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**Operator theory in function spaces.**
*(English)*
Zbl 0706.47019

This book is concerned with the theory of three important classes of linear operators, namely Toeplitz operators, Hankel operators, and (inner) composition operators in various spaces of analytic functions on the unit disc \(| z| <1\). The main emphsis is put on boundedness and compactness conditions for these operators, as well as conditions under which they belong to some Schatten class (e.g., trace class or Hilbert-Schmidt class). Many results are fairly recent and have not been published in monographic form before.

The book consists of ten chapters with the following headings: 1. Operators on Hilbert spaces. 2. Interpolation of Banach spaces. 3. Integral operators. 4. Bergman spaces. 5. Bloch and Besov spaces. 6. Toeplitz operators on the Bergman space. 7. Hankel operators on the Bergman space. 8. Hardy spaces nd BMO. 9. Hankel operators on the Hardy space. 10. Composition operators.

The book whould be of interest to all specialists and researchers in linear functional analysis and operator theory. It contains many illuminating exercises of various difficulty, which makes it valuable also to post graduate students. Unfortunately, the considerably high price will perhaps prevent the book from the wide readership it deserves.

The book consists of ten chapters with the following headings: 1. Operators on Hilbert spaces. 2. Interpolation of Banach spaces. 3. Integral operators. 4. Bergman spaces. 5. Bloch and Besov spaces. 6. Toeplitz operators on the Bergman space. 7. Hankel operators on the Bergman space. 8. Hardy spaces nd BMO. 9. Hankel operators on the Hardy space. 10. Composition operators.

The book whould be of interest to all specialists and researchers in linear functional analysis and operator theory. It contains many illuminating exercises of various difficulty, which makes it valuable also to post graduate students. Unfortunately, the considerably high price will perhaps prevent the book from the wide readership it deserves.

Reviewer: P.Zabreiko

### MSC:

47B38 | Linear operators on function spaces (general) |

47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |