The term “Holomorphic Spaces” is short for “Spaces of Holomorphic Functions.” It refers not so much to a branch of mathematics as to a common thread running through much of modern analysis—through functional analysis, operator theory, harmonic analysis, and, of course, complex analysis.

In the fall of 1995 the Mathematical Sciences Research Institute in Berkeley sponsored the program Holomorphic Spaces. Over forty participants came for periods of two weeks to a full semester; an additional forty or so attended a week-long workshop in October. Spaces of holomorphic functions arise in many contexts. The MSRI program focused predominantly on operator-theoretic aspects of the subject. A series of minicourses formed the program's centerpiece.

This volume consists of expository articles by participants in the program (plus collaborators, in two cases), including several articles based on minicourses. The opening article, by Donald Sarason, gives an overview of several aspects of the subject. The remaining articles, while more specialized, are nevertheless designed in varying degrees to be accessible to the nonexpert. A range of topics is addressed: Bergman spaces (Hakan Hedenmalm, Karl Stroethoff); Hankel operators in various guises (Vladimir Peller, Pamela Gorkin, Scott Saccone, Richard Rochberg); the Dirichlet space (Zhijian Wu); subnormal operators (John B. Conway and Liming Yang); operator models and related areas, especially interpolation problems and systems theory (Nikolai Nikolski and Vasily Vasyunin, Cora Sadosky, Nicholas Young, Alexander Kheifets, Harry Dym, James Rovnyak and coauthors). The concluding article, by Victor Vinnikov, describes an approach to certain commuting families of nonself-adjoint operators in which operator theory is linked with algebraic geometry.

Let H∞ (D) denote the algebra of bounded analytic functions on the open unit disc in the complex plane. For a function g ∈L∞ (D), the Hankel-type operator Sgis defined by Sg (f) = gf + H ∞ (D). We give here an overview of the study of the symbol of the Hankel-type operator, with emphasis on those symbols for which the operator is compact, weakly compact, or completely continuous. We conclude with a look at this operator on more general domains and several open questions.

We look at a uniform algebra A on a compact Hausdorf space X. We let M(A) denote the maximal ideal space of A. We will consider the Hankel-type operator Sg : A → C(X)/A with symbol g ∈ C(X) defined by Sg(f) = fg + A for all f ∈ A.

Even though the space L∞does not look like an algebra of continuous functions, it is possible to identify it with the space of continuous functions on its maximal ideal space X as follows: for / in L∞ define the Gelfand transform of f by f(x) = x(f) for all x G X. Since the topology on X is given by saying that a net xα converges to a: in I if and only if xα (f) converges to x(f)for all f ∈ L∞, we see that the Gelfand transform defines a continuous function on X.

The analytic properties of these forms and the associated operators have been studied extensively, with much attention given to the relationship between the properties of B and B, and those of b. The idea of bilinear forms given by a representation such as (1-1), and thus only depending on the product of the arguments, can certainly be extended to other function spaces. One investigation of those more general forms is in [Janson et al. 1987]. In the more general contexts operators based on expressions such as (1-1) are sometimes called small Hankel operators. There is another generalization, the large Hankel operators] the two types agree for the Hardy space. Recently there has also been consideration of more general classes, the Hankel forms of higher type or order. For each nonnegative integer n there is a class, iJn, of Hankel forms of type n. The elements of H1 are the traditional Hankel forms, and Hn ⊂ Hn+1 for each n.

The characteristic property of such a form is that for any polynomial, p, the new bilinear form Cp(f,g) = E(pf,g) — E(f,pg) is a Hankel form.

Thus such forms are obtained by perturbing Hankel forms in a controlled way. Higher-order forms were introduced in [Janson and Peetre 1987].

1. Introduction

This paper presents a largely self-contained expository introduction to a number of problems in interpolation theory for matrix-valued functions, including the classical problems of Schur, Nevanlinna-Pick (NP), and Caratheodory-Fejer (CF) as special cases. The development will use little more than the elementary properties of vector-valued Hardy spaces of exponent 2.

Moreover, by exercising a little care in the choice of notation, most of the analysis for all three of the classical choices of Ω+ mentioned above can be carried out in one stroke. Table 1 serves as a dictionary for the meaning of the symbol that is appropriate for the region Ω+ in use. In order to describe the BIP we need to introduce some notation.

Part I contains a revised version of the coordinate-free function model of a Hubert space contraction, based on analysis of functional embeddings related to the minimal unitary dilation of the operator. Using functional embeddings, we introduce and study all other objects of model theory, including the characteristic function, various concrete forms (transcriptions) of the model, one-sided resolvents, and so on. For the case of an inner scalar characteristic function we develop the classical H∞-calculus up to a local function calculus on the level curves of the characteristic function. The spectrum of operators commuting with the model operator, and in particular functions of the latter, are described in terms of their liftings. A simplified proof of the invariant subspace theorem is given, using the functional embeddings and regular factorizations of the characteristic function. As examples, we consider some compact convolution-type integral operators, and dissipative Schrödinger (Sturm-Liouville) operators on the half-line.

Part II, which will appear elsewhere, will contain applications of the model approach to such topics as angles between invariant subspaces and operator corona equations, generalized spectral decompositions and free interpolation problems, resolvent criteria for similarity to a normal operator, and weak generators of the commutant and the reflexivity property. Classical topics of stability of the continuous spectrum and scattering theory will also be brought into the fold of the coordinate-free model approach.

Introduction

In recent years the Dirichlet space has received a lot of attention from mathematicians in the areas of modern analysis, probability and statistical analysis. We intend to discuss some recent achievements in function theory and operator theory on the Dirichlet space. The key references are [Richter and Shields 1988; Richter and Sundberg 1992; Aleman 1992; Marshall and Sundberg 1993; Rochberg and Wu 1993; Wu 1993]. In this introductory section we state the basic results. Proofs will be discussed in the succeeding sections.

