The Theory of H(b) Spaces 2 Volume Hardback Set

An H(b) space is defined as a collection of analytic functions which are in the image of an operator. The theory of H(b) spaces bridges two classical subjects: complex analysis and operator theory, which makes it both appealing and demanding. The first volume of this comprehensive treatment is devoted to the preliminary subjects required to understand the foundation of H(b) spaces, such as Hardy spaces, Fourier analysis, integral representation theorems, Carleson measures, Toeplitz and Hankel operators, various types of shift operators, and Clark measures. The second volume focuses on the central theory. Both books are accessible to graduate students as well as researchers: each volume contains numerous exercises and hints, and figures are included throughout to illustrate the theory. Together, these two volumes provide everything the reader needs to understand and appreciate this beautiful branch of mathematics.

Volume 1: List of figures; Preface; List of symbols; Important conventions; 1. *Normed linear spaces and their operators; 2. Some families of operators; 3. Harmonic functions on the open unit disc; 4. Analytic functions on the open unit disc; 5. The corona problem; 6. Extreme and exposed points; 7. More advanced results in operator theory; 8. The shift operator; 9. Analytic reproducing kernel Hilbert spaces; 10. Bases in Banach spaces; 11. Hankel operators; 12. Toeplitz operators; 13. Cauchy transform and Clark measures; 14. Model subspaces KΘ; 15. Bases of reproducing kernels and interpolation; Bibliography; Index. Volume 2: Preface; 16. The spaces M(A) and H(A); 17. Hilbert spaces inside H2; 18. The structure of H(b) and H(b̅ ); 19. Geometric representation of H(b) spaces; 20. Representation theorems for H(b) and H(b̅); 21. Angular derivatives of H(b) functions; 22. Bernstein-type inequalities; 23. H(b) spaces generated by a nonextreme symbol b; 24. Operators on H(b) spaces with b nonextreme; 25. H(b) spaces generated by an extreme symbol b; 26. Operators on H(b) spaces with b extreme; 27. Inclusion between two H(b) spaces; 28. Topics regarding inclusions M(a) ⊂ H(b̅) ⊂ H(b); 29. Rigid functions and strongly exposed points of H1; 30. Nearly invariant subspaces and kernels of Toeplitz operators; 31. Geometric properties of sequences of reproducing kernels; References; Symbols index; Index.