The spectral theorem.

*(English)*Zbl 0615.47021
Lecture Notes in Mathematics, 1227. Berlin etc.: Springer-Verlag. VI, 104 p. DM 23.00 (1986).

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 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 \(S_ u\) of unitaries on H. Stone conjectured, and J. v. Neumann proved in Math. Ann. 104, 570-578 (1931; Zbl 0001.24703), that if the system of operators \(\{V_ t,S_ u\}\) in H is jointly irreducible, then there exists an isomorphism of H onto \(L^ 2({\mathbb{R}})\) that carries this system to the Schrödinger system \(\{V_ t',S_ u'\}\), where \[ (S_ u'f)(x)=f(x+u),\quad (V_ t'f)(x)=e^{itx}f(x),\quad f\in L^ 2({\mathbb{R}}). \] This theorem inspired essential further investigations; recall in particular G. W. Mackey’s ”systems of imprimitivity”. The concepts of ”cocycles”, ”coboundaries”, and their applications, developed mainly by the author, also originate from these studies. One of the most interesting facts discovered and exploited in the book is the relation, through these concepts, of the Stone-von Neumann theorem and the theorem of Beurling on invariant subspaces in the Hardy-Hilbert space, of the unilateral shift operator f(x)\(\mapsto zf(z)\).

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 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 \(S_ u\) of unitaries on H. Stone conjectured, and J. v. Neumann proved in Math. Ann. 104, 570-578 (1931; Zbl 0001.24703), that if the system of operators \(\{V_ t,S_ u\}\) in H is jointly irreducible, then there exists an isomorphism of H onto \(L^ 2({\mathbb{R}})\) that carries this system to the Schrödinger system \(\{V_ t',S_ u'\}\), where \[ (S_ u'f)(x)=f(x+u),\quad (V_ t'f)(x)=e^{itx}f(x),\quad f\in L^ 2({\mathbb{R}}). \] This theorem inspired essential further investigations; recall in particular G. W. Mackey’s ”systems of imprimitivity”. The concepts of ”cocycles”, ”coboundaries”, and their applications, developed mainly by the author, also originate from these studies. One of the most interesting facts discovered and exploited in the book is the relation, through these concepts, of the Stone-von Neumann theorem and the theorem of Beurling on invariant subspaces in the Hardy-Hilbert space, of the unilateral shift operator f(x)\(\mapsto zf(z)\).

Reviewer: Béla Sz.-Nagy

##### MSC:

47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47D03 | Groups and semigroups of linear operators |

28B05 | Vector-valued set functions, measures and integrals |