Introduction to:

Composition Operators and Classical Function Theory

by

Joel H. Shapiro

Springer-Verlag 1993

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.