This is a contemporary treatment of composition operators on Banach spaces of analytic functions in one complex variable. It provides a step-by-step introduction, starting with a review (including full proofs) of the key tools needed, and building the theory with a focus on Hardy and Bergman spaces. Several proofs of operator boundedness (Littlewood's principle) are given, and the authors discuss approaches to compactness issues and essential norm estimates (Shapiro's theorem) using different tools such as Carleson measures and Nevanlinna counting functions. Membership of composition operators in various ideal classes (Schatten classes for instance) and their singular numbers are studied. This framework is extended to Hardy-Orlicz and Bergman-Orlicz spaces and finally, weighted Hardy spaces are introduced, with a full characterization of those weights for which all composition operators are bounded. This will be a valuable resource for researchers and graduate students working in functional analysis, operator theory, or complex analysis.
Muu info
Presents the theory of composition operators on spaces of analytic functions, principally Hardy spaces and Bergman spaces.
Foreword;
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.
Pascal Lefèvre is a Professor at the University of Artois (France). After a Ph.D. thesis on harmonic analysis, Banach spaces and interplay with basic number theory he has worked for twenty years on composition operators. Hervé Queffélec is an Emeritus Professor at the University of Lille (France). He completed his Ph.D. thesis on harmonic analysis, Banach spaces and Dirichlet series and has worked on composition operators for the last twenty-five years.
