

for continuation of the program established herein.














CHAPTER 2
HOMOLOGICAL ALGEBRA

In the sequel, some of the results of Chapter 3 will be given homological

meaning. The homological algebra introduced here and used later, is standard with

one significant exception. In the category in which we work, we will declare a certain

class of sequences to be exact. It is relative to this notion of an exact sequence that

subsequent homological results will be stated. The development of this material is

provided for completeness. References for all of the material in this section are [20],

[6], [30], and [36].

2.1 Foundations

We begin with the

Definition 2.1.1. A category t consists of:

1. A class obcT of objects.

2. For each ordered pair of objects (M, N), there is a set, written hom(M, N).

The elements of hom(M, N) are called morphisms with domain M and

codomain N. Furthermore, if (M, N) 7 (0, P) then hom(M, N) is disjoint

from hom(O, P).

3. For each ordered triple of objects M, N, 0, there is a map, called composition,

from hom(M, N) x hom(N, 0) -+ hom(M, P), which is associative.

4. Lastly, for every object M, there is a morphism 1M E hom(M, M) satisfying

the following:








(a) For every object N and for every morphism g C hom(M, N), we have
1Mg = g.

(b) For every object N and for every morphism f E hom(N, M)' we have

flM= f.

We shall also require the

Definition 2.1.2. Let M1, M2 be objects from a category A product of M1 and

M2 is an object M from T, along with morphisms pl, and p2 from hom(M, M1)

and hom(M, M2), respectively, such that for every object N from t and morphisms

fi E hom(N, Mi), there is a morphism f C hom(N,M) such that pif = fi for

i 1,2.

Definition 2.1.3. By an additive category 91, we mean a category 91, such that

1. every finite set of objects has a product,

2. for every pair of objects (M, N), the set hom(M, N) is endowed with a binary

operation making hom(M, N) an abelian group, and

3. the composition in 3, Definition 2.1.1 above, is Z-bilinear.

Given an additive category !2, a sequence of objects M = (M,),Ez is said

to be graded or a graded object. A map p of degree r, between two graded objects

M and N, is a sequence of morphisms p = (pa) such that pn : M, --> Nn+r. By a

chain complex from 91, we mean a graded object C together with a map of degree

-1, d : C -4 C, such that d2 = 0. Here 0 stands for the identity element from each

group hom(C,, C,-l).

Definition 2.1.4. Given two chain complexes (C, d) and (C', d'), a map f : C -+ C'

of degree 0 is said to be a chain map if

d'f = fd. (2.1)








Given two chain maps f and g between (C, d) and (C', d'), we say a map h of degree
1 is a chain homotopy if

d'h + hd = f g. (2.2)

In this case we say f and g are homotopic.

Given a chain map f : C -+ C', we say f is a homotopy equivalence if there

is a chain map f' : C' -+ C such that ff' and f'f are homotopic to the identity

maps on C' and C, respectively. Let E be a class of sequences from the additive

category %1, each of the form

El 6 E E".

In particular each element of F determines a quintuple of three objects and two

morphisms. Declare the elements of F to be exact. A chain complex (C, d) such

that for every n E Z the sequence

d d
Cn+l O Cn Cn-1

is an element of F is said to be acyclic. If it is the case that the sequence {C}

is indexed by N or some finite index set, then by acyclic we understand that each

consecutive triple of objects and the connecting morphisms are an element of F. We

now make the

Definition 2.1.5. Let P be an object of %2. If for every sequence E' -+ E -+ E"

from F and morphism 4): P -E F, such that the following diagram commutes



4."0 (2.3)
k Ct
El E Ell

there exists a morphism 4': P -- E' which preserves the commutivity of the diagram,

then we say P is a projective object in %I relative to .








The following two results will establish the solution to mapping problems

which arise in the next section.

Lemma 2.1.6. Suppose that P is a projective object in 2 relative to F, a class of

exact sequences. Given the diagram
d
P d Q


f (2.4)

C/
El E ,E"

with efd = 0, and the bottom row exact, there exists a morphism g : P -+ E' which

makes the diagram commute.

Proof. Apply definition 2.1.5, taking 4 in equation (2.3) as fd. The result follows.



Lemma 2.1.7. Suppose that P is a projective object in %.t relative to E, a class of

exact sequences. Given the diagram, not necessarily commutative,
d
P d Q

f (2.5)
h
A" e' e
E l C E C E ll

where chd = ef, and the bottom row is exact, then there is a morphism k P -+ E'

such that

'k + hd = f (2.6)

Proof. Apply definition 2.1.5, taking 0 in equation (2.3) as f hd. The result then

follows. El








2.2 Resolutions

The material in Section 2.1 provides the necessary tools to establish a basic

result in homological algebra, the fundamental theorem of homological algebra. The

precise statement will be given below, but paraphrased, the theorem states that pro-

jective resolutions are unique. Just what unique means, as well as what a resolution

is, will now be addressed. First we have

Definition 2.2.1. An object M in a category T is said to be an initial object, if for

every object X G E we have hom(M, X) a singleton. On the other hand, if for every

object X E (- we have hom(X, M) a singleton, then we say M is a terminal object.

If M is both an initial and a terminal object, then we say M is a zero object. Zero
objects are usually written as simply 0.

In the most common categories, zero objects are readily available. For ex-
ample in the category of groups, the trivial group is a zero object. Likewise in the

category of complex vector spaces, the trivial vector space is a zero object. The

reason for the attention paid to zero objects is their appearance in

Definition 2.2.2. Let M be an object in an additive category 2 with a class F of

exact sequences. A resolution of M in 2 relative to P is an acyclic sequence
02 01 _____
E2 a2 1 E0 C M0 (2.7)

Arbitrary resolutions prove to be uninteresting. If we insist that there exist

j such that for each element Ei with i > j > 0 projective, then we will find that
such a resolution attains some measure of uniqueness. If Ei is projective for every

i > 0, then the resolution will be called a projective resolution. The fundamental

theorem of homological algebra follows.

Theorem 2.2.3. Let (C, d) and (C, d') be chain complexes in an additive category

2, endowed with a class 8 of exact sequences, and r be an integer. Suppose that








{ff : C -+ C}i C is projective for z > r and Ci -+ Ci--+ Ci_- is exact for i > r, then {fi} can
be extended to a chain map f : C -+ C'. Moreover this extension is unique in the
sense that any other extension f will be homotopic to f by a homotopy h such that
hi= 0 for i
Proof. We proceed by induction. Let n > r and assume that fi has been defined for
i < n in such a way so that d'fi = fi-ldi for i < n. Consider the following mapping
problem:
d d
C'+i C" Cn_
f +l! f. [f.-1 (2.8)

V di d'
C'141 Cn C,-
The existence of f,+l is given by Lemma 2.1.6.
The existence of chain map f extending {fi}i g is another such chain map extending {fi}i establish the homotopy between f and g. Let n > r and assume that hi has been
defined for i < n such that d'hi + hi-ld = fi gi. In the event that n = r take
hi = 0 for all i < r. Let t,, fn -g, for all n. Consider the following mapping
problem
d d
Cn+l "n hn-1



d" + d'
Cn+2 n+'

The existence of hn+l is given by Lemma 2.1.7.

Given two projective resolutions of an object M, we can apply Theorem
2.2.3 and conclude that the two resolutions are equivalent, in that there is a homo-
topy equivalence between the two projective resolutions. This then establishes the








uniqueness of a projective resolution. In the sequel we will consider constructions

based upon a particular projective resolution. It can be shown that the construc-

tions remain unchanged if the resolution is replaced by one to which it is homotopy

equivalent. Thus, the constructions depend only upon the existence of a projective

resolution and not the particular resolution used. This outline is justified by

Theorem 2.2.4. Given projective resolutions P and P' of an object M from an

additive category Z endowed with a class E of exact sequences, there is a chain map

f : P -- P', unique up to homotopy, which is a homotopy equivalence between P

and P'.

Proof. Consider the diagram
P2 PI P ,M 0


id


P2 P P0 M 0 (2.10)

id

P2 P1 PO M 0

Two successive applications of Theorem 2.2.3 establish the chain maps between P

and P', and P' and P necessary for a homotopy equivalence.

2.3 The Ext Functor

This section contains the constructions alluded to in Section 2.2. Given an

object M in an additive category Zf endowed with a class of exact sequences 8, and

a projective resolution P of M, we will define a functor from 921 into the category of








abelian groups. Let N be an object from %(. Consider the diagram
P2 PI P0 M ,0

(2.11)

....----------- N N N N -----------.0

which induces the sequence

hom(Po, N) hom(PI, N) -+ hom(P2,N) -+ (2.12)

Since 9 is an additive category, the horn sets are abelian groups, in particular the
sequence (2.12) is a sequence in the abelian category of Z modules. The point being

made here is that kernels, cokernels, and all finite limits in the category exist. So,

the cohomology of the sequence (2.12) is well defined. The cohomology groups of

the sequence (2.12) are called the Ext groups. The functor Ext'(M, -) from 1 to

the category of abelian groups assigns to each object N the nh homology group

of the sequence (2.12). The proof that this construction does not depend on the

particular choice of projective resolution can be found in Jacobson [20].














CHAPTER 3
OPERATOR THEORY

3.1 Introductory Remarks

The seminal paper of D. Sarason [28] excited the interest of the operator

theory community and at the same time provided the theoretical foundation for the

eventual development of H00-control theory by G. Zames [37], J.W.Helton [18], A.

Tannenbaum [35], C. Foias [15], and others in the 1980s. It was in fact work by

B. Sz. Nagy and C. Foias [34, 33], and R.G. Douglas, P.S. Muhly, and C. Pearcy

[10] in which the commutant lifting theorem was developed, providing geometrical

insight into Sarason's results. The use of the commutant lifting theorem to solve

interpolation problems relevant to control theory has been championed by C. Foias

in [16, 14]. The success of the commutant lifting approach to the many and diverse

problems to which it has been applied deserves emphasis. We will see that consid-

eration of the commutant lifting theorem invites homological questions; subsequent

chapters will address these questions.

Interpolation problems have a rich history of their own, independent of ap-

plications to control theory. R. Nevanlinna [23] and G. Pick [26], as well as C.

Carathe6dory, L. Fejer, and I. Schur, studied various problems of interpolating

data with analytic functions. D. Sarason [28] is credited with providing the op-
erator theoretic interpretation of these problems. In the sequel we will consider

the Nevanlinna-Pick interpolation problem in the classical setting and its solution.

Reproducing kernel Hilbert spaces will be defined, and the notion of a complete NP
kernel [21] will be introduced. The introduction of a complete NP kernel will then








allow the definition of the function spaces that will be of primary interest in all that

follows.

3.2 Classical Hardy Spaces

We introduce the following standard notation. Let T be the unit circle in

the complex plane and m denote (normalized) Lebesgue measure. For 1 < p _< 00

let LP(m) denote the classical function spaces on the unit circle. Let H12 denote

the Hardy space of analytic functions on the open unit disk D which have square

summable power series, and let HP0 denote the space of functions in H2 which

are bounded on D. Both I2 and H1C can be identified with subspaces of L2(m)

and LCC(m), respectively, and we utilize these identifications as is convenient. The

bilateral shift operator U is a unitary operator defined on L2 via Uf(z) = zf(z).

1H2 is invariant for U and the restriction of U to H2 is denoted by S. We refer

to S as the unilateral shift operator, or simply the shift operator, and note that

S is an isometry. These spaces have enjoyed the attention of a diverse audience,

including both operator theorists and specialists in control theory. There is a wealth

of material written about the Hardy spaces, and we refer the interested reader to

the excellent works by P.L. Duren [12] and K. Hoffman [19]. The fact that H2 is

invariant for U cannot be overstressed as can be seen in the following theorem first

demonstrated by A. Beurling [5].

Beurling-Lax-Helson Theorem If NH is a subspace of L2 invariant with respect

to the operator U, then there exist but two possibilities:

1. UN = N, in which case the is an m-measurable subset A C T, such that

N = XAL2, where XA is the characteristic function of A.

2. UN $ N, in which case there is a measurable function 0 on T with 101 = 1

(a.e.), such that N = OH2.








For a proof we refer the reader to N.K. Nikol'skiY [24]. Our immediate interest in

the Hardy Spaces stems from

Pick's Theoerem Given {X, X2,... ,X} C D, and {z, Z2..., z} C C, there

exists E HI0 with 11011o < 1 such that (xi) = zi for 1 < i < n if and only if the

matrix


*(3.1)

is positive.

Sarason demonstrates that this theorem can be obtained as a special case of

his Theorem 1 [28]:

Sarason's Theorem Let 0' be a nonconstant inner function, S as above, and
-c = i2 e obI-. If T is an operator that commutes with the projection of S onto C,

then there is a function 4) E HIP0 such that 1111 = JITlI and O(S) = T where O(S)

denotes the projection onto IC of the operator of multiplication on L2 by 4.

Indeed, let 0b be a finite Blaschke product with distinct zeros {xI, x2,.. x}.
Let AC be the space IHe OH 12. By the Beurling-Lax-Helson (BLH) theorem, we

have that AC is semi-invariant for the shift operator S. In fact we have an explicit

description of AC as the n-dimensional span of the functions gk(z) for

1 < k < n. Sarason points out that an operator T on AC commutes with the

compression of S to AC if and only if gk is an eigenvector of T* for 1 < k < n. If we
then define the operator T by T*gk = zkgk for 1 < k < n (where k is the complex

conjugate of zk), then Sarason's theorem guarantees a 4 E H0 with 14)f01 = IT1l,

and such that the compression of multiplication by 4 to AC is identical to the action

of T on )C. Since the functions gk are in fact the kernel functions for evaluation at

Xk, it is apparent then that O(xk) = zk for 1 < k < n. The requirement in Pick's

theorem that the interpolating function 4 have HP0 norm less than or equal to 1 is








then equivalent to the operator T being a contraction. This latter condition is just
the requirement that (3.1) is positive.

3.3 Dilations

Sarason's approach to interpolation focuses our attention on the space ; -

1H E 0IVI and the operator T defined by T*gk = zkgk for k = 1,... ,n. As we
saw in Section 3.2, the action of the adjoint T* upon the vectors {g} was fated

by the requirement that T commute with the compression of the shift S to 1C. In

this section we consider more closely the relation between an operator and its pro-

jection or compression onto a semi-invariant subspace. More precisely, we illustrate

circumstances under which an operator can be realized as just such a compression.

Of this point of view, N.K. Nikol'skiY [24, page 2] writes
The basic idea of the new non-classical spectral theory is to abstain
from looking at the linear operator as a sum of simple transformations
(of the type Jordan blocks) and instead consider it as "part" of a compli-
cated universal mapping, allotted (in compensation) with many auxiliary
structures.

The standard reference for the following material has become the work by B. Sz.-

Nagy and C. Foias [32].

Definition 3.3.1. Let RH and IC be two Hilbert spaces such that 7 C C. Given

two operators A : 7W -+ 7 and B : k -+ KC, we say that B is a dilation of A if the

following holds


AP = PB, (3.2)

where P is the orthogonal projection of B onto A.

