Positivity aspects of the Fantappiè transform

A Hilbert space approach to the classical Fantappiè transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappiè transforms of positive measures on the unit ball in Cn^a relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators.