Spectral theory of self-adjoint operators in Hilbert space. Translation from the 1980 Russian original by S. Khrushchëv and V. Peller. (Спектральная теория самосопряженных операторов в гильбертовом пространстве.) (English) Zbl 0744.47017

Birman, M. S.; Solomyak, M. Z.

Mathematics and Its Applications. Soviet Series. 5. Dordrecht etc.: Kluwer Academic Publishers. xvi, 301 S. (1987); translation from Leningrad: Leningrad. Univ., 264 pp. (loose errata) (1980).

This is an excellent textbook on some basic facts in operator theory. It requires a minimal background, and contains a complete exposition of the spectral theory of selfadjoint operators in the separable case, as well as several important applications and related results. Most of the material is standard; an advanced reader should consider with interest the last chapter, which presents in the language of operator theory a short and systematic account of some problems in quantum mechanics.
Chapter headings: 1. Preliminaries; 2. Geometry of Hilbert space. Linear continuous operators; 3. Linear unbounded operators; 4. Symmetric and isometric operators; 5. Spectral measures. Integration; 6. Spectral decomposition; 7. Functional models and unitary invariants of selfadjoint operators; 8. Some applications of spectral theory; 9. Theory of perturbation; 10. Semibounded operators and forms; 11. Classes of compact operators; 12. The commutation relations of quantum mechanics.

Reviewer: D. Timotin (MR 82k:47001)

MSC:

47-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
47A10	Spectrum, resolvent
47B25	Linear symmetric and selfadjoint operators (unbounded)
47B40	Spectral operators, decomposable operators, well-bounded operators, etc.
47A45	Canonical models for contractions and nonselfadjoint linear operators
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
47A55	Perturbation theory of linear operators

Keywords:

spectral theory of selfadjoint operators in the separable case; Geometry of Hilbert space; Linear unbounded operators; Symmetric and isometric operators; Spectral measures; Integration; Spectral decomposition; Functional models; unitary invariants of selfadjoint operators; perturbation; Semibounded operators and forms; compact operators; commutation relations of quantum mechanics