The spectral theorem
type: Item
instance of: scholarly article
title: The spectral theorem
zbMATH Open document ID: 0615.47021
author: Henry Helson
published in: Lecture Notes in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Chapter 1 contains the author's approach to the construction of the multiplicity function of a spectral measure in Hilbert space H. It is based on the notion of range function and has a more geometric character than e.g. those of Stone and Halmos, at least if H is separable. The spectral theorem itself is usually proved in the ''versions'': (a) for bounded self-adjoint operators, (b) for unitary operators V and the set of their iterates \(V^ n\) \((n=0,\pm 1,...)\), (c) for one (or more) parameter, groups \(V_ t\) (t\(\in {\mathbb{R}})\) of unitaries, (d) for possibly unbounded self-adjoint or normal operators. Most expositions are proceeding this way, while, in Chapter 2, the author prefers to start with (c), i.e., Stone's theorem. Namely, he reduces the study of the (continuous) group \(V_ t\) of unitaries, by the formula \[ \tau (f)=\int^{\infty}_{-\infty}f(t)V_ tdt\quad (f\in L^ 1({\mathbb{R}})), \] to the study of the operator representation \(f\mapsto \tau (f)\) of the (convolution) algebra \(L^ 1({\mathbb{R}})\). (Thereby, he kindly gives a - non fully deserved - credit to the book of Riesz-Sz. Nagy.) It follows, in Chapter 3, Bochner's theorem in its two versions: positive definiteness of the functions being defined for discrete sums or for integrals, respectively. (For the integral version right credit is given to the [Acta Litt. Sci. Szeged 6, 184-198 (1933; Zbl 0007.30903)] paper of \textit{F. Riesz} on Stone's and Bochner's theorems.) The two (rather different) versions are proved here by means of the same ideas. Chapters 4 and 5 deal with cocycles, an area in which the author has obtained particularly interesting results; see e.g. his paper in J. Lond. Math. Soc., II. Ser. 31, 473-477 (1985; Zbl 0603.28019). It was H. Weyl who, dealing with the Heisenberg-Schrödinger commutation relation in quantum mechanics, was lead to the commutation relation \(S_ uV_ t=e^{iut}V_ tS_ u\) (u,t\(\in {\mathbb{R}})\) for two continuous one parameter groups \(V_ t\) and \…
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 3996457
zbMATH Keywords: Weyl commutation relation
zbMATH Keywords: construction of the multiplicity function of a spectral measure in Hilbert space
zbMATH Keywords: range function
zbMATH Keywords: operator representation
zbMATH Keywords: Bochner's theorem
zbMATH Keywords: positive definiteness
zbMATH Keywords: cocycles
zbMATH Keywords: Stone-von Neumann theorem
zbMATH Keywords: invariant subspaces in the Hardy-Hilbert space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3996457