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## Material Information- Title:
- Homological algebra of Hilbert spaces endowed with a complete Nevanlinna-Pick kernel
- Creator:
- Clancy, Robert S., 1965-
- Publication Date:
- 1998
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- English
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- ix, 49 leaves : ill. ; 29 cm.
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Algebra ( jstor ) Functors ( jstor ) Hilbert spaces ( jstor ) Interpolation ( jstor ) Linear transformations ( jstor ) Mathematical theorems ( jstor ) Mathematics ( jstor ) Morphisms ( jstor ) Topological theorems ( jstor ) Dissertations, Academic -- Mathematics -- UF ( lcsh ) Mathematics thesis, Ph.D ( lcsh ) - Genre:
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## Notes- Thesis:
- Thesis (Ph.D.)--University of Florida, 1998.
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- Includes bibliographical references (leaves 46-48).
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- Typescript.
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- Vita.
- Statement of Responsibility:
- by Robert S. Clancy.
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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 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 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 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) = ( 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 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. REFERENCES [1] AGLER, J. Interpolation. Journal of Functional Analysis. To appear. [2] AGLER, J. The Arveson extension theorem and coanalytic models. Integral Equations Operator Theory 5 (1982), 608-631. [3] ARONSZAJN, N. Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 (1950), 337-404. [4] BALL, J. A. Rota's theorem for general functional Hilbert spaces. Proc. Amer. Math. Soc. 64, 1 (1977), 55-61. [5] BEURLING, A. On two problems concerning linear transformations in Hilbert space. Acta Math. 81 (1948), 17. 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(Translated from the French and revised). 48 [33] Sz.-NAGY, B., AND FoIAS, C. Commutants de certains operat~urs. Acta. Sci. Math. 29 (1968), 1-17. [34] Sz.-NAGY, B., AND FOIAS, C. Dilation des commutants d'operateurs. C.R. Acad. Sci. Paris, Serie A 266 (1968), 493-495. [35] TANNENBAUM, A. Feedback stabilization of linear dynamical plants with un- certanty in the gain factor,. Int. J. Control 32 (1980), 1-16. [36] VICK, J. Homology theory: An introduction to algebraic topology. Springer- Verlag, New York, 1994. [37] ZAMES, G. Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans. Automat. Control AC-26 (1981), 301-320. 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 |

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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 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 â€œanalogousâ€ 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 suggestÂ¬ 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 opÂ¬ 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 perÂ¬ spectives are necessary. The interplay between the two is rich and similar relationÂ¬ 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, (Â£,... 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 Â£(fff, X) or Â£(!H) if % = X. Elements of Â£(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 SÂ¡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 Â£ Tf, the function ph(T) = ||Tfi|| defines a seminorm on Â£(ffÂ£). The topology induced upon Â£(fff) by the family of seminorms {ph \ h â‚¬ 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 Â® Hi, respectively K,q Â® Â¥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) iâ€”^ 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 â€œforward,â€ hence the name shift operator. Now given a function / 6 M2, we may define the map M/H2 â€”$â– 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 < >=Â¿*(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)*"iâ€- (i-3) then we see that k is not an NP kernel. In fact one can explicitly calculate that b\ = 2 while b2 â€” â€” 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 Â£). 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) â€” zf to be multiplication by the polynoÂ¬ mial z. We see that Sk is bounded (see footnote 1). Given CK, a complex separable Hilbert space, we then define the operator ^Sk â€” 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) â€”Â±K of and T. By intertwining of operators, say T on % and V on fC, we mean a bounded linear map W : TÃ â€”Â» 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 â‚¬ Â£(Ai). If there is a partial isometry P : PC and C, then for every bounded intertwining f : M Â® H2{k) â€”)â– N there exists a bounded intertwining F : M Diagrammatically this is represented as MÂ®H2(k) F/ f (1-5) V p %Â®H2(k) â–º N This leads directly to Corollary 3.8.4, and Corollary 3.8.5 which we see as a generÂ¬ alization of the commutant lifting theorem. We then establish Theorem . Let M be a Hilbert space endowed with an operator C from Â£(M). SupÂ¬ pose that there is a surjective partial isometry Pjg : H?(k) -Â» 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)Â®)C â€”> 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 reproducÂ¬ 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 enÂ¬ 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 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 % Â® 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. ChapÂ¬ 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 Â£ consists of: 1. A class ob<Â¿ 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) â€”y hom(M, P), which is associative. 