Let B, B' be two dilations of A acting on K;, K;', respectively. If there is a unitary

operator : C -- such that


1. 0(h) = h Vh C 7, and








2. 0-1B'o = B,

then we say the two dilations B :C --+ C, and B' :K' -+ K' are isomorphic. One of

the best known results on dilations is

The Nagy Dilation Theorem Given C a contractive operator on a Hilbert space

W, there exists a Hilbert space '74 and an isometry U : '7 -+ such that U is a

dilation of C. Moreover this dilation U may be chosen to be minimal in the sense

that
CC
1= V u W. (3.3)
0
This minimal isometric dilation of C is then determined up to isomorphism.

In fact, P. Halmos [17] showed that a contraction can be dilated to a unitary oper-

ator. B. Sz.-Nagy [31] proved the following

The Nagy Dilation Theorem II Given C a contractive operator on a Hilbert

space H, there exists a Hilbert space 7-1 and a unitary operator U : 1 -+ 7- such that

U is a dilation of C. Moreover this dilation U may be chosen to be minimal in the

sense that
00V U7-H" (3.4)

-00
This minimal unitary dilation of C is then determined up to isomorphism.

Proofs of both of these theorems are found in B. Sz.-Nagy and C. Foias [32]. The

significance of the Nagy dilation theorem (NDT) for what follows is that

1. Contractions dilate to isometries, and

2. The restriction of an isometry to an invariant subspace is again an isometry.

It is precisely the homological perspective we assume which gives categorical signif-

icance to the above two points. This same perspective is assumed in the work of

R. Douglas and V. Paulsen [11], S. Ferguson [13], as well as J.F. Carlson and D.N.

Clark [8].








3.4 The Commutant Lifting Theorem

We begin by introducing the following notation. Let Ki, for i = 1, 2, be

Hilbert spaces, and let Ti : IC -+ ICi be operators on these Hilbert spaces. By an

intertwining of C1 and )C2, we mean a bounded linear map A :C1 -+ AC2 such that

AT1 = T2A. (3.5)

While we speak of an interwining of Hilbert spaces, equation (3.5) requires the map

A to interact with the operators Ti, for i = 1, 2 in a specific fashion. The operators

Ti for i = 1, 2, with which the intertwining A must interact via (3.5), will always

be clear from context. Note that stated in the above language, the NDT tells us

that given a contraction C on a Hilbert space R, there exist an isometry U acting

on a Hilbert space 'N, such that 'K C 'f, and the orthogonal projection P :' -* W'N

intertwines C and UXS.

Shortly after Sarason's work [28] appeared, B. Sz.-Nagy and C. Foias [34],

and then R.G. Douglas, P.S. Muhly, and C. Pearcy [10] offered what has come to

be known as

The Commutant Lifting Theorem Given contractive operators C1, C2 acting
on Hilbert spaces Hi, H2, resp., and a bounded intertwining A : H1 -+ H2, there

exists a bounded intertwining A of the minimal isometric dilations of C1, C2 such

that IlAJl < IIAfl.

Returning for the moment to the discussion of Pick's interpolations problem
and Sarason's solution, we see that the commutant lifting theorem (CLT) can be

used to provide a solution. Indeed, take C, = C2 as the projection of the shift S

onto the semi-invariant space I[ E)OH2. The matrix (3.1) is positive then if and only

if there is a contractive intertwining of the compression of the shift S with itself. In

the case where (3.1) is positive, the CLT provides a contractive intertwining of the








shift S with itself. The BLH theorem is then invoked to provide the existence of the

function E HlI in the statement of Pick's theorem [16].

3.5 Reproducing Kernels

In section 3.2 we saw that the space/C = H e owb1Fff, where 0b was a finite

Blashke product with distinct zeros x1,... ,x, had a basis gi,... ,gn where

gk(z)= '- for 1 < k < n. It was remarked then that these functions are precisely
the kernel functions for evaluation at Xk. The importance of this fact, in particular

to interpolation, will be brought to light in this section. For extensive coverage of

material related to this section, the reader may consult N. Aronszajn [3], J. Burbea

and P. Masani [7], S. Saitoh [27], J. Ball [4], J. Agler [2], and S. McCullough [22].

Let X be a set and 7, B be a Hilbert spaces. We denote the set of continuous

linear maps from RH into B by 7, B). In the case the domain and codomain

coincide we write simply (7-). We make the

Definition 3.5.1. A Hilbert space 7 of functions {ff f : X -+ B} is said to be a

reproducing kernel Hilbert space if the the following hold:

1. {fl/3C Band fo(x) =3 Vx E X} C "H,

2. The map/3 -+ fo is bounded,

3. There exists a map k : X x X --+ (B) satisfying the following:

(a) For each s G X, the map k(.,.s) : B --+ 7 via /3 '-+ k(.,s)3 is a bounded

map, and

(b) If f E 7, /3 eB, and s E X, then

< f,k(.,s)O3 >-< f(s),/3 >. (3.6)

If 7- is a reproducing kernel Hilbert space, the map k above will be referred

to as the reproducing kernel, or just kernel if clear from context.







Theorem 3.5.2. A Hilbert space 'h of functions {f f : X -4 13} satisfying proper-
ties (1) and (2) of Definition 3.5.1 is a reproducing kernel Hilbert space if and only
if for each s E X, the evaluation (/3, f) -< f(s), /3> is a bounded linear functional
on 3 e.

Proof. Let W- be a reproducing kernel Hilbert space. Then

I < f(s),/3 > I= < f,k(.,s)/3 > I < ]Hf1l Ilk(.,s)/3]] < lfe 1k(,s)l 11011, (3.7)

since k(., s) : 13 -+ W- is a bounded map, which demonstrates continuity at
0 e 0 E B D WH. Linearity then guarantees that the evaluation is bounded.
Conversely, suppose that for each s E X the evaluation (/3, f) i-+< f(s), /3 >
is a continuous map from B E W- into C. Fixing s and /3 we see that the Riesz
representation theorem then guarantees that there is an element k(., s)3 E 'H such
that

< f(s),/3 >=< f, k(-, s)/3 >. (3.8)

Continuity of the evaluation < f(s), /3 > implies that there is a constant C, such
that Ilk(.,s)/3II < C,l/3i1. Ilf 1. In particular for t G X we have
Ilk(., t)0112 < C, ilk(.,t)ol11- 110311 (3.9)

hence

Ilk(-,t)3 1 < CtI/I1. (3.10)

Define k : X x X -- BL, via k(s, )/3 = k(s, t)/3. We claim that k : X x X -+ C(B).
Given /31 and /32 from B, and w, and w2 from C, let -y w1/31 + w2/32; then we have

< f, k(-,s)wi/3i + k(.,s)w2/32 > =< f(s),w1/31 > + < f(s),w232 >
=< f(s),<- >
=< f,k(.,s)'y>. (3.11)








Since this is true for all f E 7-, we have k(.,s)wxj31 + k(.,s)w2fl2 = k(.,s)Y, and

hence in particular if we evaluate at t e X, then we see that k(t, s) is indeed linear.

Moreover, k(s,t) is bounded as

11k(s, t)0lhl2 = < k(s, t)f, k(s, t)f > I

I < k(., t)/, k(., s)k(s, t)O3> I

using (3.8). Continuity of the evaluation guarantees

I< k(-, t)O,k(-, -)k(s, t)O > Ile~sjjk(-, t) J ll.~ )O1

Ct, ljC8IIk(s,#t)0j. (3.12)

Hence

_
which shows that IIk(s,t)l < C/t, and thus k(s,t) E L(B).

Example 3.5.3. Let = H2, X = D, and k be the Szeg6 kernel, k(77,) 1-7'

Then H2 is a reproducing kernel Hilbert space when endowed with the standard
inner product.

In fact, J. Agler [2] and S. McCullough [22] have shown that the existence of

a reproducing kernel allows one to recover (operator valued) versions of Nevanlinna-

Pick interpolation problems. For the moment our interest lies in considering the the

map k alone. In our approach we assume our kernel has the form

k(z, ()=E a z n-7' (3.14)
Co
0
where ao = 1, and an > 0. We also assume k has a positive radius of convergence

about (0,0) and

aj < C2. (3.15)
aj+l








Since k(0, 0) = 1, near (0, 0) we have

= 1 Z (3.16)

and we note for future use that, for n > 1,
n
a,,= bsans. (3.17)


In this context we make the following

Definition 3.5.4. We say k is an NP kernel if b, > 0 for all n > 1.

Example 3.5.5.

1. Let a, = 1 for all n. Then k is the Szeg6 kernel described above. In this case

b = 1 and bn = 0 for all n > 2, hence the Szeg6 kernel is an NP kernel.

2. Let an = ',y. Then k is the Dirichlet kernel. While true [1, 29], it is nontrivial

to show that in this case k is an NP kernel.

3. Let a, n + 1. Then k is the Bergman kernel. In this case we see that

k(z, () = (1-c)2- One can then observe in this case that b, = 2, while b2 --1.

Hence the Bergman kernel is not an NP kernel.

Given an NP kernel k we define a bilinear form on the set of analytic poly-

nomials by

< 1Z >=fa, ifs =t; (3.18)
z0, ifs 3 t.

With deference then to example 3.5.5 we write H2(k) to denote the Hilbert space

obtained as the completion of the pre-Hilbert space structure induced by equation

(3.18). We will denote by H (k) those f C H2(k) which give rise to a bounded

multiplication operator M, : H2(k) H2(k) with symbol f. In the sequel we will

see that condition (3.15) implies that we can define an operator Sk on H2(k) via

f -4 zf.








3.6 Tensor Products and The Model ODerator


Let k be an NP kernel k(z, () = E' anZn- and C be as in equation (3.15).
For each 1 C N define si C H2(k) by s, = alz'.

Lemma 3.6.1. Relative to the inner product (3.18) with which H2(k) is endowed,
{si} is a dual basis to {zl}.

Proof. It is clear from inspection that {si} is an orthogonal set. Let M denote the
linear manifold spanned by {si}. Let h* e H2(k)* such that h*(si) = aih*(zl) = 0
for all I E N. Since H2(k) is defined as the completion of the pre-Hilbert space
induced by equation (3.18), the polynomials are dense in H2(k), hence h* = 0, and
therefore M = H2(k). Dl

We define the operator Sk : H2(k) -+ H2(k) via f '-+ zf and note that Sk is
bounded. Indeed, let f = J:__ cnzn e H2(k) and consider

IlSkf121= (Skf, Skf)
00 00
( CnZn+IYZCn Zn+I
n=0 n=0

n=O n

>j [c2 a, (3.19)
ano an+1

Equation (3.15) then implies that IlSkf 11 < ClIf I. If for 1 < 0 we intrepret sj = 0,
then


< z1-1, S*sl > =< Skz 1l s, >


=< ZIsl >= 1,


(3.20)


hence


(3.21)


Sk8l = 31.








Let M be a Hilbert space. We denote M H2(k) by M2(k). Note that each
element f E M2 (k) can be written as


f = Y Mn, & z', (3.22)

for mn E M, where the series converges in norm. Given operators T : M --+ M
and V : H2(k) -+ H2(k), we write T 8 V for the operator on M2(k) defined via

E m, 0 z' -+ E Tm, 0 Vz'. In particular we denote by MSk the operator I 0 Sk
on M2(k) via

MSkf >3 m" 0 z'+1 (3.23)

or if use is clear from context, we write Sk for MSk.

3.7 Fundamental Inequalities

In the exposition which follows we will often have the need to express the
matrix of a linear transformation relative to a given basis in block form. We associate
to a linear transformation a matrix relative to the closure of the linear manifold
spanned by an orthogonal, but not necessarily normal, set of vectors. The adjoint
of the transformation then has associated a matrix. This association is given by

Lemma 3.7.1. Let W, IC be Hilbert spaces, {vk} C 7"W, {Wk} C IC be dense sets of
mutually orthogonal vectors in -I and )C, respectively. Let T : 7- -+ C be a linear
transformation, and relative to the sets {Vk}, {Wk} we associate to T the matrix

(tij) = ( .). (3.24)


Lemma 3.7.2. With the same notation as in Lemma 3.7.1, the matrix of the trans-

pose map T* : AC -+ 7-I is given by


(i)= < vj,vj (3.25)
= i < vj j>








Proof. Calculating we have

(j) = ( < T*wv >)

(< wj, Tvi >'
= :)


iD
(< TVj,Vj > < Vin, Vj>
(ti (3.26)




Let 7-I and N2 be two Hilbert spaces, {Vik} C 1-i for i = 1, 2, and T :1 072 -+ iC.
Define T,: W1 -+ kZ via Tj(h1) = T(h 0 v2j). In this case matrix (3.24) of Lemma
3.7.1 will be written as

(... Tj1 Tj Tj+I ...) (3.27)

relative to the orthogonal decomposition

N1 0 N2 Oj (NH1 0 [V2j]) (3.28)

The matrix of the adjoint T* :A -IC 7t 0 N2 in accordance with Lemma 3.7.2
written as


(11.2112) (3.29)


Now given two complex sequences c, d : N -+ C we form the convolution
(c d), = Y cjdk. The set of all sequences then forms a semigroup with identity
j+k=n
e0 = 1, en = 0 for all n > 1. Two sequences c, d such that c d = e are said to be
an inverse pair.
For IC1 C )C2 two Hilbert spaces, let T E (AC2) and C E (1). Let N be a
Hilbert space and A: N -4 )C1, such that TCJA = Tj+IA for j > 0. Then we have
the








Lemma 3.7.3. With notation as above, let c and d be an inverse pair of sequences

such that dj > 0 for j > 0. If for all N, M C N we have
N
Z cj(CA)(CA)* < I (3.30)
j=O
and
M
SdkTkT*k < I (3.31)
k=1
then for all M E N,
M
I + E cndo(TnA)(TnA)* > 0. (3.32)
n=1
Proof. Computing we find
M
I >_ I: dT kT *k
k=1
M /M-k
> d"T k cJ(CjA)(CjA) ) *k
k=1l j=O
M M-k
5 5 dkcjTk+JAA*T*k+3, (3.33)
k=1 j=O
since TJ+lA TCjA. Reindexing we have
M n
I > E E dic-iT'AA**Tn (3.34)
n=1 1=1
which in view of the identity cdo E'-= cjdn-, for n > 1 yields

M
I + 5 cndo(Tn A)(T-A)* > 0. (3.35)
n=1



Let M be a Hilbert space, M2(k) = M & H2(k) be as in Section 3.6, K1,K2 as

above, f E (M2(k),AC1) such that

fj = cjC3fo. (3.36)







where fj is as in equation (3.27). Then Ilfl 1< 1 if and only if

1> ff. = Yafjf = ajci2cff C* (3.37)

Suppose that there is a non-negative sequence d : N -+ C such that d* ac = e, where
ac, = ajIcj12. Under these conditions we have

Corollary 3.7.4. If T E C(IC2) is a dilation of C E (1) such that T2pr1 = TC,
and
M
I > S dkT kT *k (3.38)
k=1
for all M > 1, then there is a map F E (M2(k),C2) such that PKF = f, and
JIFI < max {fiflH, ldol H1fI}.

