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.