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.
- Covers all of the material required to understand the theory and its foundations
- Suitable as a textbook for graduate courses
- Both volumes together contain over 400 exercises to test students' grasp of the material
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Reviews & endorsements
'The monograph contains numerous references to original papers, as well as numerous exercises. This monograph may be strongly recommended as a good introduction to this interesting and intensively developing branch of analysis …' Vladimir S. Pilidi, Zentralblatt MATH
'As with Volume 1, chapter notes outline historical development, and an extensive bibliography cites substantial work done in the area since 2000.' Joseph D. Lakey, MathSciNet
'… designed for a person who wants to learn the theory of these spaces and understand the state of the art in the area. All major results are included. In some situations the original proofs are provided, while in other cases they provide the 'better' proofs that have become available since. The books are designed to be accessible to both experts and newcomers to the area. Comments at the end of each section are very helpful, and the numerous exercises were clearly chosen to help master some of the techniques and tools used … In sum, these are excellent books that are bound to become standard references for the theory of H(b) spaces.' Bulletin of the American Mathematical Society
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Product details
- Date Published: October 2016
- format: Hardback
- isbn: 9781107027787
- length: 640 pages
- dimensions: 236 x 158 x 45 mm
- weight: 1.12kg
- contains: 1 b/w illus. 100 exercises
- availability: Available
Table of Contents
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.
Look Inside
Authors
Emmanuel Fricain, Université de Lille I
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.
Javad Mashreghi, Université Laval, Québec
Javad Mashreghi is a Professor of Mathematics at the Université Laval, Québec, Canada, where he has been selected Star Professor of the Year seven times for excellence in teaching. His main fields of interest are complex analysis, operator theory and harmonic analysis. He is the author of several mathematical textbooks, monographs and research articles. He won the G. de B. Robinson Award, the publication prize of the Canadian Mathematical Society, in 2004.