Author | Shapiro, Joel H. author |
---|---|
Title | Composition Operators [electronic resource] : and Classical Function Theory / by Joel H. Shapiro |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1993 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0887-7 |
Descript | XVI, 223 p. 6 illus. online resource |
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 Littlewoodโs Theorem -- 1.1 The Hardy Space H2 -- 1.2 H2 via Integral Means -- 1.3 Littlewoodโs 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-Carathรฉodory 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 Wolffโs 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 Kรถnigsโs Theorem -- 6.2 Eigenfunctions for Compact C? -- 6.3 Compactness vs. Growth of ? -- 6.4 Compactness vs. Size of ? (U) -- 6.5 Proof of Rieszโs 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 Kรถnigsโs 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