Spectral theory of operators on Hilbert space. (English) Zbl 1256.47001
Boston, MA: Birkhäuser (ISBN 978-0-8176-8327-6/hbk; 978-0-8176-8328-3/ebook). x, 197 p. (2012).
From the cover: “This work is intended to provide a concise introduction to the spectral theory of Hilbert space operators. With an emphasis on detailed proofs and recent aspects of the theory, it can serve as a modern textbook for a first graduate course in the subject. The coverage of topics is thorough, exploring various intricate points and hidden features often left untreated.
The book begins with a primer on Hilbert space theory, summarizing the basics required for the remainder of the book and establishing unified notation and terminology. After this, standard spectral results for (bounded linear) operators on Banach and Hilbert spaces, including the classical partition of the spectrum and spectral properties for specific classes of operators, are discussed. A study of the spectral theorem for normal operators follows, covering both the compact and the general case, and proving both versions of the theorem in full detail. This leads into an investigation of functional calculus for normal operators and Riesz functional calculus, which in turn is followed by Fredholm theory and compact perturbations of the spectrum, where a finer analysis of the spectrum is worked out. Here, further partitions involving the essential spectrum, including the Weyl and Browder spectra, are introduced. The final section of the book deals with Weyl’s and Browder’s theorems and provides a look at very recent results.”
The book has been written by a well-known specialist in the field. The choice of the presented material has been made with great care. As a result, the reader gets a very handy textbook containing all main theorems with complete proofs. Less important results are placed (without proofs) in “Additional Propositions”, the last section of each chapter. There are also some comments on them and references for finding proofs. At the very end of each chapter, the reader will find a collection of suggested readings. The bibliography contains all classical monographs, some important papers, and a few recent ones reflecting personal interests of the author. The book will serve as an excellent introduction to spectral theory of Hilbert space operators for graduate students. It should also be useful for all scientists using spectral theory.
The book begins with a primer on Hilbert space theory, summarizing the basics required for the remainder of the book and establishing unified notation and terminology. After this, standard spectral results for (bounded linear) operators on Banach and Hilbert spaces, including the classical partition of the spectrum and spectral properties for specific classes of operators, are discussed. A study of the spectral theorem for normal operators follows, covering both the compact and the general case, and proving both versions of the theorem in full detail. This leads into an investigation of functional calculus for normal operators and Riesz functional calculus, which in turn is followed by Fredholm theory and compact perturbations of the spectrum, where a finer analysis of the spectrum is worked out. Here, further partitions involving the essential spectrum, including the Weyl and Browder spectra, are introduced. The final section of the book deals with Weyl’s and Browder’s theorems and provides a look at very recent results.”
The book has been written by a well-known specialist in the field. The choice of the presented material has been made with great care. As a result, the reader gets a very handy textbook containing all main theorems with complete proofs. Less important results are placed (without proofs) in “Additional Propositions”, the last section of each chapter. There are also some comments on them and references for finding proofs. At the very end of each chapter, the reader will find a collection of suggested readings. The bibliography contains all classical monographs, some important papers, and a few recent ones reflecting personal interests of the author. The book will serve as an excellent introduction to spectral theory of Hilbert space operators for graduate students. It should also be useful for all scientists using spectral theory.
Reviewer: Andrzej Sołtysiak (Poznań)
MSC:
| 47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |
| 47A10 | Spectrum, resolvent |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
