A Primer, and Beyond, on Composition Operators on the Unit Disk

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.

• 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.