The study of composition operators lies at the interface of analytic function theory and operator theory. As a part of operator theory, research on composition with a fixed function acting on a space of analytic functions is of fairly recent origin, dating back to work of E. Nordgren in the mid 1960's. The first explicit reference to composition operators in the Mathematics Subject Classification Index appeared in 1990. As a glance at the bibliography will show, over the intervening years the literature has grown to a point where it would be difficult for a novice to read all of the papers in the subject. At the same time, there are themes developing so that it is possible to see important groups of papers as exploring the same theme. This book is an attempt to synthesize the achievements in the area so that those who wish to learn about it can get an overview of the field as it exists today. At the same time, we hope to bring into clearer focus the themes from the literature so it is easier to see the broad outlines of the developing theory. We have taken this opportunity to present, in addition to material that is well known to experts, some results that are appearing here for the first time. Many interesting and seemingly basic problems remain open and it is our hope that this book may point out areas in which further exploration is desirable and serve to entice others into thinking about some of these problems.

One of the attractive features of this subject is that the prerequisites are minimal. This book should be suitable for second or third year graduate students who have had basic one semester courses in real analysis, complex analysis, and functional analysis. We have included a large number of exercises with the student in mind. While these exercises vary in difficulty, they are all intended to be accessible; we have not used the exercises as a place to collect major results from the literature that space did not permit a discussion of in the text. Since the exercises both illustrate and extend the theory, we urge all readers, students and non-students alike, to consider the exercises as an integral part of the book. Rather than seeking the utmost generality, the theory is developed in a context that is comfortable and illustrates the nature of the general results.

In several places we consider composition operators acting on function spaces in the unit ball in C^N. Typically, our study of composition operators in the several variable setting is done in separate sections (or separate chapters) from the one variable theory. Exceptions to this only occur in places where the several variable reasoning is identical to that in one variable, and where the reader interested only in the latter situation would not find it burdensome to read the arguments with N set equal to 1. However, only very little from the extensive field of complex analysis in several variables is ever needed here, and we give a complete discussion of much of the several complex variable background that is central to our subject --- the automorphisms of the ball and their fixed point properties, angular derivatives and the Julia--Caratheodory theory in B_N, and iteration properties of self-maps of the ball. Most of the theory of several complex variables that we use without proof can be found in the first 45 pages of W. Rudin's "Function Theory in the Unit Ball of C^n". Statements of results used and relevant definitions are all included here, so that the reader unfamiliar with these results but willing to accept some of them without proof should find the several variable sections of the book readable. Indeed, one of our goals is to convince the reader that the unit ball of C^N is an interesting place to do function-theoretic operator theory because one can quickly get to phenomena that are not seen in the disk but which can nevertheless be handled with a minimum of technical machinery. In short, we believe this is an ideal place for a first excursion into several variable function theory.

This book is written from a philosophy that mathematics develops best from a base of well chosen examples and that its theorems describe and generalize what is true about the characteristic objects in a subject. This is a book about the concrete operator theory that arises when we study the operation of composition of analytic functions in the context of the classical spaces. In particular, we study the relationship between properties of C_\phi and properties of the symbol map \phi: the goal is to see the norm, the spectrum, normality, etc., of C_\phi as consequences of particular geometric and analytic features of the function \phi. The theory of multiplication operators, arising from the spectral theorem for normal operators, has developed and branched into the study of Toeplitz operators, subnormal operators and so on. We believe composition operators can similarly inform the development of operator theory because they are very diverse and occur naturally in a variety of problems. Composition operators have arisen in the study of commutants of multiplication operators and more general operators and play a role in the theory of dynamical systems. DeBranges' original proof of the Bieberbach conjecture depended on composition operators (he called them substitution operators) on a space of analytic functions.

Ergodic transformations are sometimes thought of as inducing composition operators on L^p spaces, for example, but not on analytic spaces; except in the introductory material, we do not discuss the theory of composition operators on non-analytic spaces because that theory seems to be developing in rather different ways. In recent years there has been broad interest in complex dynamics, in the theory of iteration of rational functions in the plane, and so on. In general, this book will not touch on these studies; their emphasis is on the sets where the iteration is chaotic, whereas our study of composition operators will emphasize the regions in which the iteration is regular.

The introductory chapter, as its title implies, sets the stage for the remainder of the book by giving the basic definitions, proving a few theorems that hold in very great generality, and posing the basic questions that will be addressed. The second chapter defines the Hardy and Bergman spaces and their generalizations that we will be working in and develops the analytic tools that are not usually covered in basic graduate courses but are needed in the study of composition operators. Readers familiar with this material may wish to skim the chapter to pick up our notation and see what we consider the basic background for our study. The third chapter contains the core material on boundedness and compactness of composition operators and estimates for their norms. Many of these computations are based on estimates arising from Carleson type measure considerations. In general, the emphasis will be on the standard spaces of analytic functions, but in chapters four and five we discuss smaller and larger spaces of analytic functions and illustrate the differences between composition operators on these spaces and the standard spaces. While the majority of the theory develops in parallel ways in one and several variables, some more subtle phenomena specific to the study of compactness and boundedness questions in several variables are investigated in chapter six. In chapter seven, computation of the spectra of composition operators is described. This description is most complete in the case of compact operators. In the case of non-compact operators, the theory is more complete in the cases in which a weighted shift analogy can be used, and less complete when the weighted shift analogy fails. For compact and invertible operators, the spectral theory is developed in one and several variables, but for the more difficult cases, we consider the theory only on one variable Hardy space and its close relatives. It turns out that composition operators are rarely normal, subnormal, or hyponormal; some results concerning such phenomena are described in chapter eight. Chapter nine consists of several sections devoted to less developed parts of the theory, such as results on equivalences, the topological structure of the space of composition operators, and an application of composition operators to a problem in polynomial approximation. After chapter three, the chapters are largely independent of each other, although chapters four and five on small and large spaces are best appreciated as a package. The results of the last three chapters are largely restricted to Hilbert spaces and especially H^2(D), while the earlier chapters include many results on Banach spaces.

While we have tried to summarize the existing literature on composition operators on spaces of analytic functions, constraints of time, space, and probably the reader's patience have prevented us from including every interesting topic. If your favorite topic has been left out, please accept our sincere regrets. To assist in the reader's further study, the bibliography attempts to be a comprehensive list of works on composition operators on analytic spaces. We apologize in advance to the authors of those papers that we have inadvertently omitted --- it was not intentional! At the end of each section, we include historical notes on the origins of the results and exercises, relationships between papers, and references for further reading.


We hope that others will find the reading of this book to be as stimulating as we have found the writing to be.

Carl C. Cowen
Barbara D. MacCluer
West Lafayette, Indiana
December 1994

Reprinted with permission from
"Composition Operators on Spaces of Analytic Functions"
by Carl C. Cowen and Barbara D. MacCluer.
Copyright CRC Press, Boca Raton, Florida.