Spaces of holomorphic functions have been a prominent theme in analysis since early in the twentieth century. Of interest to complex analysts, functional analysts, operator theorists, and systems theorists, their study is now flourishing.

This volume, an outgrowth of a 1995 program at the Mathematical Sciences Research Institute, contains expository articles by program participants describing the present state of the art. Here researchers and graduate students will encounter Hardy spaces, Bergman spaces, Dirichlet spaces, Hankel and Toeplitz operators, and a sampling of the role these objects play in modern analysis.

PREFACE
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 Stroetho ); Hankel operators in various guises (Vladimir Peller, Pamela Gorkin, Scott Saccone, Richard Rochberg); the Dirichlet space (ZhijianWu); 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.

The program committee, in addition to the editors of this volume, consisted of Joseph Ball, Nikolai Nikolski, Mihai Putinar, and Cora Sadosky. On behalf of all participants, the program committee wishes to thank the sta of MSRI, especially Director William Thurston, Associate Director Tsit-Yuen Lam, Alisa Colloms, and Kim Garrett, for their many e orts in our behalf.

This volume benefited greatly from the expertise of Silvio Levy, the series editor.

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Edited by Sheldon Axler
San Francisco State University
John E. McCarthy
Washington University, St Louis
Donald Sarason
University of California, Berkeley
Cambridge University Press

CONTENTS
Preface ix
Holomorphic Spaces: A Brief and Selective Survey 1
Donald Sarason
Recent Progress in the Function Theory of the Bergman Space 35
Hakan Hedenmalm
Harmonic Bergman Spaces 51
Karel Stroethoff
An Excursion into the Theory of Hankel Operators 65
Vladimir V. Peller
Hankel-Type Operators, Bourgain Algebras, and Uniform Algebras 121
Pamela Gorkin
Tight Uniform Algebras 135
Scott Saccone
Higher-Order Hankel Forms and Commutators 155
Richard Rochberg
Function Theory and Operator Theory on the Dirichlet Space 179
Zhijian Wu
Some Open Problems in the Theory of Subnormal Operators 201
John B. Conway and Liming Yang
Elements of Spectral Theory in Terms of the Free Function Model
Part I: Basic Constructions 211
Nikolai Nikolski and Vasily Vasyunin
Liftings of Kernels Shift-Invariant in Scattering Systems 303
Cora Sadosky
Some Function-Theoretic Issues in Feedback Stabilisation 337
Nicholas Young
The Abstract Interpolation Problem and Applications 351
Alexander Kheifets
A Basic Interpolation Problem 381
Harry Dym
Reproducing Kernel Pontryagin Spaces 425
Daniel Alpay, Aad Dijksma, James Rovnyak, and
Hendrik S. V. de Snoo
Commuting Operators and Function Theory on a Riemann Surface 445
Victor Vinnikov