OPERATOR-VALUED POSITIVE-DEFINITE KERNELS ON TUBES

In this paper we generalize the concept of an infinite positive measur e on a sigma-algebra to a vector valued setting, where we consider mea sures with values in the compactification of a convex cone C which can be described as the set of monoid homomorphisms of the dual cone C i nto [0, infinity]. Applying these concepts to measures on the dual of a vector space leads to generalizations of Bochner's Theorem to operat or valued positive definite functions on locally compact abelian group s and likewise to generalizations of Nussbaum's Theorem on positive de finite functions on cones. In the latter case we use the Laplace trans form to realize the corresponding Hilbert spaces by holomorphic functi ons on tube domains.