Proof. Without loss of generality assume that If 1 < 1, hence equation (3.37) holds.
Consider then the map F : M2(k) -+ )C2 whose matrix relative to the decomposition
of K2 = (K72 e K(1) e K has the form
F=co(go ) j= 3(gi) = cjT fo j > 1. (3.39)
&=cAf F j _

for some go : M -+ C2 G Ki. Since T is a dilation of C we have PiT3 = CJPK1 for
all j > 0, hence PKF = f. Moreover JIFfl < 1 if and only if

I > FF* = E aFjF (3.40)
j=0
= aolc 2FOFJ + 5 ajIcji2TifofoTJ*. (3.41)
j=1
Since T2PK1 = TC, we have by induction

TJfo = TCJ-lfo. (3.42)

Recall that equation (3.37) holds, so we can apply Lemma 3.7.3 with A = fo to
conclude that

1 E ajcj12doTjfof0TJ*jj < 1 (3.43)
j=1








Thus the operator matrix

(cofo cif1 .) (3.44)

is bounded with norm If 11 and the operator matrix

(clTfo c2T2fo ...) (3.45)

has norm bounded above by ldo' 1. The Parrott theorem can then be applied to the
operator matrix

cogo Clg1 (3.46)
cofo cifl ..4

to conclude that there exist go such that

1FII_ max {1, Idol1}. (3.47)



3.8 Constructions

Let { M }iO be a sequence of Hilbert spaces such that M i+1 D M i. Denote
by Pi the orthogonal projection from Mi+1 onto Mi. Let M-1 = 0, and let M
4--
denote the Hilbert space EO=0(Mi G Mi-1) whose elements are the vectors
00
m = (mo, ml,...) with lm > Imjll2 -< c. (3.48)
0
We denote by Pi the projection from the Hilbert space M to the space Mi.

Lemma 3.8.1. Let 7 be a Hilbert space and fi :74 -+ Mi be a sequence of bounded
maps. If there is a constant C such that for all i 0,1,2,... we have 11fill < C
and pifi+l = fi, then there is a bounded map F : 7 -+ M such that IFH] < C and
P e-
P~=f

Proof. Define F via


h -+ {foh, (f fo)h,... (fi fi-1)h,... }.


(3.49)








Since f?+i is a dilation of fi, fi+l fi maps h into Mi+i e Mi. Inspection shows
that

Il{foh, (f, fo)h, ,(fi- fi-1)h, 0,...}II = jjfjhII < C]jhHI for all i > 1. (3.50)

hence IIFhl
Combining Corollary 3.7.4, and Lemma 3.8.1 we have

Theorem 3.8.2. Let {Mi}i'1l be a sequence of Hilbert Spaces such that Mj+j D
Mi, T E (M ), Ti+j is a dilation of Tj, and T2+lPMj = Tj+ITj. Let :X be a Hilbert
space, and f :(-2(k), Mo), such that

f, = c3Tofo. (3.51)

If there is a non-negative sequence d : N -+ C such that d ac = e, where acj = ajcj,
and equation (3.38) holds, then there is a map F E C(1H2(k),M) with J]Fl I ffH
and PMo F = f.

Let X and M be two Hilbert spaces, and AV be a Hilbert space endowed with
an operator C E L(AO.

Theorem 3.8.3. With the above notation, suppose that there is a partial isometry
P : -2(k) -+ IV which interwines the operators iSk and C. Then for every bounded

intertwining f : M2(k) -+ AF there exists a bounded intertwining F : M2(k) -*
'k2(k) such that JIFHI < f 1 and PF = f.

Proof. If P = 0 then set F = 0. Hence assume P # 0. Let f : M2(k) -+ N be an
intertwining of MSk and C. Without loss of generality assume that IIffl < 1. We
proceed recursively. Define o = P(NK). Since P is a partial isometry, P = PP*P
and thus P* when restricted to the image of P is an isometry. Hence we abuse
notation by referring to the subspace P*(Af%) c 1-2(k) as Ao. For i > 1 define ji to
be the least such integer such that X 0 [kji] is not a subspace of AYi_1. Then we set Ai








to be the closure of the span of the subspaces AfY-I and X [kj,]. Let Co = CiAr0 and
note that P intertwines XSk and C. Let i > 1, and for each A( define an operator

Ci = PAri HSk. Since xS*(J 0 [kq]) C X{ 0 [kq-1] we have Cj(A/j e N/-1) = 0.
Hence Cj+iPArg = Ci+ICi. One then verifies that K" -(k), and applies Theorem
3.8.2. F]

Let X- and M be as above, and N, N, N" be Hilbert spaces endowed with
operators C', C, C" from (N'), L(N), (N"), respectively. Assume further that
there exist partial isometrics 7r' : K' -+ K and 7r : Kr -+ K" which intertwine the
operators C', C, and C", and irir' = 0. Lastly suppose that there is a surjective
partial isometry Pg, : 7-12(k) --* A which intertwines the operators C' and XSk.

Under these conditions we have

Corollary 3.8.4. Iff E -(.M2(k),NK) interwines MSk and C and 7rf = 0, then

there is an intertwining F: M2(k) --+ K' such that J[F[[ 11f 1 and 7r'F f.

Proof. Apply Theorem 3.8.3 to the composition 7r'Pg, and the map

f E (M2(k),K).

Since PV, is surjective the composition 7r'Pg, is a partial isometry. Indeed, let
x G ker(r'P,)'. In particular x E ker(P,)', since

ker(P, ) C ker(7r'PAr,).

Therefore we have
< P'V',(x), P',(x) > =< x, x >. (3.52)

Moreover we have Pg,(ker(r'PAr,)) Cker(7r)'. Indeed, PV, is a surjective map, so
for y Eker(7T') we compute

< Pr,(x), Y > =< P,(X),PAO,() > Pr'(Y) =Y
=< x, > E ker(r'Pr,)


(3.53)


= 0.








Hence

< 7r'P,(x),7r'PI(x) > =< PA,(x),Px,(x) >

=< X, X >

establishing the claim.

Let X- and Vt be as above, and N', N be a Hilbert spaces endowed with operators

C', C from (N), (N), respectively. Lastly suppose that there are surjective

partial isometries PAr' : -2(k) -* K' and PAr : M2(k) --* K which intertwine the
operators C' and .Sk, and C and MSk, respectively. Under these conditions we have

Corollary 3.8.5. If there exist a map g : A -- K which intertwines the operators

C' and C, then there is an intertwining F: V(k) --+ M'2(k) such that JIFH1 < HgH

and F'Ig = g*.

Proof. Apply Theorem 3.8.3 to the composition gP, : WH2(k) -+ K". Since Pg
M2 (k) -- A is a surjective partial isometry, Theorem 3.8.3 insures the existence of
an intertwining F: W2(k) -+ M2(k), satisfying HFfl < 1gPv,1H < flgj. Using the

fact that PgF = gPg,, and taking adjoints then shows that F*Ig = g*. L

3.9 Kernels

Let XK be as above, and N be a Hilbert space endowed with an operator C
from C(N). Suppose that there is a surjective partial isometry Pg 7-2(k) -* K

which intertwines the operators C and 'KSk. Under these conditons we make the

Definition 3.9.1. Let Dc = I _- 1 bC'C*l. If in the event that N = 7/2(k),

then we write Dn for D Sk.

Our first observaton is


Lemma 3.9.2. DC is a positive contraction for all n E N.








Proof. Recall that 3.6.1 established that H2(k) has a basis {si} dual to the basis

{zl} with respect to the inner product (3.18). We begin with the case X = C, i.e.

V(k) = H2(k). Fix n and consider


teth bjSkSj) sq, Sr >=E bj < sq-j, sr-j > (3.54)
j=l j=l

If q r then the sum is 0. Otherwise, if q = r, then (3.54) is E'=, bjaq-j. Inter-

preting a, = 0 for I < 0 we have
n q
S bjaq- 5 bja_ = aq. (3.55)
j=l j=1

Hence we conclude
In
E 5* S k < 1. (3.56)
j=1

In the case that X C, it is clear that (3.56) holds with Sk replaced with Sk.

Since PV intertwines C and Sk, and PA is a partial isometry, we have

C = PArSkP r. Hence PArgSPkPA = CPAr = PgSk. Likewise PjPgvSk*P = ,*p;,

upon taking adjoints. Hence we have


CjCj = PArSk -P. (3.57)

Since bj > 0 it then follows that (3.56) holds with C in place of Sk. The result then

follows. El

Lemma 3.9.3. The limit

lim Dc = DC (3.58)
n-+oo

exists in the strong operator topology.

Proof. Since bj > 0, it is clear from inspection that I > DC > DC, > 0 for all

n E N. Hence for m < n we have II D C-DC II < 1. Together with several applications








of the Cauchy-Schwarz inequality this yields

IID'x- D'xlj4= 11 < (DC Dc)x, (DC Dc)x > 112
(DC DC)x,x >< (D DC)2x, (DC- D )x >

(DC DC)x,x > II(D D C)2xI II(DC D)xjl

(< Dx,x > < Dcx,x >)lxH12 (3.59)

Since I > DC > Dn+1 > O, Dx, x > is a bounded decreasing sequence of

numbers. The above calculation shows that Dcx is a Cauchy sequence. Define
Dcx = limno Dix. Then an application of the Banach Steinhaus theorem will

show that DC is a bounded positive operator. E

Lemma 3.9.2 establishes that for each n E N the operator DC has a positive square

root, which we denote by Be.

Corollary 3.9.4. The limit

lim BO BC (3.60)
n_+400

exists in the strong operator topology. Moreover

(BC)2 = DC. (3.61)

Proof. The proof of Lemma 3.9.3 applies mutatis mutandis to show that BC is a

bounded positive operator. The functional calculus for self adjoint operators then

guarantees (3.61). El

Definition 3.9.5. For n E N, define V : A -+ H2(k) 0 A! via
n
m si 0 B(_(C*)lm, (3.62)
i=0

Wn : Af-4 H2(k) 0 A via
n
m + s, 0 BC(C*)'m, (3.63)
1=0








and W, : -(k) -+ H2(k) 0 W2(k) via


h s, B5S*'h (3.64)
1=0
Theorem 3.9.6. With W,, as above, W = lin+, W,, exists in the strong operator

topology. Moreover, W is an isometry.

Proof. We need to show that for fixed m, Wm = lim,,_+ Wnm exists, and

jjWml = ink.l An application of the Banach Steinhaus theorem then gives con-

vergence in the strong operator topology. Towards this end we make the following

observations.

Lemma 3.9.7. W lin + W,, exists in the strong operator topology and W is an

isometry.

Proof. Indeed, for fixed h = S"-o h' 0 Sn, and n > m, we have
n
-(W. Wm)hll2 ( aiSkDsSk'h, h) (3.65)
1=m-1
n
22
= S 'h'2
l=m+l

Hence Wnh is a Cauchy sequence in H2(k) 07j2(k). Moreover limn, IWnhI 11hl,

as the following calculation shows


IWhH = aSkD Sk'h, h) (3.66)
1=0
n
Y alh'l2.
1=0
So in fact W is an isometry El


Lemma 3.9.8. For each n E N, V is an isometry.








Proof. Again, calculating


n

1=0

n i -


=(\ E anbj-nCi(C*)im, m)
i-0 n=O
( m).


(3.67)


n n-I
7*)lm n) =(Y albjCl3(C*)+r, m)
1=0 j=O


Hence V,, is an isometry.

Now, we have the following


n2 00
0 > IW2mll -I IVmll2 E al bjCl+JDC(C*)1+jm, m)
1=0 j=n+l-I
n 00

1=0 j=n+l-1
n 00

l=0 j-n+1-1
=IWnM 12 _-.I112,


which shows that lim_,0 IIW0 ll = f1112H, and hence { lWmll} is Cauchy. The form

of W, then implies IlWjm WiYm112 = I IIWjml2 IWimll2 I, and we see that Wnm
is a Cauchy sequence, thus establishing the theorem. LI

Theorem 3.9.9. Let M be a Hilbert space endowed with an operator C from L(M).

Suppose that there is a surjective partial isometry P : -2(k) -* M which inter-
twines the operators C and {Sk. Let IC denote the kernel of the partial isometry

Pg. Then there is a partial isometry PC : H2(k) 0 C -+ IC which intertwines the

operators K$k and H$kl-,c.

Proof. Theorem 3.9.6 shows that W maps IC into H2(k) 0 K isometrically. Take

W* : H2(k) 0 C -+ KC as the partial isometry. Let T =j Sk Jic. Then we have the


(3.68)





37


following


SWk= Z s, 0 DT (T*)l+lk (3.69)
1=0



WT*k.

Then upon taking adjoints we see that W* is in fact an intertwining, as was to be

shown. El














CHAPTER 4
HOMOLOGICAL MEANING

4.1 Introduction

In this chapter we show that the results of Chapter 3 establish the solution of

several mapping problems in homological algebra. In particular we show that in the

category in which we will work, projective objects exist. Moreover there are enough

projectives in this category, in the sense that every object can be realized as the

image of a projective. We go on to demonstrate that every object in the category

then has a projective resolution. The discussion in Chapter 2 guarantees then the

essential uniqueness of such a projective resolution. As a result it is then possible

to define an Ext functor from this category to the category of abelian groups. In

all that follows k denotes a fixed NP kernel, and H2(k) is defined as in Section 3.5.

4.2 The Category 5)2

In this section we define the category in which we will work. Recall from

Section 2.1 that in Definition 2.1.1 a category provides a class of objects. Objects

in the category 5)2 are pairs (M]f, T) where

1. T is a bounded operator on the separable Hilbert space M,

2. there exists a separable Hilbert space X- such that MYC is a subspace of

9{0 H2(k), and

3. the orthogonal projection PM : 0 H2(k) -+ X intertwines the operators T

and Sk.

Morphisms between objects (M, T) and (N, V) are bounded linear intertwinings.

With these definitions is is routine to verify that S)2 forms a category. It is convenient








to establish the following nomencalture. Given an object (M, T) from b2 and the

space X 0 H2(k) in item 2 above, we say that M is a *-submodule of X- 0 H2(k).

It is in fact easy to verify that given two objects (3V[, T) and (N, V) from

bj2 the object (M e N, T e V) is a product in the category. Indeed, there exist

objects (X{1 H2(k), S 1)) and (X{2 0 H2(k), S(2)) of which M\{, N are *-submodules,

respectively. One then checks that (M E N, T e V) is a *-submodule of

(1) J{ 2) 0 H2 (k). Since addition of bounded intertwings produces bounded inter-

twinings, we have established

Theorem 4.2.1. The category S52 is an additive category.

Let F be the class of all sequences

.... PM I M Y M ---------------- (4.1)


in which each object is an object from 2)2, each morphism is an intertwining partial

isometry, and for y' and p successive morphisms in the sequence we have

image(tz') = kernel(y).

Since we have not established the existence of (co)kernels in the category S52, this last

requirement on the morphisms p' and y is established in the category of (separable)

Hilbert spaces. We declare the elements of F to be the exact sequences in the

category S52.

