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Théorie spectrale. (French) Zbl 0532.47014

The main purpose of this monograph is to expose the theory of spectral measures with applications to hermitian and normal operators on Hilbert spaces. It is organized in six chapters.
Chapter 1 is devoted to the continuous functional calculus on *-Banach algebras. In Chapter 2 the continuous functional calculus for a bounded normal operator is extended to all Baire functions. Chapter 3 is of a more general level, dealing with spectral and quasi-spectral measures and their relation with the spectrum. Chapter 4 may be considered as the heart of the monograph. The functional calculus is constructed in all generality, for unbounded operators. The results ar applied to the extension theory of symmetric operators. The last two chapters are for applications. Chapter 5 contains some elements of the classical moment problem of Hamburger. Chapter 6 is an introduction to the formalism of quantum mechanics.
The monograph is intended to open the way to more advanced research problems. The results are classical and there are overlaps with other well known monographs (specially K. Maurin: Methods of Hilbert spaces (1967; Zbl 0166.102) and N. I. Ahiezer and I. M. Glazman: The theory of linear operators on Hilbert spaces (Russian) (1950; Zbl 0041.229).
Reviewer: Şt.Frunză

MSC:

47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
47B25 Linear symmetric and selfadjoint operators (unbounded)
47A60 Functional calculus for linear operators
47A10 Spectrum, resolvent
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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