4. Lastly, for every object M, there is a morphism 1m Â£ hom{M1 M) satisfying the following: 6 7 (a) For every object N and for every morphism g (b) For every object N and for every morphism / â‚¬ hom(N, M)â€™ we have /1M - /â€¢ 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 (Â£, along with morphisms pi, and P2 from hom(M, M\) and hom(M, M2), respectively, such that for every object N from (Â£ and morphisms fi Â£ hom(N, MÂ¿ ), there is a morphism / G hom(N, M) such that pif â€” 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,^ â€”v By a chain complex from 21, we mean a graded object C together with a map of degree â€” 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 â€”> C such that ff and /'/ are homotopic to the identity maps on C and C. respectively. Let Â£ be a class of sequences from the additive category 21, each of the form E' E â€”- E". In particular each element of Â£ determines a quintuple of three objects and two morphisms. Declare the elements of Â£ 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 Â£ 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 Â£. We now make the Definition 2.1.5. Let P be an object of 21. If for every sequence E' E â€”Â»â€¢ E" from Â£ and morphism (f> : P -> E, such that the following diagram commutes there exists a morphism if; : P â€”> E' which preserves the commutivity of the diagram, then we say P is a projective object in 21 relative to Â£. 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 Â£, 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 â–¡ Lemma 2.1.7. Suppose that P is a projective object in 21 relative to Â£., a class of exact sequences. Given the diagram, not necessarily commutative, (2.5) where ehd â€” ef, and the bottom row is exact, then there is a morphism k : P â€”> E' such that e'k + hd â€” f (2.6) Proof. Apply definition 2.1.5, taking 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 proÂ¬ 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 Â£ â‚¬ we have hom(M, X) a singleton. On the other hand, if for every object X â‚¬ â‚¬ 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 exÂ¬ 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 Â£ of exact sequences. A resolution of M in 21 relative to Â£ is an acyclic sequence E2 â€”â€”- Ei â€”â€”- E0 ^ M - * 0 (2.7) Arbitrary resolutions prove to be uninteresting. If we insist that there exist j such that for each element E,Â¿ 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 Â£ of exact sequences, and r be an integer. Suppose that 11 {fi â– â– Ci -+ C'} Â¿ be extended to a chain map f : C â€”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 /Â¿ has been defined for i < n in such a way so that dÂ¿/Â¿ = 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 {/Â¿}Â¿ defined for i < n such that d'hi + = /Â¿ â€” gt. In the event that n â€” r take hi = 0 for all i < r. Let tn â€” fn â€” 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 homoÂ¬ 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 construcÂ¬ 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 â€”>â– P', unique up to homotopy, which is a homotopy equivalence between P and P'. Proof. Consider the diagram â€” â–º F 2 -F 51 â–º F >0â€” â–ºA Ã â–º id _ v ^ r v V . n/r 2 x i o Â° 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 Â£, 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) â€”Â» hom(P\, N) â€”V hom(P2, N) â€”> â– â– â– (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: â€”) 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 HÂ°Â°-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 Sarasonâ€™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 considÂ¬ 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 apÂ¬ plications to control theory. R. Nevanlinna [23] and G. Pick [26], as well as C. CaratheÃ³dory, L. Fejer, and I. Schur, studied various problems of interpolating data with analytic functions. D. Sarason [28] is credited with providing the opÂ¬ 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 HÂ°Â° 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 LÂ°Â°(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) â€” 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 mâ€”measurable subset A C T, such that H â€” 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\ â€” 1 (a.e.), such that H = 0M2. 16 For a proof we refer the reader to N.K. Nikolâ€™skii [24]. Our immediate interest in the Hardy Spaces stems from Pickâ€™s Theoerem . Given {xi,x2,... ,a;n} C D. and {zi,z2,..- ,zn} C C, there exists matrix (3.1) is positive. Sarason demonstrates that this theorem can be obtained as a special case of his Theorem 1 [28]: Sarasonâ€™s Theorem . Let ip be a nonconstant inner function, S as above, and K, â€” H2 Â© ipH2. If T is an operator that commutes with the projection of S onto JC, then there is a function || â€” ||Tj| and denotes the projection onto K of the operator of multiplication on L2 by 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) â€” 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 â€” zj.gk for 1 < k < n (where zj. is the complex conjugate of Zk), then Sarasonâ€™s theorem guarantees a Â£ HÂ°Â° with 11 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
theorem that the interpolating function 6 have H00 norm less than or equal to 1 is
k=l
then for all MgN,
M
I + "^2 cnd0{TnA){TnA)* > 0.
Uâ€” 1
Proof. Computing we find
M
I > Â£ dtTiT'k
k= 1
M /M-k
>^dkTk ( J2 Cj(CjA){C3Ay ) T
k= 1
M M-k
. jâ€”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.
nâ€”1
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
â–¡
Let M be a Hilbert space, A42(k) = M Â® H2(k) be as in Section 3.6, K.\,K,2 as
above, / â‚¬ C{M2(k),K\) such that
/; = cjCjf0.
(3.36)
28
where fj is as in equation (3.27). Then ||/|| < 1 if and only if
/>//â€¢ = = ^aÂ¿icÂ¿!2cyâ€ž/0*c'*
(3.37)
Suppose that there is a non-negative sequence d : N â€”^ C such that d*ac = e, where
acj â€” cij jcj j2. Under these conditions we have
Corollary 3.7.4. IfT â‚¬ Â£(/C2) Â¿s a dilation of C E Â£(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) â€”> 1C2 whose matrix relative to the decomposition
of 1C2 â€” (1C2 Â© /Ci) Â© Afi has the form
ft = CO ( 7 ] r, = Cj (Â®) = CjT'/o i > 1. (3.39)
go
Jo,
for some g0 : M â€”> )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
â€˜F0FZ + J2ai M^/o/oV2*
3=0
â€” 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 â€¢ â€¢ â– )
is bounded with norm \\f\\ and the operator matrix
(3.44)
(c,Tfâ€ž c2T2f0 â– â– â– )
(3.45)
has norm bounded above by |
As we stated in the beginning of this section, every object in f)2 can be |