The tail of a positivity-preserving semigroup

Abstract

With certain assumptions a representation theorem is proved for the elements of $$ \cap _{\sigma \in \Sigma } $$ σS, where Σ is an abelian semigroup of, endomorphisms of a real vector space, andS is a convex antisymmetric cone. Application is made to chacterization of nonnegative harmonic functions on bounded Lipschitz domains, of Hausdorff-Stieltjes moment sequences, and of “bilateral Laplace transforms” on locally compact abelian groups, Euclidean motion groups, and noncompact semi-simple Lie groups. Uniqueness of the representation is proved in both the Euclidean motion and the semi-simple cases.