The study of composition operators lies at the interface of analytic function theory and operator theory. Composition Operators on Spaces of Analytic Functions synthesizes the achievements of the past 25 years and brings into focus the broad outlines of the developing theory. It provides a comprehensive introduction to the linear operators of composition with a fixed function acting on a space of analytic functions. This new book both highlights the unifying ideas behind the major theorems and contrasts the differences between results for related spaces.
Nine chapters introduce the main analytic techniques needed, Carleson measure and other integral estimates, linear fractional models, and kernel function techniques, and demonstrate their application to problems of boundedness, compactness, spectra, normality, and so on, of composition operators. Intended as a graduate-level textbook, the prerequisites are minimal. Numerous exercises illustrate and extend the theory. For students and non-students alike, the exercises are an integral part of the book.
By including the theory for both one and several variables, historical notes, and a comprehensive bibliography, the book leaves the reader well grounded for future research on composition operators and related areas in operator or function theory.
A Menagerie of Spaces
Some Theorems on Integration
Geometric Function Theory in the Disk
Iteration of Functions in the Disk
The Automorphisms of the Ball
Julia-Carathéodory Theory in the Ball
Boundedness in Classical Spaces on the Disk
Compactness and Essential Norms in Classical Spaces on the Disk
Composition Operators with Closed Range
Boundedness on Hp (BN)
Compactness on Small Spaces
Boundedness on Small Spaces
Boundedness on Large Spaces
Compactness on Large Spaces
Special Results for Several Variables
Invertible Operators on the Classical Spaces on the Disk
Invertible Operators on the Classical Spaces on the Ball
Spectra of Compact Composition Operators
Spectra: Boundary Fixed Point, j'(a)