Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups

Abstract

The theory of symmetric local semigroups due to A. Klein and L. Landau ( J. Funct. Anal. 44 (1981) , 121–136) is generalized to semigroups indexed by subsets of R n for n > 1. The result implies a similar result of A. E. Nussbaum ( J. Funct. Anal. 48 (1982) , 213–223). It is further generalized to semigroups that are symmetric local in some directions and unitary in others. The results are used to give a simple proof of A. Devinatz's ( Duke Math. J. 22 (1955) , 185–192) and N. I. Akhiezer's (“the Classical Moment Problem and Some Related Questions,” Hafner, New York, 1965) generalization of a theorem of Widder concerning the representation of functions as Laplace integrals. This result is extended to the representation as a Laplace integral of a function taking values in B ( R ), the set of bounded linear operators on a Hilbert space R . Also, a theorem is proved encompassing both the result of Devinatz and Akhiezer, and Bochner's theorem on the representation of positive definite functions as Fourier integrals.