Composition Operators: and Classical Function Theory / Edition 1

by Joel H. Shapiro

Overview

The study of composition operators forges links between fundamental properties of linear operators and beautiful results from the classical theory of analytic functions. This book provides a self-contained introduction to both the subject and its function-theoretic underpinnings. The development is geometrically motivated, and accessible to anyone who has studied basic graduate-level real and complex analysis. The work explores how operator-theoretic issues such as boundedness, compactness, and cyclicity evolve - in the setting of composition operators on the Hilbert space H2 into questions about subordination, value distribution, angular derivatives, iteration, and functional equations. Each of these classical topics is developed fully, and particular attention is paid to their common geometric heritage as descendants of the Schwarz Lemma.

Product Details

ISBN-13:	9780387940670
Publisher:	Springer New York
Publication date:	08/26/1993
Series:	Universitext / Universitext: Tracts in Mathematics Series
Edition description:	Softcover reprint of the original 1st ed. 1993
Pages:	239
Product dimensions:	9.21(w) x 6.14(h) x 0.51(d)

Table of Contents

0 Linear Fractional Prologue.- 0.1 First Properties.- 0.2 Fixed Points.- 0.3 Classification.- 0.4 Linear Fractional Self-Maps of U.- 0.5 Exercises.- 1 Littlewood’s Theorem.- 1.1 The Hardy Space H2.- 1.2 H2 via Integral Means.- 1.3 Littlewood’s Theorem.- 1.4 Exercises.- 1.5 Notes.- 2 Compactness: Introduction.- 2.1 Compact Operators.- 2.2 First Class of Examples.- 2.3 A Better Compactness Theorem.- 2.4 Compactness and Weak Convergence.- 2.5 Non-Compact Composition Operators.- 2.6 Exercises.- 2.7 Notes.- 3 Compactness and Univalence.- 3.1 The H2 Norm via Area Integrals.- 3.2 The Theorem.- 3.3 Proof of Sufficiency.- 3.4 The Adjoint Operator.- 3.5 Proof of Necessity.- 3.6 Compactness and Contact.- 3.7 Exercises.- 3.8 Notes.- 4 The Angular Derivative.- 4.1 The Definition.- 4.2 The Julia-Carathéodory Theorem.- 4.3 The Invariant Schwarz Lemma.- 4.4 A Boundary Schwarz Lemma.- 4.5 Proof that (JC 1)—(JC 2).- 4.6 Proof that (JC 2)—(JC 3).- 4.7 Angular derivatives and contact.- 4.8 Exercises.- 4.9 Notes.- 5 Angular Derivatives and Iteration.- 5.1 Statement of Results.- 5.2 Elementary Cases.- 5.3 Wolff’s Boundary Schwarz Lemma.- 5.4 Contraction Mappings.- 5.5 Grand Iteration Theorem, Completed.- 5.6 Exercises.- 5.7 Notes.- 6 Compactness and Eigenfunctions.- 6.1 Königs’s Theorem.- 6.2 Eigenfunctions for Compact C?.- 6.3 Compactness vs. Growth of—.- 6.4 Compactness vs. Size of— (U).- 6.5 Proof of Riesz’s Theorem.- 6.6 Exercises.- 6.7 Notes.- 7 Linear Fractional Cyclicity.- 7.1 Hypercyclic Fundamentals.- 7.2 Linear Fractional Hypercyclicity.- 7.3 Linear Fractional Cyclicity.- 7.4 Exercises.- 7.5 Notes.- 8 Cyclicity and Models.- 8.1 Transferenc from Models.- 8.2 From Maps to Models.- 8.3 A General Hypercyclicity Theorem.- 8.4 Exercises.- 8.5 Notes.- 9 Compactness from Models.- 9.1 Review of Königs’s Model.- 9.2 Motivation.- 9.3 Main Result.- 9.4 The Hyperbolic Distance on U.- 9.5 The Hyperbolic Distance on G.- 9.6 Twisted Sectors.- 9.7 Main Theorem: Down Payment.- 9.8 Three Lemmas.- 9.9 Proof of the No-Sectors Theorem.- 9.10 Exercises.- 9.11 Notes.- 10 Compactness: General Case.- 10.1 Motivation.- 10.2 Inadequacy of Angular Derivatives.- 10.3 Non-Univalent Changes of Variable.- 10.4 Decay of the Counting Function.- 10.5 Proof of Sufficiency.- 10.6 Averaging the Counting Function.- 10.7 Proof of Necessity.- 10.8 Exercises.- 10.9 Notes.- Epilogue.- References.- Symbol Index.- Author Index.

Editorial Reviews

The study of composition operators links fundamental properties of linear operators and results from the classical theory of analytic functions. The author provides a self-contained introduction to both the subject and its function-theoretic underpinnings. He explores how operator-theoretic issues such as boundedness, compactness, and cyclicity evolve into questions about subordination, value distribution, angular derivatives, iteration, and functional equations. Problems are included in each chapter. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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