Integral Representation of Semi-Groups of Unbounded Self-Adjoint Operators

Abstract

1. In this paper we obtain the spectral integral representation for semi-groups {Tx} of, in general, unbounded self-adjoint operators in Hilbert space. The index x ranges over a locally compact semi-group e which can be imbedded in a locally compact group ( which satisfies some other additional conditions which will be listed in Section 2. Our result generalizes theorems by E. Hille [6], B. v. Sz. Nagy [9], A. Devinatz [2], A. E. Nussbaum [11] and a recent theorem by A. Devinatz and the author [3]. In all these previous results it was assumed either that x ranges over a semi-group which is a Cartesian product I.+ x E+ where I+ is the set of vectors in rn-dimensional euclidean space with non-negative integer components and E + is the set of vectors in n-dimensional euclidean space with non-negative components (cf. A. Devinatz and A. E. Nussbaum loc. cit.) or that the operators Tx are uniformly bounded (cf. A. E. Nussbaum [11]). The method of proof consists, in essence, in imbedding the semi-group of operators in an algebra of unbounded operators (cf. J. M. G. Fell and J. L. Kelley [5]) and an adaptation to our situation of a method used by R. S. Phillips [12] in his demonstration of M. H. Stone's theorem on the integral representation of groups of unitary operators (cf. [11]). It must be remarked, however, that the imbedding of the semi-group of self-adjoint operators in an algebra of unbounded operators is only possible by virtue of a theorem concerning the permutability of self-adjoint operators proved by the present author in collaboration with A. Devinatz and J. von Neumann [4]. 2. We start with the definition of a locally compact full semi-group. (cf. [11]. Note that we do not assume here that X is abelian). DEFINITION 1. A semi-group (5 is called a locally compact full semigroup if (a) e can be imbedded in a locally compact group ( (b) e is locally compact in the relative topology of (, and (c) every non-empty bounded open set in (5 has non-zero measure with respect to the left (or right) Haar measure of (. Throughout this paper we shall always assume that e is a locally compact full semi-group.