4.3 Projective Modules

In this section we demonstrate that projective objects exist in the category
b2. We then show that every object in S52 is the image of a projective object. We

begin with the

Theorem 4.3.1. Let XC be a complex separable Hilbert space. The object

(X 0 H2(k), Sk) is projective in the category S52.








Proof. We have to solve the mapping problem described by (2.3):
Jf 0H 2(k)

0" (4.2)

El C E C Elf

Since E' is in the category, there is a Hilbert space M such that E' is a *-submodule
of M 02) H2(k). Note that 0 maps 9J 0 H2(k) into the image of the partial isometry

C'. Hence if we can solve the mapping problem

M H2(k).. X H2(k)

PE' 0 (43)


where PE, denotes the orthogonal (intertwining) projection onto E', then taking

the composition PE,4 = 4 will solve the mapping problem (4.2). By applying

Theorem 3.8.3 with e'PE, as P and 0 as f, we see that there exists a bounded

intertwining 4 which solves the problem (4.3). In fact, Theorem 3.8.3 guarantees

that 1 1 < I11 11. 1:.

As we stated in the beginning of this section, every object in Y)2 can be

realized as the image of a projective object. Indeed, by fiat, objects in the category
J2 are precisely those pairs (M, T) for which Mvf could be realized as a *-submodule

of some J- 0 H2(k). As we have just seen, XC 0 H2(k) is projective in S)2, hence

there are "enough" projectives in the category 2.

4.4 The Commutant Lifting Theorem

In this section we show that the Commutant Lifting Theorem appears in the
category J2 as the solution to a mapping problem. Specifically, let M and N be two

objects from Sj2, and *-submodules of 9-1 0 H2(k) and X2 0 H2(k), respectively. In

this situation we have








Theorem 4.4.1. For every morphism y : J -+ N there exists a morphism :
X1 0 H2(k) -+ 9/2 0 H2(k) making the following diagram commute

J9/1 H(k) M


Y5 (4.4)
9/2 0 H2 (k) P

Proof. The bottom row in the diagram (4.4) can be extended to end in 0. This
extended bottom row is then an element of F the class of exact sequences. Since

HJ1 0 H2(k) is a projective object in the category, there exists a morphism / solving
the diagram (4.2) with yPMt in place of 0 from (4.2). LI

4.5 The Existence of Resolutions

In this section we show that every object in the category S12 has a projective
resolution. The key point in this demonstration is establishing that the kernel of

the orthogonal projection from an object Ho 0 H2(k) onto a *-submodule M is still
in the category S52. Once the kernel is known to be in the category, we know that

there is an object J-C1 0 H2(k) for which the kernel is a *-submodule. An induction
then establishes the existence of the projective resolution of M. The requirement

that the kernel X of the orthogonal projection PM : Ho 0 H2(k) -+ M is again in

the category is given by Theorem 3.9.9. The beginning of the resolution, and the

base case for the induction is represented as
... ..X e HI(k) X/o o HI(k) P -M .0


Px (4.5)

X
Since X 0 H2(k) is again a projective object it serves as the second element in the

projective resolution. The next element in the projective resolution is derived by

repeating the above process.





42


With the establishment that every object in S'2 has a projective resolution
we can there define an Ext functor from the category J)2 to the category of abelian

groups as outlined in Section 2.3.














CHAPTER 5
CONCLUSION
5.1 Summary

We have shown that given a Nevanlinna-Pick kernel k we are able to construct

a Hilbert space H2(k) of functions for which k is a reproducing kernel. We have

shown that multiplication by the polynomial z is a bounded operator, denoted Sk, on

H2(k). Hence we are able to view H2(k) as a module over the algebra of multipliers

of H2(k)- i.e. the algebra consisting of those elements f G H2(k) such that the

map g -+ fg is a bounded map. From this point of view it is natural to investigate

the representations of this algebra of multipliers. Such an investigation is a study

of the module homomorphisms, or bounded intertwinings. The condition that an

intertwining be continuous can be seen as essential as H2(k) is endowed with a norm
structure.

Our approach focused upon a particular class of modules, which turned out

to be projective objects in our category. Namely, given a complex separable Hilbert

space M, the operator mSk = I 0 Sk : 0M H2(k) -+ M 0 H2(k) is bounded. We

showed that under the hypothesis of Theorem 3.8.3 we were able to establish the

existence of a bounded intertwining F which solved the following mapping problem
X D H2(k)

ff (5.1)
'PAr
M & H'(k) 3 A, 0

provided PAg is a partial isometry intertwining the operators MSk on M 0 H2(k),
and C on A/.








As a result we were able to establish Corollaries 3.8.4 and 3.8.5. In Chapter

4 we defined the category -b2 and demonstrated that Corollary 3.8.4 means that

the objects of the form (J-( H2(k),Sk) in the category 552 are projective. We

also showed in Chapter 4 that Corollary 3.8.5 is a generalization of the Commutant

Lifting Theorem.

Additionally we established Theorem 3.9.9 in Chapter 3. This result estab-

lished the existence of a bounded intertwining which in Chapter 4 we showed meant

that objects in the category 52 had projective resolutions. The essential unique-

ness of a projective resolution in an additive category relative to a class of exact

sequences was established in Chapter 2. Thus an Ext functor from the category -52

to the category of Abelian groups is well defined.

5.2 The Horizon

We now briefly outline some directions for further research. With the ex-

istence of an Ext functor established, one would like to be able to calculate Ext

groups. In particular, one would like to be able to demonstrate that there exist a

Nevanlinna-Pick kernel k and objects (7-, T), and (KC, C) such that Ext2(7-, 1C) 7 0.

It is worthwhile to point out that in the classical case when k is the Szeg6 kernel

that this will never be the case. Indeed, in this case, every object (R-, T) has a
projective resolution of the form

0 P1 P0 --t. (5.2)


Whether this is true or not for the Dirichlet kernel is not known at this time.

Similarly, given a kernel k one would like to be able to make a calculation

of the homological dimension for the ring of multipliers of H2(k). A less ambitious

goal than developing the tools necessary to answer this question for all k would be

simply to find conditions upon k which would imply the homological dimension was
greater than 2, 3 ....








If it can be established that higher Ext groups do exist, one might then delve

into the structure of the Ext groups themselves. These groups are in fact groups of

bounded linear intertwinings, and hence carry more structure then just that of an

Abelian group. In particular, they can be endowed with a topology, and therefore

carry at least the structure of a topological vector space (TVS).

Perhaps most ambitious of all might be the problem of realizing a particular

TVS as a particular Ext group in a given dimension. Of course all that can be said

of such a program now is that it lies on the horizon.

Other directions for research exist as well, in particular the further develop-

ment of the categorical foundations. Specifically, we believe it possible to show that

the category 5)2 is in fact abelian. Once established, several standard homological

results such as the Snake Lemma will follow. The significance of the category -92

being abelian is that homology groups may be defined directly from complexes in

-b2.

We believe these questions are both interesting and instructive. Answers

will lead to new insights, and we believe these insights will be both useful and

productive.














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BIOGRAPHICAL SKETCH


Robert Stephen Clancy was born in Madrid, Spain, on July 22, 1965. He was

adopted at birth by his father and mother, Robert and Wanda Clancy. He grew up

in Central Florida with his sister Christine five years his senior, where he graduated

from Palm Bay Senior High School in 1983. He attended the University of Florida

and received a Bachelor of Science degree in physics in 1989. He continued on at

the University of Florida in the Department of Mathematics earning a Masters of

Science degree in 1991. In October of 1993, Robert met his birth mother Mary

(Bromaghim) Bostwick, with whom he now has a lasting bond. In August of 1996,

he met Jennifer Lynne Airoldi, with whom he now endeavors to spend as much time

as possible.








I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.


Scott McCullough Clhairman
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Phi sophy.


Li-Chien Shen
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor f Philosophy.


Jorg Martinez
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully ade i s,
quality, as a dissertation for the degree of or-o ilo hy.





This dissertation was submitted to the Grad te Faculty of the Department
of Mathematics in the College of Liberal Arts and Sciences and to the Graduate
School and was accepted as partial fulfillment of the requirements for the degree of
Doctor of Philosophy.

August 1998
Dean, Graduate School




Full Text
HOMOLOGICAL ALGEBRA OF HILBERT SPACES
ENDOWED WITH A COMPLETE NEVANLINNA-PICK KERNEL
By
ROBERT S. CLANCY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

Robert S. Clancy

This work is dedicated to the memory of my father Robert James Clancy,
who provided me an unassailable example of what a man should strive to be.

ACKNOWLEDGEMENTS
I would like to thank my advisor for his patience and generosity. More than
a mentor, he has inspired me. I am in his debt. I would also like to thank the
members of my committee for their contribution to this work and to my education.
I would like to thank the staff of the Math Department, especially Sandy, who has
made my time at Florida much easier. I am especially grateful to my close friends
who have lent me their support throughout this project. Finally, this work could
not have been completed had it not been for my mother and Jennifer. I will never
be able to thank you both enough.
IV

PREFACE
Homological algebra has long been established as a field of research separate
from the topological problems of the late nineteenth and early twentieth century
which spawned the subject. Indeed, by the middle of this century a rich body of
knowledge had been developed and cast in the abstract setting of category theory.
Concurrent with the abstraction of these results from the topological setting was
a diversification in their application. Although sometimes ridiculed for the level of
abstraction, category theory thrived as new and exciting homological applications
appeared in group theory, lie algebras, and logic, among other areas as well. Still,
the geometric insight afforded by the original toplogical problems remains a powerful
influence. Each of the areas mentioned has been cross fertilized by interactions with
the other areas by the ability of category theory to give a precision to “analogous”
results in different areas of research.
In the following work operator theoretic results which we establish are given
homological meaning. The homological framework that has been developed serves
not only to provide what we believe to be the proper perspective from which to
view these operator theoretic results, but enriches the operator theory by suggest¬
ing directions for further research. It is our opinion that when the results which
homological algebra seek are established, they will provide fruitful insight into op¬
erator theory itself.
It would be unfair however to represent the development of the following
results as a strict application of homological algebra to operator theory. As each

area of mathematics develops, there arises an organization of the subject providing
some implicit valuation to certain results above others. The organization of the
material takes place both by the logical ordering of the work, and also subjectively
by the community of mathematicians working in the area. One result in operator
theory which has achieved some status in the latter regard is the commmutant lifting
theorem. Operator theorists have been very successful in employing the commutant
lifting theorem to solve many problems within their discipline. Hence the proof of a
generalization of the commutant lifting theorem in our setting provides an operator
theoretic rationale for this approach. Additionally, specialists working in closely
related areas, e.g. control theory, are taking more and more abstract approaches to
difficult problems in their disciplines.
In short, our opinion is that both operator theoretical and homological per¬
spectives are necessary. The interplay between the two is rich and similar relation¬
ships have proven to be very powerful in other areas of mathematics.
Our notation is standard for the most part. The held of complex numbers
is denoted by C. We use the math fraktur font 21, 23, (£,... to denote categories.
Hilbert spaces are always complex and separable and usually written in math script
J-C, or calligraphy % . The set of bounded linear maps between Hilbert spaces fff and
X is written £(fff, X) or £(!H) if % = X. Elements of £(3fi, X) are referred to as
operators; in particular, operators are bounded. Roman majuscules T, V, W,... will
typically be used to denote operators. An important exception to this convention is
the model operator S¡t defined in Section 3.6. The definition of a Nevanlinna-Pick
reproducing kernel is given in Chapter 3, after which we reserve k to denote a (fixed)
Nevanlinna-Pick kernel and refer to k as an NP kernel. Given a Hilbert space Tf
and an element h £ Tf, the function ph(T) = ||Tfi|| defines a seminorm on £(ff£).
The topology induced upon £(fff) by the family of seminorms {ph \ h € TC} is called
the strong operator topology.
vi

We assume the reader is familiar with the standard results of functional
analysis, such as is covered in Conway [9]. Specifically such results as the Banach-
Steinhaus theorem, the principle of uniform boundedness, and various convergence
criteria in the strong operator topology are assumed. Perhaps less well known, but
of great importance in the sequel is the Parrott theorem.
The Parrott Theorem . Let H and K, be Hilbert spaces with decompositions
'Ho ® Hi, respectively K,q ® ¥L\ and let Mx be the bounded transformation from H
into K- with operator matrix
Mx
X B
C A
(1)
with respect to the above decompositions. Then
This result first appeared in a paper of S. Parrott [25], in which it is used to obtain a
generalization of the Nagy-Foias dilation theorem and interpolations theorems. As
such, the expert will not be surprised at the utility this theorem has afforded us.
m/||Mx|| = max
x
0 0
C A
0 B
0 A
vii

TABLE OF CONTENTS
ACKNOWLEDGEMENTS iv
PREFACE v
ABSTRACT ix
CHAPTERS
1 INTRODUCTION 1
2 HOMOLOGICAL ALGEBRA 6
2.1 Foundations 6
2.2 Resolutions 10
2.3 The Ext Functor 12
3 OPERATOR THEORY 14
3.1 Introductory Remarks 14
3.2 Classical Hardy Spaces 15
3.3 Dilations 17
3.4 The Commutant Lifting Theorem 19
3.5 Reproducing Kernels 20
3.6 Tensor Products and The Model Operator 24
3.7 Fundamental Inequalities 25
3.8 Constructions 29
3.9 Kernels 32
4 HOMOLOGICAL MEANING 38
4.1 Introduction 38
4.2 The Category fj2 38
4.3 Projective Modules 39
4.4 The Commutant Lifting Theorem 40
4.5 The Existence of Resolutions 41
5 CONCLUSION 43
5.1 Summary 43
5.2 The Horizon 44
REFERENCES 46
BIOGRAPHICAL SKETCH 49
viii

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
HOMOLOGICAL ALGEBRA OF HILBERT SPACES
ENDOWED WITH A COMPLETE NEVANLINNA-PICK KERNEL
By
Robert S. Clancy
August 1998
Chairman: Dr. Scott McCullough
Major Department: Mathematics
In this work, the representation of operators upon a Hilbert space endowed
with a Nevanlinna-Pick reproducing kernel. A generalization of the commutant
lifting theorem in this context. It is shown that in an appropriate category there
are projective objects. Furthermore it is demonstrated that objects in this category
have projective resolutions.
IX

CHAPTER 1
INTRODUCTION
Our results concern Hilbert spaces endowed with a reproducing kernel k(z, Q =
Yjn=oanznln such that aQ = 1, an > 1, \{z, C) = 1 - Kzn(n, and bn > 0 for
n > 1. Following Agler [2] we call such a kernel k a Nevanlinna-Pick (NP) kernel.1
A classical example of a Hilbert space endowed with an NP kernel is the Hardy
space H2 with the Szego kernel k(z,() = JZTr- In this case an = 1 for all n > 0,
while 6i = 1 and bn = 0 for all n > 2. Classical results of complex analysis teach us
that the Szego kernel is a reproducing kernel for H2 â–  The multiplication Mz defined
via f(z) i—^ z- f(z) defines an isometric operator on the space H2. It is also true that
the set of analytic polynomials is dense in H2. In fact {zn}%L0 form an orthonormal
basis for H2. Then relative to this basis, the effect of the multiplication operator Mz
is to shift the Fourier coefficients “forward,” hence the name shift operator. Now
given a function / 6 M2, we may define the map M/H2 —$■ H2 via g K> fg. If Mf
is bounded, then we speak of the (multiplication) operator with symbol f. While
it is clear that the operators Mf commute with the shift operator Mz, Beurling [5]
showed that this essentially characterized the commutant of the shift operator Mz.
Another classical Hilbert space of interest is the Dirichlet space. Here our
kernel is given by
n + 1
(1.1)
1 Actually, we also require k to have a positive radius of convergence about (0,0), and there
exists a constant C such that < C2. The rationale for these conditions is found in Section
3.6.
a3 +1
1

