The Theory of H(b) Spaces: Volume 1

In two volumes, this comprehensive treatment covers all that is needed to understand and appreciate this beautiful branch of mathematics.

Emmanuel Fricain (Author), Javad Mashreghi (Author)

9781107027770, Cambridge University Press

Hardback, published 26 May 2016

702 pages, 30 b/w illus. 320 exercises
23.6 x 15.8 x 5 cm, 1.2 kg

'… 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

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.

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 K?
15. Bases of reproducing kernels and interpolation
Bibliography
Index.

Subject Areas: Complex analysis, complex variables [PBKD], Calculus & mathematical analysis [PBK], Mathematics [PB]