Convergence of semigroups of measures on a Lie group

Details Convergence of semigroups of measures on a Lie group Convergence of semigroups of measures on a Lie group

2010-09-20

arxiv.org/abs/1009.3821v1

Mathematics

Functional Analysis

6 pages, http://www.math.uni.wroc.pl/~glowacki

Scientific paper

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.

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