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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.

Introduction

Analysis Background

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

Norms

Boundedness in Classical Spaces on the Disk

Compactness and Essential Norms in Classical Spaces on the Disk

Hilbert-Schmidt Operators

Composition Operators with Closed Range

Boundedness on Hp (BN)

Small Spaces

Compactness on Small Spaces

Boundedness on Small Spaces

Large Spaces

Boundedness on Large Spaces

Compactness on Large Spaces

Hilbert-Schmidt Operators

Special Results for Several Variables

Compactness Revisited

Wogen's Theorem

Spectral Properties

Introduction

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)

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.

Introduction

Analysis Background

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

Norms

Boundedness in Classical Spaces on the Disk

Compactness and Essential Norms in Classical Spaces on the Disk

Hilbert-Schmidt Operators

Composition Operators with Closed Range

Boundedness on Hp (BN)

Small Spaces

Compactness on Small Spaces

Boundedness on Small Spaces

Large Spaces

Boundedness on Large Spaces

Compactness on Large Spaces

Hilbert-Schmidt Operators

Special Results for Several Variables

Compactness Revisited

Wogen's Theorem

Spectral Properties

Introduction

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)

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.

Introduction

Analysis Background

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

Norms

Boundedness in Classical Spaces on the Disk

Compactness and Essential Norms in Classical Spaces on the Disk

Hilbert-Schmidt Operators

Composition Operators with Closed Range

Boundedness on Hp (BN)

Small Spaces

Compactness on Small Spaces

Boundedness on Small Spaces

Large Spaces

Boundedness on Large Spaces

Compactness on Large Spaces

Hilbert-Schmidt Operators

Special Results for Several Variables

Compactness Revisited

Wogen's Theorem

Spectral Properties

Introduction

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)

Introduction

Analysis Background

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

Norms

Boundedness in Classical Spaces on the Disk

Compactness and Essential Norms in Classical Spaces on the Disk

Hilbert-Schmidt Operators

Composition Operators with Closed Range

Boundedness on Hp (BN)

Small Spaces

Compactness on Small Spaces

Boundedness on Small Spaces

Large Spaces

Boundedness on Large Spaces

Compactness on Large Spaces

Hilbert-Schmidt Operators

Special Results for Several Variables

Compactness Revisited

Wogen's Theorem

Spectral Properties

Introduction

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)

Introduction

Analysis Background

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

Norms

Boundedness in Classical Spaces on the Disk

Compactness and Essential Norms in Classical Spaces on the Disk

Hilbert-Schmidt Operators

Composition Operators with Closed Range

Boundedness on Hp (BN)

Small Spaces

Compactness on Small Spaces

Boundedness on Small Spaces

Large Spaces

Boundedness on Large Spaces

Compactness on Large Spaces

Hilbert-Schmidt Operators

Special Results for Several Variables

Compactness Revisited

Wogen's Theorem

Spectral Properties

Introduction

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)

Introduction

Analysis Background

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

Norms

Boundedness in Classical Spaces on the Disk

Compactness and Essential Norms in Classical Spaces on the Disk

Hilbert-Schmidt Operators

Composition Operators with Closed Range

Boundedness on Hp (BN)

Small Spaces

Compactness on Small Spaces

Boundedness on Small Spaces

Large Spaces

Boundedness on Large Spaces

Compactness on Large Spaces

Hilbert-Schmidt Operators

Special Results for Several Variables

Compactness Revisited

Wogen's Theorem

Spectral Properties

Introduction

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)