2
The Dirichlet space is the completion of the pre-Hilbert space consisting of analytic
polynomials endowed with the bilinear form
< >=¿*(3 + 1). (1.2)
Agler [2] has shown that the Dirichlet kernel defined in 1.1 is in fact an NP kernel
for the Dirichlet space. Again, multiplication by z is a bounded operator on the
Dirichlet space.
One is tempted then to establish mutatis mutandis much of the same body
operatory theory for the Dirichlet kernel that exists for the Szego kernel. Indeed, we
believe this idea visible in much of what follows, although the following (counter)
example cautions us to choose the generalizations of the Szego kernel judiciously.
Specifically, if we consider the Bergmann kernel where
*(*.o = E(n+1)*"i”- (i-3)
then we see that k is not an NP kernel. In fact one can explicitly calculate that
b\ = 2 while b2 — — 1. In the sequel, we will see that the non-negativity of the bn
for n > 1 is crucial in establishing our results.
We arrive at our results by first fixing an NP kernel £). We define a
pre-Hilbert space on the space of analytic polynomials via the bilinear form
s t ^ st
< ZS,Zt > = .
as
We denote by H2(k) the completion of this pre-Hilbert space. Upon the Hilbert
space H2(k) we define the operator Sk{f) — zf to be multiplication by the polyno¬
mial z. We see that Sk is bounded (see footnote 1). Given CK, a complex separable
Hilbert space, we then define the operator ^Sk — I 0 Sk on the space CK 0 H2(k).
Of particular interest to us are those Hilbert spaces /C and operators T G C()C) for
which there is an intertwining W : CK0 H2(k) —±K of and T. By intertwining
of operators, say T on % and V on fC, we mean a bounded linear map W : Tí —» V
such that WT = VW. One of our fundamental results is
(1.4)

3
Theorem . Let IN", PC and M be complex separable Hilbert spaces and C € £(Ai).
If there is a partial isometry P : PC H2{k) —> D\T which intertwines the operators
and C, then for every bounded intertwining f : M ® H2{k) —)■ N there exists a
bounded intertwining F : M H2(k) —> PC such that ||F|| < ||/|| and PF = f.
Diagrammatically this is represented as
M®H2(k)
F/ f (1-5)
V p
%®H2(k) ► N
This leads directly to Corollary 3.8.4, and Corollary 3.8.5 which we see as a gener¬
alization of the commutant lifting theorem. We then establish
Theorem . Let M be a Hilbert space endowed with an operator C from £(M). Sup¬
pose that there is a surjective partial isometry Pjg : H?(k) -» A4 which intertwines
the operators C and wSk- Let 1C denote the kernel of the partial isometry Ptf. Then
there is a partial isometry Pk : H2(k)®)C —> 1C which intertwines the operators
and iKchcl/c-
The homological importance of this last result is that we will be able to show
that objects will have projective resolutions. In the classical case of the forward shift
on the Hilbert space M2, this result reduces to the observation that upon restricting
an isometry to an invariant subspace, the restricted operator is again an isometry.
Thus when our kernel is the Szego kernel, we obtain a proof that every object will
have a two step projective resolution. In other words we obtain a proof that the
Ext" for n > 2 functors are trivial. Achieving the same level of knowledge when
the Szego kernel is replaced by the Dirichlet kernel has proven to be a challenging
problem which at the present remains open.

4
In what follows we briefly describe the order of presentation. In Chapter 2
we introduce the necessary Homological Algebra we will require. The treatment is
very specific to our needs and we only establish that part of the theory which we
will later employ. In particular, in Section 2.1 we define a category, and products.
No discussion is made of more general limits. We then define an additive category,
in order to describe chain complexes and homotopy. A significant development in
this material is the treatment of exact sequences, which we briefly explain. We show
that one may decree a class of sequences to be exact. Once done, we can define a
projective object, and establish what is meant by an acyclic chain complex. We then
establish the solvability of two mapping problems which arise in Section 2.2. It is in
Section 2.2 where we establish that projective resolutions (relative to our definiton
of an exact sequence) are essentially unique. More precisely, projective resolutions
are unique up to a homotopy equivalence. This uniqueness then allows the definiton
of the Ext functor in Section 2.3.
Chapter 3 contains the operator theoretic results described above. Following
some introductory remarks which place the results in context, we define a reproduc¬
ing kernel Hilbert space in Section 3.5, and then define an NP kernel. Given an NP
kernel fc, we define the Hilbert space H2(k), which by its construction will be en¬
dowed with the given NP kernel as a reproducing kernel. In Section 3.6 we define our
model operator ^Sk and show that it is a bounded operator on the Hilbert space
!K technical results necessary for the proof of the theorems in Section 3.8.
In Sections 3.8 and 3.9 are found the statements and proofs of the results to
which we give homological meaning in Chapter 4. We use the Parrott Theorem in
the proof of Theorem 3.8.3, from which we are able to establish Corollary 3.8.4. We
show in Chapter 4 that Corollary 3.8.4 implies the objects of the form % ® H2(k)
are projective in the category fj2 defined in Section 4.2. Theorem 3.8.3 also allows

5
us to establish Corollary 3.8.5, which is our generalization of the commutant lifting
theorem. In Section 3.9 we establish the existence of several limits of sequences of
operators in the strong operator topology. The operators so defined are then used
in the proof of Theorem 3.9.9.
In Chapter 4 we establish the category in which we work and then give the
homological meaning of some of our results. Notably, we show that Theorem 3.9.9
demonstrates that projective resolutions exist for every object in the category Sj2 in
which we work. These results show that an Ext functor can then be defined. Chap¬
ter 5 addresses specific questions that remain to be answered, as well as directions
for continuation of the program established herein.

CHAPTER 2
HOMOLOGICAL ALGEBRA
In the sequel, some of the results of Chapter 3 will be given homological
meaning. The homological algebra introduced here and used later, is standard with
one significant exception. In the category in which we work, we will declare a certain
class of sequences to be exact. It is relative to this notion of an exact sequence that
subsequent homological results will be stated. The development of this material is
provided for completeness. References for all of the material in this section are [20],
[6], [30], and [36].
2.1 Foundations
We begin with the
Definition 2.1.1. A category £ consists of:
1. A class ob<¿ of objects.
2. For each ordered pair of objects (M,N), there is a set, written hom(M, N).
The elements of hom(M, N) are called morphisms with domain M and
codomain N. Furthermore, if (M, N) ^ (O, P) then hom(M, N) is disjoint
from hom(0, P).
3. For each ordered triple of objects M, N, O, there is a map, called composition,
from hom(M,N) x hom(N,0) —y hom(M, P), which is associative.
4. Lastly, for every object M, there is a morphism 1m £ hom{M1 M) satisfying
the following:
6

7
(a) For every object N and for every morphism g 1 Mg - g-
(b) For every object N and for every morphism / € hom(N, M)’ we have
/1M - /•
We shall also require the
Definition 2.1.2. Let Mi, M2 be objects from a category C. A product of Mi and
M2 is an object M from (£, along with morphisms pi, and P2 from hom(M, M\)
and hom(M, M2), respectively, such that for every object N from (£ and morphisms
fi £ hom(N, M¿ ), there is a morphism / G hom(N, M) such that pif — fi for
i = 1,2.
Definition 2.1.3. By an additive category 21, we mean a category 21, such that
1. every finite set of objects has a product,
2. for every pair of objects (M, N), the set /iom(M, N) is endowed with a binary
operation making hom(M, N) an abelian group, and
3. the composition in 3, Definition 2.1.1 above, is Z-bilinear.
Given an additive category 21, a sequence of objects M = (Mn)nez is said
to be graded or a graded object. A map p of degree r, between two graded objects
AF and A/ is a sequence of morphisms p (^p^'j such that pn * M,^ —v By a
chain complex from 21, we mean a graded object C together with a map of degree
— 1, d : C -+ C1 such that d2 = 0. Here 0 stands for the identity element from each
group hom(Cn,Cn-1).
Definition 2.1.4. Given two chain complexes (C, d) and (C", d'), a map / : C C
of degree 0 is said to be a chain map if
d'f = fd.
(2.1)

8
Given two chain maps / and g between (C, d) and (C", d'). we say a map h of degree
1 is a chain homotopy if
d'h + hd = f - g. (2.2)
In this case we say / and g are homotopic.
Given a chain map f : C C". we say / is a homotopy equivalence if there
is a chain map /' : C —> C such that ff and /'/ are homotopic to the identity
maps on C and C. respectively. Let £ be a class of sequences from the additive
category 21, each of the form
E' E —- E".
In particular each element of £ determines a quintuple of three objects and two
morphisms. Declare the elements of £ to be exact. A chain complex (C,d) such
that for every n E Z the sequence
c,
d
71+1
cn
is an element of £ is said to be acyclic. If it is the case that the sequence {Cn}
is indexed by N or some finite index set, then by acyclic we understand that each
consecutive triple of objects and the connecting morphisms are an element of £. We
now make the
Definition 2.1.5. Let P be an object of 21. If for every sequence E' E —»• E"
from £ and morphism (f> : P -> E, such that the following diagram commutes
there exists a morphism if; : P —> E' which preserves the commutivity of the diagram,
then we say P is a projective object in 21 relative to £.

9
The following two results will establish the solution to mapping problems
which arise in the next section.
Lemma 2.1.6. Suppose that P is a projective object in 21 relative to £, a class of
exact sequences. Given the diagram
9
f
E'
t
/
E E
(2.4)
with efd = 0, and the bottom row exact, there exists a morphism g : P -> E' which
makes the diagram commute.
Proof. Apply definition 2.1.5, faking in equation (2.3) as fd. The result follows.
â–¡
Lemma 2.1.7. Suppose that P is a projective object in 21 relative to £., a class of
exact sequences. Given the diagram, not necessarily commutative,
(2.5)
where ehd — ef, and the bottom row is exact, then there is a morphism k : P —> E'
such that
e'k + hd — f
(2.6)
Proof. Apply definition 2.1.5, taking in equation (2.3) as / — hd. The result then
follows. â–¡

10
2.2 Resolutions
The material in Section 2.1 provides the necessary tools to establish a basic
result in homological algebra, the fundamental theorem of homological algebra. The
precise statement will be given below, but paraphrased, the theorem states that pro¬
jective resolutions are unique. Just what unique means, as well as what a resolution
is, will now be addressed. First we have
Definition 2.2.1. An object M in a category C is said to be an initial object, if for
every object X £ € we have hom(M, X) a singleton. On the other hand, if for every
object X € € we have hom(X, M) a singleton, then we say M is a terminal object.
If M is both an initial and a terminal object, then we say M is a zero object. Zero
objects are usually written as simply 0.
In the most common categories, zero objects are readily available. For ex¬
ample in the category of groups, the trivial group is a zero object. Likewise in the
category of complex vector spaces, the trivial vector space is a zero object. The
reason for the attention paid to zero objects is their appearance in
Definition 2.2.2. Let M be an object in an additive category 21 with a class £ of
exact sequences. A resolution of M in 21 relative to £ is an acyclic sequence
E2 ——- Ei ——- E0 ^ M - * 0 (2.7)
Arbitrary resolutions prove to be uninteresting. If we insist that there exist
j such that for each element E,¿ with i > j > 0 projective, then we will find that
such a resolution attains some measure of uniqueness. If Ei is projective for every
i > 0, then the resolution will be called a projective resolution. The fundamental
theorem of homological algebra follows.
Theorem 2.2.3. Let (C, d) and (C,d') be chain complexes in an additive category
21, endowed with a class £ of exact sequences, and r be an integer. Suppose that

11
{fi ■■ Ci -+ C'} ¿ Ci is projective for i > r and C'i+1 —> C[ —> C'i_l is exact for i > r, then {/¿} can
be extended to a chain map f : C —y C. Moreover this extension is unique in the
sense that any other extension f will be homotopic to f by a homotopy h such that
hi = 0 for i < r.
Proof. We proceed by induction. Let n > r and assume that /¿ has been defined for
i < n in such a way so that d¿/¿ = for i < n. Consider the following mapping
problem:
a
n+1
Cn
d
Cn-r
fn+1
cUi
d!
fn
CL
d!
fn-1
(2.8)
C-i
The existence of fn+1 is given by Lemma 2.1.6.
The existence of chain map / extending {/¿}¿ g is another such chain map extending {/¿}¿ establish the homotopy between f and g. Let n > r and assume that hi has been
defined for i < n such that d'hi + = /¿ — gt. In the event that n — r take
hi = 0 for all i < r. Let tn — fn — gn for all n. Consider the following mapping
problem
The existence of hn+1 is given by Lemma 2.1.7.
(2.9)
â–¡
Given two projective resolutions of an object M, we can apply Theorem
2.2.3 and conclude that the two resolutions are equivalent, in that there is a homo¬
topy equivalence between the two projective resolutions. This then establishes the

12
uniqueness of a projective resolution. In the sequel we will consider constructions
based upon a particular projective resolution. It can be shown that the construc¬
tions remain unchanged if the resolution is replaced by one to which it is homotopy
equivalent. Thus, the constructions depend only upon the existence of a projective
resolution and not the particular resolution used. This outline is justified by
Theorem 2.2.4. Given projective resolutions P and P' of an object M from an
additive category 21 endowed with a class E of exact sequences, there is a chain map
f : P —>■ P', unique up to homotopy, which is a homotopy equivalence between P
and P'.
Proof. Consider the diagram
— ► F
2 -F
51 â–º F
>0— ►A
Í ►
id
_ v ^ r
v V . n/r
2 x
i
o °
id
â–º F
>2 â–º Pi /
>0 *
0 (2.10)
0
Two successive applications of Theorem 2.2.3 establish the chain maps between P
and Pand P' and P necessary for a homotopy equivalence. â–¡
2.3 The Ext Functor
This section contains the constructions alluded to in Section 2.2. Given an
object M in an additive category 21 endowed with a class of exact sequences £, and
a projective resolution P of M, we will define a functor from 21 into the category of

13
abelian groups. Let N be an object from 21. Consider the diagram
which induces the sequence
(2-11)
hom(P0, N) —» hom(P\, N) —V hom(P2, N) —> ■ ■ ■ (2.12)
Since 21 is an additive category, the horn sets are abelian groups, in particular the
sequence (2.12) is a sequence in the abelian category of Z modules. The point being
made here is that kernels, cokernels, and all finite limits in the category exist. So,
the cohomology of the sequence (2.12) is well defined. The cohomology groups of
the sequence (2.12) are called the Ext groups. The functor Extn(M: —) from 21 to
the category of abelian groups assigns to each object N the nth homology group
of the sequence (2.12). The proof that this construction does not depend on the
particular choice of projective resolution can be found in Jacobson [20].

