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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
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.

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