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.
- Covers all of the material required to understand the theory and its foundations
- Suitable as a textbook for graduate courses
- Contains over 200 exercises to test students' grasp of the material covered
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Reviews & endorsements
'This two volume monograph is a compendium of the H(b) spaces that will be of interest to both graduate students and practicing mathematicians interested in function-theoretic operator theory. There are 31 chapters between the two volumes and a detailed bibliography consisting of 766 entries. The first volume is devoted to general function-theoretic operator theory (and indeed is a useful reference in its own right) while the second volume is more specialized and contains an in-depth survey of H(b)H(b) theory and related ideas.' Steve Deckelman, MAA Reviews
'… 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: May 2016
- format: Hardback
- isbn: 9781107027770
- length: 702 pages
- dimensions: 236 x 158 x 50 mm
- weight: 1.2kg
- contains: 30 b/w illus. 320 exercises
- availability: Available
Table of Contents
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.
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Authors
Emmanuel Fricain, Université de Lille 1
Emmanuel Fricain is a Professor in the Laboratoire Paul Painlevé at Université Lille 1. A part of his research focuses on the interaction between complex analysis and operator theory, which is the main matter of this book. He has a long experience of teaching numerous graduate courses on different aspects of analytic Hilbert spaces and 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 Laval University, Québec, where he has been selected Star Professor of the Year five times for excellence in teaching. His main fields of interest are complex analysis, operator theory and harmonic analysis.