CHAPTER 3
OPERATOR THEORY
3.1 Introductory Remarks
The seminal paper of D. Sarason [28] excited the interest of the operator
theory community and at the same time provided the theoretical foundation for the
eventual development of H°°-control theory by G. Zames [37], J.W.Helton [18], A.
Tannenbaum [35], C. Foias [15], and others in the 1980s. It was in fact work by
B. Sz. Nagy and C. Foias [34, 33], and R.G. Douglas, P.S. Muhly, and C. Pearcy
[10] in which the commutant lifting theorem was developed, providing geometrical
insight into Sarason’s results. The use of the commutant lifting theorem to solve
interpolation problems relevant to control theory has been championed by C. Foias
in [16, 14]. The success of the commutant lifting approach to the many and diverse
problems to which it has been applied deserves emphasis. We will see that consid¬
eration of the commutant lifting theorem invites homological questions; subsequent
chapters will address these questions.
Interpolation problems have a rich history of their own, independent of ap¬
plications to control theory. R. Nevanlinna [23] and G. Pick [26], as well as C.
Caratheódory, L. Fejer, and I. Schur, studied various problems of interpolating
data with analytic functions. D. Sarason [28] is credited with providing the op¬
erator theoretic interpretation of these problems. In the sequel we will consider
the Nevanlinna-Pick interpolation problem in the classical setting and its solution.
Reproducing kernel Hilbert spaces will be defined, and the notion of a complete NP
kernel [21] will be introduced. The introduction of a complete NP kernel will then
14

15
allow the definition of the function spaces that will be of primary interest in all that
follows.
3.2 Classical Hardy Spaces
We introduce the following standard notation. Let T be the unit circle in
the complex plane and m denote (normalized) Lebesgue measure. For 1 < p < oc
let Lp(m) denote the classical function spaces on the unit circle. Let H2 denote
the Hardy space of analytic functions on the open unit disk D which have square
summable power series, and let H°° denote the space of functions in H2 which
are bounded on D. Both H2 and H00 can be identified with subspaces of L2(m)
and L°°(m), respectively, and we utilize these identifications as is convenient. The
bilateral shift operator U is a unitary operator defined on L2 via Uf(z) — zf(z).
H2 is invariant for U and the restriction of U to H2 is denoted by S. We refer
to S as the unilateral shift operator, or simply the shift operator, and note that
S is an isometry. These spaces have enjoyed the attention of a diverse audience,
including both operator theorists and specialists in control theory. There is a wealth
of material written about the Hardy spaces, and we refer the interested reader to
the excellent works by P.L. Duren [12] and K. Hoffman [19]. The fact that H2 is
invariant for U cannot be overstressed as can be seen in the following theorem first
demonstrated by A. Beurling [5].
Beurling-Lax-Helson Theorem . If H is a subspace of L2 invariant with respect
to the operator U, then there exist but two possibilities:
1. UH = %, in which case the is an m—measurable subset A C T, such that
H — XaL2, where xa 25 the characteristic function of A.
2. U7i H, in which case there is a measurable function 9 on T with \0\ — 1
(a.e.), such that H = 0M2.

16
For a proof we refer the reader to N.K. Nikol’skii [24]. Our immediate interest in
the Hardy Spaces stems from
Pick’s Theoerem . Given {xi,x2,... ,a;n} C D. and {zi,z2,..- ,zn} C C, there
exists ||c>o < 1 such that (p(xi) — Zi for 1 < i < n if and only if the
matrix
(3.1)
is positive.
Sarason demonstrates that this theorem can be obtained as a special case of
his Theorem 1 [28]:
Sarason’s Theorem . Let ip be a nonconstant inner function, S as above, and
K, — H2 © ipH2. If T is an operator that commutes with the projection of S onto JC,
then there is a function

|| — ||Tj| and denotes the projection onto K of the operator of multiplication on L2 by Indeed, let ip be a finite Blaschke product with distinct zeros {aq, x2,... , xn}.
Let K be the space H2 0 ipM2. By the Beurling-Lax-Helson (BLH) theorem, we
have that K, is semi-invariant for the shift operator S. In fact we have an explicit
description of 1C as the n-dimensional span of the functions gk(z) — for
1 < k < n. Sarason points out that an operator T on 1C commutes with the
compression of S to 1C if and only if is an eigenvector of T* for 1 < k < n. If we
then define the operator T by T*gk — zj.gk for 1 < k < n (where zj. is the complex
conjugate of Zk), then Sarason’s theorem guarantees a £ H°° with 11||oo = ||T||,
and such that the compression of multiplication by

of T on K. Since the functions gk are in fact the kernel functions for evaluation at
Xk, it is apparent then that theorem that the interpolating function 6 have H00 norm less than or equal to 1 is
1 -ZjZ\
1 —XiX\

17
then equivalent to the operator T being a contraction. This latter condition is just
the requirement that (3.1) is positive.
3.3 Dilations
Sarason’s approach to interpolation focuses our attention on the space 1C —
H2 0 ^H2 and the operator T defined by T*gk = z*kgk for k = 1,... , n. As we
saw in Section 3.2, the action of the adjoint T* upon the vectors {g^} was fated
by the requirement that T commute with the compression of the shift S to 1C. In
this section we consider more closely the relation between an operator and its pro¬
jection or compression onto a semi-invariant subspace. More precisely, we illustrate
circumstances under which an operator can be realized as just such a compression.
Of this point of view, N.K. Nikol’skii [24, page 2] writes
The basic idea of the new non-classical spectral theory is to abstain
from looking at the linear operator as a sum of simple transformations
(of the type Jordan blocks) and instead consider it as “part” of a compli¬
cated universal mapping, allotted (in compensation) with many auxiliary
structures.
The standard reference for the following material has become the work by B. Sz.-
Nagy and C. Foias [32].
Definition 3.3.1. Let B and 1C be two Hilbert spaces such that B C 1C. Given
two operators A : Tí —> Ti and B : 1C —>■ 1C, we say that B is a dilation of A if the
following holds
AP = PB, (3.2)
where P is the orthogonal projection of B onto A.
Let B,B' be two dilations of A acting on 1C, 1C', respectively. If there is a unitary
operator : 1C -» 1C1 such that
1. 4>(h) — h V7i £ PL, and

18
2. <¡b~íB' then we say the two dilations B : —> 1C, and B' : K' —> 1C are isomorphic. One of
the best known results on dilations is
The Nagy Dilation Theorem . Given C a contractive operator on a Hilbert space
Tt, there exists a Hilbert space H and an isometry U : H —>■ H such that U is a
dilation of C. Moreover this dilation U may be chosen to be minimal in the sense
that
OO
H = \J UnH. (3.3)
o
This minimal isometric dilation of C is then determined up to isomorphism.
In fact, P. Halmos [17] showed that a contraction can be dilated to a unitary oper¬
ator. B. Sz.-Nagy [31] proved the following
The Nagy Dilation Theorem II . Given C a contractive operator on a Hilbert
space H, there exists a Hilbert space Tl and a unitary operator U : TL —V H such that
U is a dilation of C. Moreover this dilation U may be chosen to be minimal in the
sense that
CO
H = \J UnTi. (3.4)
— OO
This minimal unitary dilation of C is then determined up to isomorphism.
Proofs of both of these theorems are found in B. Sz.-Nagy and C. Foias [32]. The
significance of the Nagy dilation theorem (NDT) for what follows is that
1. Contractions dilate to isometries, and
2. The restriction of an isometry to an invariant subspace is again an isometry.
It is precisely the homological perspective we assume which gives categorical signif¬
icance to the above two points. This same perspective is assumed in the work of
R. Douglas and V. Paulsen [11], S. Ferguson [13], as well as J.F. Carlson and D.N.
Clark [8].

19
3.4 The Commutant Lifting Theorem
We begin by introducing the following notation. Let /C¿, for i = 1,2, be
Hilbert spaces, and let T¿ : 1C, —> be operators on these Hilbert spaces. By an
intertwining of Ki and lC2l we mean a bounded linear map A : /Ci —>■ )C2 such that
ATi = T2A. (3.5)
While we speak of an interwining of Hilbert spaces, equation (3.5) requires the map
A to interact with the operators T¿, for i — 1,2 in a specific fashion. The operators
Ti for i — 1,2, with which the intertwining A must interact via (3.5), will always
be clear from context. Note that stated in the above language, the NDT tells us
that given a contraction Con a Hilbert space H. there exist an isometry U acting
on a Hilbert space H, such that PL GPL, and the orthogonal projection P : PL -» PL
intertwines C and UXS.
Shortly after Sarason’s work [28] appeared, B. Sz.-Nagy and C. Foias [34],
and then R.G. Douglas, P.S. Muhly, and C. Pearcy [10] offered what has come to
be known as
The Commutant Lifting Theorem . Given contractive operators C\, C2 acting
on Hilbert spaces Hi, H2, resp., and a bounded intertwining A : Hi —>■ H2, there
exists a bounded intertwining A of the minimal isometric dilations of Ci, C2 such
that ||y4|[ < ||A||.
Returning for the moment to the discussion of Pick’s interpolations problem
and Sarason’s solution, we see that the commutant lifting theorem (CLT) can be
used to provide a solution. Indeed, take C\ = C2 as the projection of the shift S
onto the semi-invariant space H2 ©^H2. The matrix (3.1) is positive then if and only
if there is a contractive intertwining of the compression of the shift S with itself. In
the case where (3.1) is positive, the CLT provides a contractive intertwining of the

20
shift S with itself. The BLH theorem is then invoked to provide the existence of the
function £ H°° in the statement of Pick’s theorem [16].
3.5 Reproducing Kernels
In section 3.2 we saw that the space K, — H2 © ipH2, where ip was a finite
Blashke product with distinct zeros Xi,... ,xn had a basis Qi, â–  â–  â–  ,gn where
gk(z) = 1_]c«z for 1 < k < n. It was remarked then that these functions are precisely
the kernel functions for evaluation at Xk. The importance of this fact, in particular
to interpolation, will be brought to light in this section. For extensive coverage of
material related to this section, the reader may consult N. Aronszajn [3], J. Burbea
and P. Masani [7], S. Saitoh [27], J. Ball [4], J. Agler [2], and S. McCullough [22].
Let AT be a set and 3-1, B be a Hilbert spaces. We denote the set of continuous
linear maps from % into B by jC(7i,B). In the case the domain and codomain
coincide we write simply £{%). We make the
Definition 3.5.1. A Hilbert space 3~t of functions {/| / : X —» B} is said to be a
reproducing kernel Hilbert space if the the following hold:
1. {fp\/3 £ B and fp(x) = ¡3 Vx G X} C H,
2. The map ¡3 is bounded,
3. There exists a map k : X x X —> C{B) satisfying the following:
(a) For each s £ X, the map k(-,s) : B —> H via (3 k(-,s)/3 is a bounded
map, and
(b) If f £%, (3 £ B, and s £ X, then
< /, k(‘, s)(3 >=< /(s), (3 > . (3.6)
If 31 is a reproducing kernel Hilbert space, the map k above will be referred
to as the reproducing kernel, or just kernel if clear from context.

21
Theorem 3.5.2. A Hilbert space H of functions {/1 / : X —B} satisfying proper¬
ties (1) and (2) of Definition 3.5.1 is a reproducing kernel Hilbert space if and only
if for each s € X, the evaluation (/?,/) t-7< f(s),/3 > is a bounded linear functional
on B © PL.
Proof. Let PL be a reproducing kernel Hilbert space. Then
| < f(s),l3 > | = | < },k(;s)0 > | < ll/ll ||t('.<)/S|l < ll/ll IIM-.^II 11/311. (3.7)
since k(-,s) : B —>• PI is a bounded map, which demonstrates continuity at
0 © 0 E B ® PI. Linearity then guarantees that the evaluation is bounded.
Conversely, suppose that for each the evaluation (/?,/) HX f(s),/3 >
is a continuous map from B ® Pi into C. Fixing s and ¡3 we see that the Riesz
representation theorem then guarantees that there is an element k(-,s)/3 € Pi such
that
=. (3.8)
Continuity of the evaluation < f(s),/3 > implies that there is a constant Cs such
that ||fc(-,s)/3|| < CS\\P\\ ■ ll/ll. In particular for t £ X we have
Wki-^pf hence
\\k(-,t)P\\ < Ct\\P\\. (3.10)
Define k : X x X —> BB via k(s,t)P = k(s,t)P. We claim that k : X x X —» C(B).
Given Pi and p2 from B: and w\ and w2 from C, let 7 = W\P\ + w2p2, then we have
< /, k(-,s)w1p1 + k(-,s)w2p2 > =< f(s), W1P1 > + < f(s), w2p2 >
=< /(«), 7 >
=< f,k(-,s)7>. (3.11)

22
Since this is true for all f £ %, we have k(-,s)wi/3i + k(-,s)w2p2 = fc(-,s)7, and
hence in particular if we evaluate at t £ X, then we see that k(t, s) is indeed linear.
Moreover, k(s,t) is bounded as
\\k(s,t)/3\\2 = | < k(s,t)(3,k(s,t)/3 > |
= | < k(-,t)/3,k(-,s)k(s,t)P > |
using (3.8). Continuity of the evaluation guarantees
| < k(-,t)(3,k(-,s)k(s,t)(3 > \leCs\\k(-, t)/3\\ â–  \\k(s,t)/3\\
— Ct\\f3\\Cs\\k(s,t)f3\\. (3.12)
Hence
\\k(s,m\ which shows that ||fc(s,t)|| < CsCt, and thus k(s,t) £ C(B). □
Example 3.5.3. Let % = H2, X = D, and k be the Szego kernel, k(rj,() =
Then H2 is a reproducing kernel Hilbert space when endowed with the standard
inner product.
In fact, J. Agler [2] and S. McCullough [22] have shown that the existence of
a reproducing kernel allows one to recover (operator valued) versions of Nevanlinna-
Pick interpolation problems. For the moment our interest lies in considering the the
map k alone. In our approach we assume our kernel has the form
OO
k(z,0 = Y.a”z “C”. (3.14)
0
where ao = 1, and an > 0. We also assume k has a positive radius of convergence
about (0,0) and
aj+1
(3.15)

23
Since A;(0,0) — 1, near (0,0) we have
OO
-(z, C) = l-£>z"C”, (3.16)
n=1
and we note for future use that, for n > 1,
n
ctn — ^ ] bson—s. (3.17)
5—1
In this context we make the following
Definition 3.5.4. We say k is an NP kernel if bn > 0 for all n > 1.
Example 3.5.5.
1. Let an — 1 for all n. Then k is the Szego kernel described above. In this case
bi — 1 and bn — 0 for all n > 2, hence the Szego kernel is an NP kernel.
2. Let an — ^L_. Then k is the Dirichlet kernel. While true [1, 29], it is nontrivial
to show that in this case k is an NP kernel.
3. Let an = n + 1. Then k is the Bergman kernel. In this case we see that
k(z, £) = (1_N)2. One can then observe in this case that bx = 2, while b2 = — 1.
Hence the Bergman kernel is not an NP kernel.
Given an NP kernel k we define a bilinear form on the set of analytic poly¬
nomials by
< >=
, if s — t;
(3.18)
0, if s / t.
With deference then to example 3.5.5 we write H2(k) to denote the Hilbert space
obtained as the completion of the pre-Hilbert space structure induced by equation
(3.18). We will denote by H°°(k) those / £ H2{k) which give rise to a bounded
multiplication operator Mf : H2(k) -* H2(k) with symbol /. In the sequel we will
see that condition (3.15) implies that we can define an operator Sk on H2(k) via
f^zf.

