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An H(b) space is defined as a collection of analytic functions that 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. Volume 1 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. Volume 2 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.
List of contents
Preface; 16. The spaces M(A) and H(A); 17. Hilbert spaces inside H2; 18. The structure of H(b) and H(b¿ ); 19. Geometric representation of H(b) spaces; 20. Representation theorems for H(b) and H(b¿); 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(b¿) ¿ 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.
About the author
Emmanuel Fricain is Professor of Mathematics at Laboratoire Paul Painlevé, Université Lille 1, France. Part of his research focuses on the interaction between complex analysis and operator theory, which is the main content of this book. He has a wealth of experience teaching numerous graduate courses on different aspects of analytic Hilbert spaces, and he has published several papers on H(b) spaces in high-quality journals, making him a world specialist in this subject.
Summary
This comprehensive treatment in two volumes is accessible to graduate students as well as researchers. It covers all of the preliminary subjects required to fully understand and appreciate this beautiful branch of mathematics, such as Hardy spaces, Fourier analysis and Carleson measures. Volume 2 focuses on the central theory.