1. Invariant Subspaces

The codimension-one property for invariant subspaces of the Dirichlet space is related to the cellular indecomposibility of the operator (Mz,D). This concept was first introduced and studied by Olin and Thomson [1984] for more general Hubert spaces. Later Bourdon [1986] proved in that if the operator Mz is cellular indecomposable on a Hubert space Hof analytic functions on 𝔻 with certain properties, then every nonzero invariant subspace for (Mz, H) has the codimension-one property.

1. The Hardy and Bergman Spaces: A Comparison

The Hardy space H2 consists of all holomorphic functions on the open unit disk 𝔻 such that. where 𝕋 stands for the unit circle and ds is arc length measure, normalized so that the mass of 𝕋 equals 1. In terms of Taylor coefficients, the norm takes a more appealing form. The Bergman space, on the other hand, consists of all holomorphic functions on 𝔻 where dS is area measure, normalized so that the mass of 𝔻 equals 1.

where x, y are elements of a vector space Xy D is a positive semidefinite quadratic form on X, T1 and T2 are linear operators on X, and M1, M2 are linear operators from X to the separable Hubert spaces E1, E2. After introducing the de Branges-Rovnyak function space Hw associated with w, we can formulate the interpolation conditions thus: w is a solution to the interpolation problem if and only if there exists a linear mapping F : X → Hw

The solutions w turn out to be the scattering matrices of the unitary colligations that extend the isometric colligation defined by the Lyapunov identity. These extensions and their scattering functions can be described using a “universal” extension and its scattering operator function. The description formula for solutions looks like.

The matrix S is defined essentially uniquely by the data of the problem and is called the scattering matrix of the problem. Using the functional model and the Fourier representation of the “universal” extension one can investigate analytic properties of the scattering matrices S for classes of interpolation problems.

Lecture 1. The Abstract Interpolation Problem

I will begin with the formal setup of the Abstract Interpolation Problem, or AIP, then consider several examples and discuss the role of the AIP in their investigation.

1. Introduction

We present here the main results of the theory of reproducing kernel Pontryagin spaces [Schwartz 1964; Sorjonen 1975] including some recent improvements [Alpay et al. 1997]. The paper is expository and is intended for nonspecialists in the indefinite theory. We presume knowledge of the Hubert space case, that is, Aronszajn's theory [1950] of reproducing kernel Hubert spaces. The main point is that much of the experience with the Hubert space theory is transferable to Pontryagin spaces. Section 2 presents background from operator theory. A key result here is a theorem to construct contraction operators by specifying their action on dense sets; we give a new proof that reduces the result to the isometric case. The main results on reproducing kernels are in Section 3.

Scalar-valued functions are assumed throughout. See [Alpay et al. 1997] for the extension to vector-valued functions and a detailed account of the theory of generalized Schur functions and associated colligations and reproducing kernel Pontryagin spaces.

2. Contraction operators on Pontryagin spaces

Inner products are assumed to be linear and symmetric and defined on a complex vector space. Orthogonality and direct sum are defined for any inner product space as in the Hubert space case.

The operator S is pure if S has no normal summand and is irreducible if S is not unitarily equivalent to a direct sum of two nonzero operators. The theory of subnormal operators provides rich applications in many areas, since many natural operators that arise in complex function theory, differential geometry, potential theory, and approximation theory are subnormal operators. Many deep results have been obtained since Halmos introduced the concept of a subnormal operator. In particular, Thomson's solution of the long-standing problem on the existence of bounded point evaluations reveals a structure theory of cyclic subnormal operators. Thomson's work answers many questions that had been open for a long time and promises to enable researchers to answer many more; see [Thomson 1991] or [Conway 1991]. The latter is a general reference for the theory of subnormal operators.

Here we will present some research problems on subnormal operators and discuss some possibilities for their solution.

1. Introduction

B. Cole and T. W. Gamelin [1982] introduced a generalized notion of analyticity, which they called tightness. If K is a compact space and X ⊂ C(K) is a closed subspace (in the uniform norm) we say X is a tight subspace if the Hankeltype operator Sg : X → C(K)/X defined by / i→ fg + X is weakly compact for every g ∈ C(K). Recall that a uniform algebra A on K is a closed, separating subalgebra of C(K) which contains the constants. We say a uniform algebra A on K is a tight uniform algebra if it is a tight subspace of C(K). The following prototypical example from [Cole and Gamelin 1982] illustrates how tightness could be thought of as an abstract version of the solvability of a -problem with a mild gain in smoothness.

In the late 70's M. S. Livsic has discovered that a pair of commuting nonselfadjoint operators in a Hubert space, with finite nonhermitian ranks, satisfy a polynomial equation with constant (real) coefficients; in particular the joint spectrum of such a pair of operators lies on a certain algebraic curve in the complex plane, the so called discriminant curve of the pair of operators. More generally, it turns out that much in the same way as the study of a single nonself adjoint operator is intimately related to the function theory on the complex plane, more specifically on the upper half-plane, the study of a system of commuting nonself adjoint operators, at least with finite nonhermitian ranks, is related to the function theory on a compact Riemann surface of a higher genus, more specifically on a compact real Riemann surface. Prom a different perspective, while the study of a single nonselfadjoint operator leads to one-variable continuous time linear systems, the study of a pair of commuting nonselfadjoint operators leads to two-variable continuous time systems, which are necessarily overdetermined, hence must be considered together with an additional structure of compatibility conditions at the input and at the output. In this survey we give an introduction to the spectral theory of commuting nonselfadjoint operators and its interplay with system theory and the theory of Riemann surfaces and algebraic curves, including some recent results and open problems.