Composition Operators on Spaces of Analytic Functions

Carl C. Cowen Jr.

• Publisher: Routledge
• Year: 2019
• Print ISBN: 9780849384929
• eBook ISBN: 9781351459136
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

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.

1

Introduction

The classical Banach spaces are all realized by norming a collection of real or complex valued functions on a set X. If φ maps X into itself, it is natural to consider the composition operator C_φ defined by

$\left(C_{\phi }f\right)\left(x\right)=f\left(\phi \left(x\right)\right)$

for x in X and functions f in the Banach space. Such operators are clearly linear. We will study other properties of these operators for various Banach spaces and φ’s, principally Banach spaces of analytic functions and analytic maps φ, but always for spaces of complex valued functions. If, in addition, ψ is a complex valued function defined on X, the weighted composition operator W_φψ is defined by (W_φψf)(x) = ψ(x)f(φ(x)).

Although the substitution operation is basic to mathematics and studies involving composition of functions have been pursued for a very long time (for example, iteration of rational functions or ergodic theory), the study, as a part of operator theory, of linear operators induced by composition with fixed functions has a relatively short history. This is an interesting contrast to the much older study of linear operators induced by multiplication by fixed functions, which has its roots in the spectral theorem and is being pursued today in such guises as the theory of subnormal operators and the theory of Toeplitz operators.

Especially in view of the relative neglect of the study of composition operators, it is reasonable to point out some of the reasons for their study now. The pat answer to this question is that they are “natural objects” that are “interesting in themselves”. While true, this answer satisfies no one not already convinced of their interest. A better answer, one that illuminates their naturality, is that they are surprisingly general and occur in settings other than the obvious ones. For example, a backward shift of any multiplicity can be represented as a composition operator. The Hilbert space ℓ²(N) can be regarded as a space of complex valued functions on the set of non-negative integers N, and for φ defined on N by φ(n) = n + 1 the composition operator C_φ on ℓ²(N) is

$\left(f\left(0\right),f\left(1\right),\mathrm{\dots }\right)\mapsto \left(f\left(1\right),f\left(2\right),\mathrm{\dots }\right)$

which is a backward shift of multiplicity one. Forelli [Fo64] proved that all isometries of the Hardy spaces H^P(D) for p ≠ 2 are weighted composition operators. They arise naturally in studying other questions from operator theory. For a multiplication operator, (M_hf)(x) = h(x)f(x), consider the question “Which operators A satisfy AM_h = M_hA?’ Clearly, other multiplication operators do. Less obvious is that if there is φ so that h ∘ φ = h then C_φ commutes with M_h:

$C_{\phi }{M}_{h}f=\left(h\circ \phi \right)\left(f\circ \phi \right)=h\left(f\circ \phi \right)M_h{C}_{\phi }f$

Although there need not always be non-trivial φ that satisfy this identity, if h is a reasonably nice bounded analytic function, then the commutant of M_h, at least on H²(D), is generated by the multiplication operators and the composition operators that arise in...

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Introduction
• 2 Analysis Background
• 3 Norms
• 4 Small Spaces
• 5 Large Spaces
• 6 Special Results for Several Variables
• 7 Spectral Properties
• 8 Normality
• 9 Miscellanea
• Bibliography
• Symbol Index
• Index