24
3.6 Tensor Products and The Model Operator
Let k be an NP kernel k(z, £) = anznXl and C be as in equation (3.15).
For each l G N define s¡ G H2(k) by s¡ = a¡zl.
Lemma 3.6.1. Relative to the inner product (3.18) with which H2{k) is endowed,
{5/} is a dual basis to {V}.
Proof. It is clear from inspection that {5/} is an orthogonal set. Let M. denote the
linear manifold spanned by {s;}. Let h* G H2(k)* such that h*(s¡) = a¡h*(zl) — 0
for all l G N. Since H2{k) is defined as the completion of the pre-Hilbert space
induced by equation (3.18), the polynomials are dense in H2{k)1 hence h* = 0, and
therefore M. — H2(k). □
We define the operator Sk : H2(k) —>■ H2(k) via / i-> zf and note that Sk is
bounded. Indeed, let / = cnzn G H2(k) and consider
\\Skf\\2 = (Skf,Skf)
00 00
= CnZ
\
77.—0
OO I 19
r |Z
'~"n
71=0
— \ ^ 'n
n a”+l
71=0
= E
n—0
-n un
&n ^ra+1
(3.19)
Equation (3.15) then implies that || then
< zl \S*kst > =< Skzl \si >
—< z\si >= 1,
hence
S*kSl = 5,.
(3.20)
(3.21)

25
Let M be a Hilbert space. We denote M® H2(k) by M.2(k). Note that each
element / 6 M2(k) can be written as
/ = ^mn®zn, (3.22)
for mn 6 M, where the series converges in norm. Given operators T : M —>■ M
and V : H2[k) —> H2(k), we write T ® V for the operator on M.2(k) defined via
mn ® zn i->- 'YjTrrin ® V zn. In particular we denote by j^Sk the operator I ® Sk
on M.2(k) via
(3.23)
or if use is clear from context, we write Sk for M^k-
3.7 Fundamental Inequalities
In the exposition which follows we will often have the need to express the
matrix of a linear transformation relative to a given basis in block form. We associate
to a linear transformation a matrix relative to the closure of the linear manifold
spanned by an orthogonal, but not necessarily normal, set of vectors. The adjoint
of the transformation then has associated a matrix. This association is given by
Lemma 3.7.1. Let 7-L,IC be Hilbert spaces, {ufe} C Li, {w/,.} C 1C be dense sets of
mutually orthogonal vectors in Li and 1C, respectively. Let T : LL K. be a linear
transformation, and relative to the sets {«&}, {wk} we associate to T the matrix
(b'j) —
< Tvj,Wj >\
< W{,Wi > J
(3.24)
Lemma 3.7.2. With the same notation as in Lemma 3.7.1, the matrix of the trans¬
pose map T* : 1C Li is given by
â– JZ
< Vj,Vj >
< Vi, Vi >
(3.25)

26
Proof. Calculating we have
< T*Wj,Vj >\
)
f < Wj,Tvj >\
V < Vi, Vi > )
í < Tvi, Wj > < vj, Vj >
V < Vj,Vj > < Vi, Vi >
(in Vj,vi >N)
V < Vi,Vi> )
(3.26)
â–¡
Let PL\ and Pi2 be two Hilbert spaces, C PLi for ¿ = 1,2, and T : PLi ® PL2 —> £■
Define Tj : Pii —¥ 1C via Tj(h\) — T(hi ® V2j). In this case matrix (3.24) of Lemma
3.7.1 will be written as
(... Tj-t Tj Tj+i ...) (3.27)
relative to the orthogonal decomposition
PLl®'H2 = @3 {Hi 0 K']) • (3.28)
The matrix of the adjoint T* : K, —>■ PLi 0 PL2 in accordance with Lemma 3.7.2
written as
( ' \
ifSf • (3-29)
\ ; /
Now given two complex sequences c, d : N —> C we form the convolution
(c*d)n= Y, Cjdk. The set of all sequences then forms a semigroup with identity
j+k=n
eo = 1, en = 0 for all n > 1. Two sequences c,d such that c* d — e are said to be
an inverse pair.
For /Ci C /C2 two Hilbert spaces, let T 6 £(/C2) and C £ C(Ki). Let Pi be a
Hilbert space and A : Pi —y K\, such that TC^A = TJ+1A for j > 0. Then we have
the

27
Lemma 3.7.3. With notation as above, let c and d be an inverse pair of sequences
such that dj > 0 for j > 0. If for all N, M G N we have
N
Y,cj(C’A)(C’Ay i=o
and
M
Y,dkTkT*k k=l
then for all MgN,
M
I + "^2 cnd0{TnA){TnA)* > 0.
U— 1
Proof. Computing we find
M
I > £ dtTiT'k
k= 1
M /M-k
>^dkTk ( J2 Cj(CjA){C3Ay ) T
k= 1
M M-k
. j—O
= dkCjTk+jAA*T*k+j,
k= 1 j=0
since T3+1A = TC3A. Reindexing we have
M n
dlCn-,TnAA*T*
n=1 1=1
which in view of the identity cnd0 = Y^]=o cjdn-j for n > 1 yields
M
I + J2 Cnd0{TnA)(TnA)* > 0.
n—1
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
â–¡
Let M be a Hilbert space, A42(k) = M ® H2(k) be as in Section 3.6, K.\,K,2 as
above, / € C{M2(k),K\) such that
/; = cjCjf0.
(3.36)

28
where fj is as in equation (3.27). Then ||/|| < 1 if and only if
/>//• = = ^a¿ic¿!2cy„/0*c'*
(3.37)
Suppose that there is a non-negative sequence d : N —^ C such that d*ac = e, where
acj — cij jcj j2. Under these conditions we have
Corollary 3.7.4. IfT € £(/C2) ¿s a dilation of C E £(XT) such that T2Pjc1 = PC',
and
M
/ > ^ dkTkT*k
(3.38)
fc=i
/or all M > 1, then there is a map F E C(M2(k), K,2) such that P^F = f, and
imi< ^{11/11,1^111/11}.
Proof. Without loss of generality assume that ||/|| < 1, hence equation (3.37) holds.
Consider then the map F : M.2(k) —> 1C2 whose matrix relative to the decomposition
of 1C2 — (1C2 © /Ci) © Afi has the form
ft = CO ( 7 ] r, = Cj (®) = CjT'/o i > 1. (3.39)
go
Jo,
for some g0 : M —> )C2 0 K\. Since T is a dilation of C we have P^T3 = C3 P^ for
all j > 0, hence P>clF = /. Moreover ||F|| < 1 if and only if
‘F0FZ + J2ai M^/o/oV2*
3=0
— ao Co
(3.40)
(3.41)
j= 1
Since T2P^ = TC, we have by induction
T3f0 = TC3~J0.
(3.42)
Recall that equation (3.37) holds, so we can apply Lemma 3.7.3 with A = f0 to
conclude that
2_j aj\cj\doTJfofoTJ*\\ < 1
3=1
(3.43)

29
Thus the operator matrix
(c0/o Ci/i • • ■)
is bounded with norm \\f\\ and the operator matrix
(3.44)
(c,Tf„ c2T2f0 ■■■)
(3.45)
has norm bounded above by | operator matrix
Co go c-i,gi
co/o clfi
to conclude that there exist go such that
||F|| < max {1, |c?o 1|}-
(3.46)
(3.47)
â–¡
3.8 Constructions
Let {Mi}^ be a sequence of Hilbert spaces such that A4¿+i D Mi. Denote
by pi the orthogonal projection from Mi+i onto M{. Let M_i = 0, and let M
denote the Hilbert space ©jfi0(Aii © A4,_i) whose elements are the vectors
OO
m = (mo, mi,...) with ||m|| = ^ ||m,j|2 < oo. (3.48)
o
We denote by P¡ the projection from the Hilbert space M to the space M¿.
Lemma 3.8.1. Let Tí be a Hilbert space and fi : Tí Mi be a sequence of bounded
maps. If there is a constant C such that for all i = 0,1,2,... we have |¡/¿|| < C
and Pifi+i = fi, then there is a bounded map F : Tí —> M such that ||F|| < C and
PiF = fi
Proof. Define F via
h ^ {/oh, (/i - f0)h,... , (fi - fi-i)h,
(3.49)

30
Since fi+í is a dilation of /¿, /¿+i — fi maps h into M¿+1 © Mi. Inspection shows
that
I\{foh, (A - /o)fc,/¡.,)A, 0,... }¡| = ||/,A|| < C\\h\\ for all i > 1. (3.50)
hence \\Fh\\ < (7||fi||. F is therefore well defined and bounded with ||F|| < C. â–¡
Combining Corollary 3.7.4, and Lemma 3.8.1 we have
Theorem 3.8.2. Let {Mi^ft^ be a sequence of Hilbert Spaces such that M¡+1 D
Mi, Ti £ C(Mi), Ti+i is a dilation ofTi, and T2+1Pm, = P+iP. Let % be a Hilbert
space, and f : C(H2(k),Mo), such that
fj = CjTofo- (3.51)
If there is a non-negative sequence d : N —y C such that d* ac — e, where acj = ajCj,
and equation (3.38) holds, then there is a map F £ C(H2{k),M) with ||F|| < |[/||
and PMoF = f.
Let % and M be two Hilbert spaces, and AÍ be a Hilbert space endowed with
an operator C £ £(Af).
Theorem 3.8.3. With the above notation, suppose that there is a partial isometry
P : H2(k) —y M which interwines the operators intertwining f : M2(k) —»■ Af there exists a bounded intertwining F : M2(k) —>
TF^k) such that ||F|| < ||/|| and PF — f.
Proof. If P = 0 then set F — 0. Hence assume P / 0. Let / : M2(k) —> ÁÍ be an
intertwining of j^Sk and C. Without loss of generality assume that ||/|| < 1. We
proceed recursively. Define Af0 = P(Af). Since P is a partial isometry, P = PP*P
and thus P* when restricted to the image of P is an isometry. Hence we abuse
notation by referring to the subspace P*(W0) C TL2{k) as Af0. For i > 1 define jt to
be the least such integer such that [fcy] is not a subspace of . Then we set Mt

31
to be the closure of the span of the subspaces A/¿_i and Let Co = C\aí0 and
note that P intertwines •nSk and C. Let i > 1, and for each A/¿ define an operator
Ci = PAii-nSk- Since oiSlifK (2) [kq]) C PC Hence C2+1/V; = C¿+iC¿. One then verifies that A" = "H2(/e), and applies Theorem
3.8.2. â–¡
Let TC and M be as above, and AT', AT, Af" be Hilbert spaces endowed with
operators C', C, C" from £(A['), £(Af), £(>1"), respectively. Assume further that
there exist partial isometries 7r' : AÍ' —^ AT and 7r : AÍ —> Af" which intertwine the
operators C, C, and C", and 7r7r' = 0. Lastly suppose that there is a surjective
partial isometry Pap : H2(k) -» AP which intertwines the operators C and wSk-
Under these conditions we have
Corollary 3.8.4. If f G £(A42(fc), Af) interwines A/[Sk and C and nf — 0, then
there is an intertwining F : M2(k) -* AP such that ||F|| < ||/|[ and tt'F = f.
Proof. Apply Theorem 3.8.3 to the composition n'Pjj-i and the map
Since Pat> is surjective the composition it1 Pap is a partial isometry. Indeed, let
x G ker(7r'Pv/)x. particular x G ker(Py/)x, since
ker(PV') C ker(Tr'P^).
Therefore we have
< Pap(x), Paí'{x) >=< x,x > . (3.52)
Moreover we have Py/(ker(7r,Py')X) Cker(7r/)J-. Indeed, Pap is a surjective map, so
for y Gker(7r/) we compute
< PaT'{x), y>=< Paí'{x), PaT'Ív) > PArfy) = y
=< x, y > y G ker(7r'Py/)
= 0.
(3.53)

32
Hence
< Tr'P^f,(x)1Tr'PAfl(x) > =< Pjip(x), Pm-'(x) >
= < X, X >
establishing the claim. â–¡
Let J~C and M be as above, and Al7, AT be a Hilbert spaces endowed with operators
C', C from £(3\T'), £(Af), respectively. Lastly suppose that there are surjective
partial isometries Py/ : 1-L2(k) -» J\P and Py : A42(k) -» Af which intertwine the
operators C and o Corollary 3.8.5. If there exist a map g : AC —> AÍ which intertwines the operators
C and C, then there is an intertwining F : 7F2(k) A42(k) such that ||P|| < ||p||
and F*|y = g*.
Proof. Apply Theorem 3.8.3 to the composition pPy/ : P?{k) —> AP. Since Py :
A42(k) -» AÍ is a surjective partial isometry, Theorem 3.8.3 insures the existence of
an intertwining F : H2(k) —» A42(k), satisfying ||F|| < ||#Py/|| < ||g||. Using the
fact that PyP = 3.9 Kernels
Let “K be as above, and AT be a Hilbert space endowed with an operator C
from C(H). Suppose that there is a surjective partial isometry Py : FC2(k) -» Af
which intertwines the operators C and wSk- Under these conditons we make the
Definition 3.9.1. Let Df = I — b¡ClC*1. If in the event that AT = 'H2(fc),
then we write Dn for D%Sk.
Our first observaton is
Lemma 3.9.2. Df is a positive contraction for all n € N.

