Positive Reynolds operators and moment theory

Abstract

In this paper, positive moment operators on the space C(S) are considered, where S is a compact abelian semigroup possessing a separating set of continuous semicharacters. These operators have the convenient property that every semicharacter is an eigenvector. A criterion on the corresponding set of eigenvalues, the moments, is introduced, which is satisfied for the moment family of a positive moment operator T if, and only if, T is a Reynolds operator. It is shown that these operators are uniquely determined by a set of semicharacters and corresponding moments, if the semicharacters separate S and the moments are nonzero. In this context, all derivations are characterized, which generate a semigroup and commute with translations. Finally, all positive Reynolds and moment operators on the multiplicative unit interval are determined.