Book contents
- Frontmatter
- Dedication
- Contents
- Foreword
- Preface
- 1 Introduction: Toolbox
- 2 Boundedness of Composition Operators
- 3 A First Contact with Compactness Issues
- 4 Fundamental Examples
- 5 Carleson Embedding Point of View
- 6 Compactness via Nevanlinna Counting Functions and Essential Norms
- 7 Carleson versus Nevanlinna
- 8 Spectrum
- 9 Schatten Classes for Composition Operators
- 10 Approximation Numbers on Hardy and Bergman Spaces
- 11 Composition Operators on Hardy–Orlicz and Bergman–Orlicz Spaces
- 12 Composition Operators on Weighted Hardy Spaces
- Appendix A Schur Test Lemma
- Appendix B Multiplier
- Appendix C Toeplitz Operators
- Appendix D Van der Corput and the Stationary Phase
- Appendix E Several Variables
- Appendix F Reminders on Subordination of Sequences
- Appendix G Singular Numbers and Their Comparisons
- Appendix H Orlicz Spaces
- Appendix I Exercises
- Appendix J A Small List of Open Problems
- Bibliography
- Index
3 - A First Contact with Compactness Issues
Published online by Cambridge University Press: 15 May 2026
Book contents
- Frontmatter
- Dedication
- Contents
- Foreword
- Preface
- 1 Introduction: Toolbox
- 2 Boundedness of Composition Operators
- 3 A First Contact with Compactness Issues
- 4 Fundamental Examples
- 5 Carleson Embedding Point of View
- 6 Compactness via Nevanlinna Counting Functions and Essential Norms
- 7 Carleson versus Nevanlinna
- 8 Spectrum
- 9 Schatten Classes for Composition Operators
- 10 Approximation Numbers on Hardy and Bergman Spaces
- 11 Composition Operators on Hardy–Orlicz and Bergman–Orlicz Spaces
- 12 Composition Operators on Weighted Hardy Spaces
- Appendix A Schur Test Lemma
- Appendix B Multiplier
- Appendix C Toeplitz Operators
- Appendix D Van der Corput and the Stationary Phase
- Appendix E Several Variables
- Appendix F Reminders on Subordination of Sequences
- Appendix G Singular Numbers and Their Comparisons
- Appendix H Orlicz Spaces
- Appendix I Exercises
- Appendix J A Small List of Open Problems
- Bibliography
- Index
Summary
We begin this chapter with a general principle. Originally, it was used in the framework of some specific spaces of analytic functions, often Hilbertian, mainly the Hardy space H2 or the Bergman space B2, but we shall apply it in many spaces, so it is worth stating as a general criterion.
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- A Primer, and Beyond, on Composition Operators on the Unit Disk , pp. 131 - 151Publisher: Cambridge University PressPrint publication year: 2026
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