Joel H. Shapiro is an adjunct professor of Mathematics and Statistics at Portland State University. He received his PhD from the University of Michigan.

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, and features a development accessible to anyone who has studied basic graduate level real and complex analysis. The work traces how such operator-theoretic issues as boundedness, compactness, and cyclicity, when studied in the context of composition operators, evolve into questions about subordination, value-distribution, angular derivatives, iteration, and functional equations; and it carefully develops each of these classical topics.

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 Compactnessfrom 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.