33
Proof. Recall that 3.6.1 established that H2{k) has a basis {sj dual to the basis
{z1} with respect to the inner product (3.18). We begin with the case CK = C, i.e.
n2(k) = H2(k). Fix n and consider
(n \ n
Y,hiSkSk I sq,sr >— y ^ bj < Sq—j, Sr—j > . (3.54)
i—1 / i—1
If q r then the sum is 0. Otherwise, if q = r, then (3.54) is J2j=1 bjQq-j- Inter¬
preting a¡ — 0 for l < 0 we have
n q
y ^ bj(tq-j < y ^ bjOq—j = aq. (3.55)
j-i i=i
Hence we conclude
n
(3-56)
i=i
In the case that TC C, it is clear that (3.56) holds with Sk replaced with Sk-
Since P_\f intertwines C and Sk, and Pjg is a partial isometry, we have
C — P\rS[;Pp. Hence PjgSkPffPj^r — 0Pm — PMSk. Likewise PfrPx'S^Pfr = S*;Pfr,
upon taking adjoints. Hence we have
CjC*j = PjtfSiSZPfr. (3.57)
Since bj > 0 it then follows that (3.56) holds with C in place of Sk- The result then
follows. â–¡
Lemma 3.9.3. The limit
lim Dcn = Dc (3.58)
71—>00
exists in the strong operator topology.
Proof. Since bj > 0, it is clear from inspection that / > D% > > 0 for all
n G N. Hence form < n we have ||Z)^ —Z)^|| < 1. Together with several applications

34
of the Cauchy-Schwarz inequality this yields
II D°mx - Dcnx ||4 = || < (Dcm - Dcn)x, (Dcm - Dcn)x > ||2
<<(DCm~ Dcn)x,x >< (Dcm - Dcn)\,(Dcm - D°n)x >
<<{Dcm- Dcn)x,x > || (Dcm - Dcnfx || || (Dcm - Dcn)x ||
< (< DmXiX > ~ < DnXiX >) Ikll2 (3-59)
Since / > D^ > D^+1 > 0, < D%x,x > is a bounded decreasing sequence of
numbers. The above calculation shows that D^x is a Cauchy sequence. Define
Dcx = linin—^oo D^x. Then an application of the Banach Steinhaus theorem will
show that Dc is a bounded positive operator. â–¡
Lemma 3.9.2 establishes that for each n £ N the operator D% has a positive square
root, which we denote by .
Corollary 3.9.4. The limit
lim B° = Bc (3.60)
n—too
exists in the strong operator topology. Moreover
(.B°f = Dc. (3.61)
Proof. The proof of Lemma 3.9.3 applies mutatis mutandis to show that Bc is a
bounded positive operator. The functional calculus for self adjoint operators then
guarantees (3.61). â–¡
Definition 3.9.5. For iíéN, define Vn : AT ^ H2{k) (g) Af via
n
m ^ Sl ® (3-62)
1=0
Wn : Af H2(k) N via
n
m i-> Si (g) Bc (C*)lm,
l=o
(3.63)

35
and Wn : Pt2(k) -> H2{k) (8) 7f2(A:) via
n
BskStlh (3.64)
l-o
Theorem 3.9.6. IWth Wn as above, W — linv^.00 Wn exists in the strong operator-
topology. Moreover, W is an isometry.
Proof. We need to show that for fixed m, Wm = Wnm exists, and
||Wm|| = ||m||. An application of the Banach Steinhaus theorem then gives con¬
vergence in the strong operator topology. Towards this end we make the following
observations.
Lemma 3.9.7. W = lim^oo W„ exists in the strong operator topology and W is an
isometry.
Proof. Indeed, for fixed h — Y^=o hn <8> •§„, and n > m, we have
n
||(W„ - Wm)/.||2 = { y a,SltDtSfh,h) (3.65)
l=m-\-1
= ¿ ai\h‘\2-
Hence Wnh is a Cauchy sequence in H2(k)®FL2{k). Moreover lim^oo ||Wn/i|| — ||fi||,
as the following calculation shows
n
\\V),h\\ = (Y,aiSlDÍSt 1=0
= Yjal\h‘\2-
1=0
So in fact W is an isometry â–¡
Lemma 3.9.8. For each n £ N, Vn is an isometry.

36
Proof. Again, calculating
n
HUmf (3.67)
1=0
n / n—l \ n n—l
= 1=0 \j=o / /=o j=o
n ¿
i—0 n=0
={m, m}.
Hence is an isometry. â–¡
Now, we have the following
n oo
0>||Wnm||2-||Km||2 = 6JC'+iZ)c(^/+im,m) (3.68)
/—0 j=n+l —l
n oo
= j5i+'Pv(S;)'+Jm,m)
/=0 j=n+l—/
><í>
i=0 j=n+l—£
= ||'Wnm
which shows that lim^oo || Wnm|| = ||m||, and hence {||kCnra||} is Cauchy. The form
of Wn then implies ||W}m — lTjm||2 = | ||VLjm||2 — |¡||2 |, and we see that Wnm
is a Cauchy sequence, thus establishing the theorem. â–¡
Theorem 3.9.9. Let M be a Hilbert space endowed with an operator C from £(M).
Suppose that there is a surjective partial isometry P^r : H?{k) -» M. which inter¬
twines the operators C and ocSk- Let K2 denote the kernel of the partial isometry
Pjy. Then there is a partial isometry P¡c : H2(k) ® 1C —> fC which intertwines the
operators K:Sk and ^Sk\)c-
Proof. Theorem 3.9.6 shows that W maps ¡C into H2(k) 1C isometrically. Take
W* : H2(k) ® /C —>■ K. as the partial isometry. Let T — — m
biSl+J(SZ)'+’m,m)

37
following
OO
xSiWk = si ® DT (T*)l+1k (3.69)
1=0
oo
= Y/*i®DT(T*)lT*k
1=0
= WT*k.
Then upon taking adjoints we see that W* is in fact an intertwining, as was to be
shown. â–¡

CHAPTER 4
HOMOLOGICAL MEANING
4.1 Introduction
In this chapter we show that the results of Chapter 3 establish the solution of
several mapping problems in homological algebra. In particular we show that in the
category in which we will work, projective objects exist. Moreover there are enough
projectives in this category, in the sense that every object can be realized as the
image of a projective. We go on to demonstrate that every object in the category
then has a projective resolution. The discussion in Chapter 2 guarantees then the
essential uniqueness of such a projective resolution. As a result it is then possible
to define an Ext functor from this category to the category of abelian groups. In
all that follows k denotes a fixed NP kernel, and H2(k) is defined as in Section 3.5.
4.2 The Category 352
In this section we define the category in which we will work. Recall from
Section 2.1 that in Definition 2.1.1 a category provides a class of objects. Objects
in the category are pairs (M, T) where
1. T is a bounded operator on the separable Hilbert space M,
2. there exists a separable Hilbert space % such that M is a subspace of
R (g> H2(k), and
3. the orthogonal projection Py[ : TC ® H2(k) —» M intertwines the operators T
and Sk-
Morphisms between objects (M, T) and (Af, V) are bounded linear intertwinings.
With these definitions is is routine to verify that Sj2 forms a category. It is convenient
38

39
to establish the following nomencalture. Given an object (M, T) from fj'2 and the
space ‘K ® H2{k) in item 2 above, we say that M is a *-submodule of % ® H2(k).
It is in fact easy to verify that given two objects (M, T) and (IN, V) from
f)2 the object (3VC ® IN, T © V) is a product in the category. Indeed, there exist
objects (Hi ® H2(k),Sl^) and (‘H2® H2(k),S^) of which M, IN' are *-submodules,
respectively. One then checks that (M ©IN', T © V) is a *-submodule of
(9fi ©IK2) ®H2(k). Since addition of bounded intertwings produces bounded inter-
twinings, we have established
Theorem 4.2.1. The category f)2 is an additive category.
Let £ be the class of all sequences
M' —► 'M —- M" ► • • • (4.1)
in which each object is an object from Sj2, each morphism is an intertwining partial
isometry, and for y! and y successive morphisms in the sequence we have
image(/i7) = kernel(^i).
Since we have not established the existence of (co)kernels in the category ft2, this last
requirement on the morphisms y' and y is established in the category of (separable)
Hilbert spaces. We declare the elements of £ to be the exact sequences in the
category fj2.
4.3 Projective Modules
In this section we demonstrate that projective objects exist in the category
5)2. We then show that every object in fj2 is the image of a projective object. We
begin with the
Theorem 4.3.1. Let TC be a complex separable Hilbert space. The object
(3i ® H2(k),Sk) is projective in the category fj2.

40
Proof. We have to solve the mapping problem described by (2.3):
PC ® H\k)
E'
(4.2)
E"
Since E' is in the category, there is a Hilbert space 3VI such that E' is a *-submodule
of M® H2{k). Note that (p maps PC ® H2{k) into the image of the partial isometry
e'. Hence if we can solve the mapping problem
ip
3VC (4.3)
E"
where Pe> denotes the orthogonal (intertwining) projection onto E\ then taking
the composition Pe'iP — ip will solve the mapping problem (4.2). By applying
Theorem 3.8.3 with PPe' as P and

intertwining ip which solves the problem (4.3). In fact, Theorem 3.8.3 guarantees
that II^H < \\ As we stated in the beginning of this section, every object in f)2 can be
realized as the image of a projective object. Indeed, by fiat, objects in the category
f)2 are precisely those pairs (M, T) for which M could be realized as a ^-submodule
of some PC ® H2(k). As we have just seen, PC ® H2{k) is projective in f)2, hence
there are “enough” projectives in the category ij2.
4.4 The Commutant Lifting Theorem
In this section we show that the Commutant Lifting Theorem appears in the
category f)2 as the solution to a mapping problem. Specifically, let M and AT be two
objects from f)2, and ^-submodules of PCi ® H2(k) and PC2 <8> H2{k), respectively. In
this situation we have

41
Theorem 4.4.1. For every morphism p : JVC —> X there exists a morphism p :
® H2(k) —> Ji2 H2(k) making the following diagram commute
‘Ki 0 H2(k)
2/i \ Pm
t1
T A
%2 H (k)
M
h
X
(4.4)
Proof. The bottom row in the diagram (4.4) can be extended to end in 0. This
extended bottom row is then an element of £ the class of exact sequences. Since
TCi (g) H2{k) is a projective object in the category, there exists a morphism fi solving
the diagram (4.2) with pPj& in place of f from (4.2). â–¡
4.5 The Existence of Resolutions
In this section we show that every object in the category S)2 has a projective
resolution. The key point in this demonstration is establishing that the kernel of
the orthogonal projection from an object PC0 ® H2(k) onto a *-submodule M is still
in the category 5)2. Once the kernel is known to be in the category, we know that
there is an object !Hi ® H2(k) for which the kernel is a *-submodule. An induction
then establishes the existence of the projective resolution of JVC. The requirement
that the kernel % of the orthogonal projection Pjvc : Xo <8> H2(k) —»■ M is again in
the category is given by Theorem 3.9.9. The beginning of the resolution, and the
base case for the induction is represented as
X 0 H2(k) —* Xo 0 H2(k) M
0
(4.5)
Since X 0 H2(k) is again a projective object it serves as the second element in the
projective resolution. The next element in the projective resolution is derived by
repeating the above process.

42
With the establishment that every object in f)2 has a projective resolution
we can there define an Ext functor from the category f)2 to the category of abelian
groups as outlined in Section 2.3.

CHAPTER 5
CONCLUSION
5.1 Summary
We have shown that given a Nevanlinna-Pick kernel k we are able to construct
a Hilbert space H2{k) of functions for which A: is a reproducing kernel. We have
shown that multiplication by the polynomial 2 is a bounded operator, denoted Sk, on
H2(k). Hence we are able to view H2(k) as a module over the algebra of multipliers
of H2{k)- i.e. the algebra consisting of those elements / £ H2(k) such that the
map g i-> fg is a bounded map. From this point of view it is natural to investigate
the representations of this algebra of multipliers. Such an investigation is a study
of the module homomorphisms, or bounded intertwinings. The condition that an
intertwining be continuous can be seen as essential as H2(k) is endowed with a norm
structure.
Our approach focused upon a particular class of modules, which turned out
to be projective objects in our category. Namely, given a complex separable Hilbert
space M, the operator M$k — I Sk : M H2(k) —»■ M (g) H2(k) is bounded. We
showed that under the hypothesis of Theorem 3.8.3 we were able to establish the
existence of a bounded intertwining F which solved the following mapping problem
%®H2{k)
(5.1)
provided Py is a partial isometry intertwining the operators j^Sk on M® H2(k),
and C on AÍ.
43

44
As a result we were able to establish Corollaries 3.8.4 and 3.8.5. In Chapter
4 we defined the category 9)2 and demonstrated that Corollary 3.8.4 means that
the objects of the form (% ® H2{k)^Sk) in the category Sj2 are projective. We
also showed in Chapter 4 that Corollary 3.8.5 is a generalization of the Commutant
Lifting Theorem.
Additionally we established Theorem 3.9.9 in Chapter 3. This result estab¬
lished the existence of a bounded intertwining which in Chapter 4 we showed meant
that objects in the category f)2 had projective resolutions. The essential unique¬
ness of a projective resolution in an additive category relative to a class of exact
sequences was established in Chapter 2. Thus an Ext functor from the category f)2
to the category of Abelian groups is well defined.
5.2 The Horizon
We now briefly outline some directions for further research. With the ex¬
istence of an Ext functor established, one would like to be able to calculate Ext
groups. In particular, one would like to be able to demonstrate that there exist a
Nevanlinna-Pick kernel k and objects (%, T), and (/C, C) such that Ext2(7L, 1C) ^ 0.
It is worthwhile to point out that in the classical case when k is the Szego kernel
that this will never be the case. Indeed, in this case, every object () has a
projective resolution of the form
0 Pi P0 U. (5.2)
Whether this is true or not for the Dirichlet kernel is not known at this time.
Similarly, given a kernel k one would like to be able to make a calculation
of the homological dimension for the ring of multipliers of H2(k). A less ambitious
goal than developing the tools necessary to answer this question for all k would be
simply to find conditions upon k which would imply the homological dimension was
greater than 2,3,. ...

45
If it can be established that higher Ext groups do exist, one might then delve
into the structure of the Ext groups themselves. These groups are in fact groups of
bounded linear intertwinings, and hence carry more structure then just that of an
Abelian group. In particular, they can be endowed with a topology, and therefore
carry at least the structure of a topological vector space (TVS).
Perhaps most ambitious of all might be the problem of realizing a particular
TVS as a particular Ext group in a given dimension. Of course all that can be said
of such a program now is that it lies on the horizon.
Other directions for research exist as well, in particular the further develop¬
ment of the categorical foundations. Specifically, we believe it possible to show that
the category T)2 is in fact abelian. Once established, several standard homological
results such as the Snake Lemma will follow. The significance of the category i}2
being abelian is that homology groups may be defined directly from complexes in
£2.
We believe these questions are both interesting and instructive. Answers
will lead to new insights, and we believe these insights will be both useful and
productive.

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BIOGRAPHICAL SKETCH
Robert Stephen Clancy was born in Madrid, Spain, on July 22, 1965. He was
adopted at birth by his father and mother, Robert and Wanda Clancy. He grew up
in Central Florida with his sister Christine five years his senior, where he graduated
from Palm Bay Senior High School in 1983. He attended the University of Florida
and received a Bachelor of Science degree in physics in 1989. He continued on at
the University of Florida in the Department of Mathematics earning a Masters of
Science degree in 1991. In October of 1993, Robert met his birth mother Mary
(Bromaghim) Bostwick, with whom he now has a lasting bond. In August of 1996,
he met Jennifer Lynne Airoldi, with whom he now endeavors to spend as much time
as possible.
49

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
¡M cC ¿Lu ¿1
Scott McCullough , Chairman
Associate Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of PhUqsophy.
Li-Chien Shen
Associate Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor pi Philosophy.
/MfitmiZsi
Jorgé Martinez
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in spG
quality, as a dissertation for the degree of Doqtoirof~P;hilq§op'hyv
This dissertation was submitted to the Graduáte Faculty of the Department
of Mathematics in the College of Liberal Arts and Sciences and to the Graduate
School and was accepted as partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
August 1998
Dean, Graduate School



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