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**Unbounded self-adjoint operators on Hilbert space.**
*(English)*
Zbl 1257.47001

Graduate Texts in Mathematics 265. Dordrecht: Springer (ISBN 978-94-007-4752-4/hbk; 978-94-007-4753-1/ebook). xx, 432 p. (2012).

This book is a graduate-level textbook on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schrödinger operators) and analysis. On the whole, the book can be used for teaching a graduate course in spectral theory to students and is also suitable for self-study.

The book consists of six parts, each of which is subdivided into several chapters, each chapter ending with a section containing exercises. The chapter headings are as follows:

Chapter 1. Closed and Adjoint Operators.

Chapter 2. The Spectrum of a Closed Operator.

Chapter 3. Some Classes of Unbounded Operators.

Chapter 4. Spectral Measures and Spectral Integrals.

Chapter 5. Spectral Decompositions of Self-adjoint and Normal Operators.

Chapter 6. One-Parameter Groups and Semigroups of Operators.

Chapter 7. Miscellanea.

Chapter 8. Perturbations of Self-adjoint Operators.

Chapter 9. Trace Class Perturbations of Spectra of Self-adjoint Operators.

Chapter 10. Semibounded Forms and Self-adjoint Operators.

Chapter 11. Sectorial Forms and \(m\)-Sectorial Operators.

Chapter 12. Discrete Spectra of Self-adjoint Operators.

Chapter 13. Self-adjoint Extensions: Cayley Transform and Krein Transform.

Chapter 14. Self-adjoint Extensions: Boundary Triplets.

Chapter 15. Sturm-Liouville Operators.

Chapter 16. The One-Dimensional Hamburger Moment Problem.

The book concludes with appendices devoted to basic topics in analysis and the theory of bounded operators.

There are already a lot of comparable textbooks, for example, N. I. Akhiezer and I. M. Glazman [Theory of linear operators in Hilbert space. Transl. from the Russian and with a preface by Merlynd Nestell. New York, NY: Dover Publications (1993; Zbl 0874.47001)], M. Reed and B. Simon [Methods of modern mathematical physics. II: Fourier analysis, self-adjointness. New York, San Francisco, London: Academic Press (1975; Zbl 0308.47002)], M. S. Birman and M. Z. Solomyak [Spectral-theory of self-adjoint operators in Hilbert space. Transl. from the Russian. Dordrecht etc.: Kluwer Academic Publishers (1987; Zbl 0744.47017)]. The distinguishing feature of the present book is that it treats a number of important subjects (such as Krein’s spectral shift, boundary triplets, the Krein-Birman-Vishik theory of positive self-adjoint extensions, and others) which are rarely if ever presented in textbooks. Unfortunately, however, the progress made in the field over the last years is not covered.

The book consists of six parts, each of which is subdivided into several chapters, each chapter ending with a section containing exercises. The chapter headings are as follows:

Chapter 1. Closed and Adjoint Operators.

Chapter 2. The Spectrum of a Closed Operator.

Chapter 3. Some Classes of Unbounded Operators.

Chapter 4. Spectral Measures and Spectral Integrals.

Chapter 5. Spectral Decompositions of Self-adjoint and Normal Operators.

Chapter 6. One-Parameter Groups and Semigroups of Operators.

Chapter 7. Miscellanea.

Chapter 8. Perturbations of Self-adjoint Operators.

Chapter 9. Trace Class Perturbations of Spectra of Self-adjoint Operators.

Chapter 10. Semibounded Forms and Self-adjoint Operators.

Chapter 11. Sectorial Forms and \(m\)-Sectorial Operators.

Chapter 12. Discrete Spectra of Self-adjoint Operators.

Chapter 13. Self-adjoint Extensions: Cayley Transform and Krein Transform.

Chapter 14. Self-adjoint Extensions: Boundary Triplets.

Chapter 15. Sturm-Liouville Operators.

Chapter 16. The One-Dimensional Hamburger Moment Problem.

The book concludes with appendices devoted to basic topics in analysis and the theory of bounded operators.

There are already a lot of comparable textbooks, for example, N. I. Akhiezer and I. M. Glazman [Theory of linear operators in Hilbert space. Transl. from the Russian and with a preface by Merlynd Nestell. New York, NY: Dover Publications (1993; Zbl 0874.47001)], M. Reed and B. Simon [Methods of modern mathematical physics. II: Fourier analysis, self-adjointness. New York, San Francisco, London: Academic Press (1975; Zbl 0308.47002)], M. S. Birman and M. Z. Solomyak [Spectral-theory of self-adjoint operators in Hilbert space. Transl. from the Russian. Dordrecht etc.: Kluwer Academic Publishers (1987; Zbl 0744.47017)]. The distinguishing feature of the present book is that it treats a number of important subjects (such as Krein’s spectral shift, boundary triplets, the Krein-Birman-Vishik theory of positive self-adjoint extensions, and others) which are rarely if ever presented in textbooks. Unfortunately, however, the progress made in the field over the last years is not covered.

Reviewer: Michael Perelmuter (Kyïv)

### MSC:

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

47B25 | Linear symmetric and selfadjoint operators (unbounded) |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

47Axx | General theory of linear operators |

47E05 | General theory of ordinary differential operators |

47F05 | General theory of partial differential operators |

35Pxx | Spectral theory and eigenvalue problems for partial differential equations |