Convergence of semigroups of measures on a Lie group

TL;DR

This paper provides a simplified proof of Siebert's theorem on the convergence of semigroups of measures on Lie groups, extending the results to Banach space representations and avoiding the use of unitary representations.

Contribution

The paper offers a straightforward proof of Siebert's theorem and extends its applicability to Banach space representations, simplifying the original approach.

Findings

Generated functionals converge pointwise on C^2(G)

The convergence result is extended to Banach space representations

Simplified proof avoids using unitary representations

Abstract

A theorem of Siebert asserts that if a sequence of semigroups of probability measures on a Lie group G is weakly convergent to a semigroup of the same type, then the corresponding generating functionals are convergent in the weak operator topology in every unitary representation of the group.The aim of this note is to give a simple proof of the theorem and propose some improvements. In particular, we completely avoid employing unitary representations by showing simply that under the same hypothesis the generating functionals are convergent pointwise on C^2(G). As a corollary,the above thesis of Siebert is extended to strongly continuous representations of G on Banach spaces.