Composition Operators on Spaces of Analytic Functions

✍ Scribed by Cowen Jr., Carl C.; MacCluer, Barbara I

• Publisher: Routledge;CRC
• Year: 2019
• Tongue: English
• Leaves: 401
• Series: Studies in advanced mathematics
• Category: Library

✦ Synopsis

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.  Read more...

Abstract: 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

✦ Table of Contents

Content: Cover
Title Page
Copyright Page
Table of Contents
Preface
1: Introduction
Exercises
Notes
2: Analysis Background
2.1 A menagerie of spaces
Spaces of functions of several variables
Exercises
Notes
2.2 Some theorems on integration
Carleson measure theorems
Exercises
Note
2.3 Geometric function theory in the disk
Exercises
Notes
2.4 Iteration of functions in the disk
Lemmas on iteration near the boundary
Exercises
Notes
2.5 The automorphisms of the ball
Exercises
Notes
2.6 Julia-Caratheodory theory in the ball
Exercises
Notes
3: Norms 3.1 Boundedness in classical spaces on the diskExercises
Notes
3.2 Compactness and essential norms in classical spaces on the disk
Exercises
Notes
3.3 Hilbert-Schmidt operators
Exercises
Notes
3.4 Composition operators with closed range
Exercises
Notes
3.5 Boundedness on Hp(BN)
Exercises
Notes
4: Small Spaces
4.1 Compactness on small spaces
Exercises
Notes
4.2 Boundedness on small spaces
Exercises
Notes
5: Large Spaces
5.1 Boundedness on large spaces
Exercises
Notes
5.2 Compactness on large spaces
Exercises
Notes
5.3 Hilbert-Schmidt operators
Exercises
Notes Notes7.8 Spectra: inner functions
Exercises
Notes
8: Normality
8.1 Normal and hyponormal composition operators
Exercises
Notes
8.2 Subnormality of adjoints
Exercises
Notes
9: Miscellanea
9.1 Adjoints of some composition operators
A norm calculation
Exercises
Notes
9.2 Equivalence of composition operators
Exercises
Notes
9.3 Topological structure
Exercises
Notes
9.4 Polynomial approximation
Exercises
Notes
Bibliography
Symbol Index
Index

✦ Subjects

Analytic spaces;Composition operators;MATHEMATICS;Functional Analysis