The study of composition operators links some of the most basic
questions you can ask about linear operators with beautiful classical
results from analytic-function theory. The process invests old
theorems with new meanings, and bestows upon functional analysis
an intriguing class of concrete linear operators. Best of all,
the subject can be appreciated by anyone with an interest in function
theory or functional analysis, and a background roughly equivalent
to the following twelve chapters of Rudin's textbook *Real and
Complex Analysis: *Chapters 1--7 (measure and integration,
L^p spaces, basic Hilbert and Banach space theory), and
10--14 (basic function theory through the Riemann Mapping Theorem).

In this book I introduce the reader to both the theory of composition operators, and the classical results that form its infrastructure. I develop the subject in a way that emphasizes its geometric content, staying as much as possible within the prerequisites set out in the twelve fundamental chapters of Rudin's book.

Although much of the material on operators is quite recent, this book is not intended to be an exhaustive survey. It is, quite simply, an invitation to join in the fun. The story goes something like this.

The setting is the simplest one consistent with serious ``function-theoretic
operator theory:'' the unit disc U of the complex plane, and the
Hilbert space H^2 of functions holomorphic on U with square summable
power series coefficients. To each holomorphic function p that
takes U into itself we associate the *composition operator*
C_phi defined by

(C_phi) f (z)= f(phi(z)) ( f holomorphic on U, z in U),

and set for ourselves the goal of discovering the connection between the function theoretic properties of p and the behavior of C_phi on H^2.

Indeed, it is already a significant accomplishment to show that each composition operator takes H^2 into itself; this is essentially Littlewood's famous Subordination Principle. Further investigation into properties like compactness, spectra, and cyclicity for composition operators leads naturally to classical results like the Julia-Carathéodory Theorem on the angular derivative, the Denjoy-Wolff iteration theorem, Koengs's solution of Schroeder's functional equation, the Koebe Distortion Theorem, and to hidden gems like the Linear Fractional Model Theorem, and Littlewood's ``Counting Function'' generalization of the Schwarz Lemma. I list below a more detailed outline.

**Chapter 0.** This is a prologue, to be consulted as needed,
on the basic properties and classification of linear fractional
transformations. Linear fractional maps play a vital role in our
work, both as agents for changing coordinates and transforming
settings (e.g., disc to half-plane), and as a source of examples
that are easily managed, yet still rich enough to exhibit surprisingly
diverse behavior. This diversity of behavior foreshadows the Linear
Fractional Model Theorem, which asserts that *every* univalent
self-map of the disc is conjugate to a linear fractional self-map
of some, usually more complicated, plane domain.

**Chapter 1.** After developing some of the basic properties
of H^2, we show that every composition operator acts boundedly
on this Hilbert space. As pointed out above, this is essentially
Littlewood's Subordination Principle. We present Littlewood's
original proof---a beautiful argument that is perfectly transparent
in its elegance, but utterly baffling in its lack of geometric
insight. Much of the sequel can be regarded as an effort to understand
the geometric underpinnings of this theorem.

**Chapters 2 through 4: ** Having established
the boundedness of composition operators, we seek to characterize
those that are *compact.* Chapter 2 sets out the motivation
for this problem, and in Chapter 3 we discover that the geometric
soul of Littlewood's Theorem is bound up in the Schwarz Lemma.
Armed with this insight, we are able to characterize the *univalently
induced *compact composition operators, obtaining
a compactness criterion that leads directly to the theorem on
the angular derivative. In Chapter 4 we prove this theorem in
a way that emphasizes its geometric content, especially its connection
with the Schwarz Lemma. At the end of Chapter 4 we give further
applications to the compactness problem.

**Chapters 5 and 6: **Closely related to the Julia-Carathéodory
Theorem is the Denjoy-Wolff Iteration Theorem (Chapter 5), which
plays a fundamental role in much of the further theory of composition
operators. In Chapter 6 we use the Denjoy-Wolff Theorem as a tool
in determining the *spectrum* of a compact composition
operator. Our work on the spectrum leads to a connection between
the Riesz Theory of compact operators and Koenigs' classical work
on holomorphic solutions of Schroeder's functional equation---the
eigenvalue equation for composition operators. We develop the
relevant part of the Riesz Theory, and show how, in helping to
characterize the spectrum of a compact composition operator, it
also provides a growth restriction on the solutions of Schroeder's
equation for the inducing map. When the inducing map is univalent
this growth restriction has a compelling geometric interpretation,
which re-emerges in Chapter 9 to provide, under appropriate hypotheses,
a geometric characterization of compactness based on linear fractional
models.

**Chapter 7 and 8: **The study of eigenfunctions begun in
Chapter 6 leads in two directions. In these chapters we show how
it suggests the idea of using simple linear fractional transformations
as ``models'' for holomorphic self-maps of the disc. These *linear
fractional models* provide an important tool for investigating
other properties of composition operators. We illustrate their
use in the study of *cyclicity.* Taking as inspiration
earlier work of Birkhoff, Seidel, and Walsh, we show that certain
composition operators exhibit *hypercyclicity;* they
have a dense orbit (in the language of dynamics, they are *topologically
transitive.* Our method involves characterizing the hypercyclic
composition operators induced by linear fractional self-maps of
U (Chapter 7), and then transferring these results, by means of
an appropriate linear fractional model, to more general classes
of inducing maps (Chapter 8).

**Chapter 9 ** The second direction for the study of
eigenfunctions involves turning around the growth condition discovered
in Chapter 6 to provide a compactness criterion: If a holomorphic
self-map phi of U obeys reasonable hypotheses, and the solutions
of Schroeder's equation for phi do not grow too fast, then C_phi
is compact. The requisite growth condition has a simple geometric
interpretation in terms of the linear fractional model for phi.
Again the key to this work is the Schwarz Lemma, this time in
the form of hyperbolic distance estimates that follow from the
Koebe Distortion Theorem.

**Chapter 10: **The solution to the compactness problem
for general composition operators leads into value distribution
theory, most notably the asymptotic behavior of the Nevanlinna
Counting Function. Here our story comes full circle: the crucial
inequality on the Counting Function is due to Littlewood, and
when reinterpreted it becomes a striking generalization of the
Schwarz Lemma.