logo_acta

Acta Mathematica Vietnamica

DIFFERENTIABLE FUNCTIONS AND THE GENERATORS ON A HILBERT-LIE GROUP

icon-email ERDAL COSKUN

Abstract

A convolution semigroup plays an important role in the theory of probability measure on Lie groups. The basic problem is that one wants to express a semigroup as a Lévy-Khinchine formula. If (μt)tR+ is a continuous semigroup of probability measures on a Hilbert-Lie group G, then we define
Tμtf:=faμt(da)(fC(G); t>0).
It is apparent that (Tμt)tR+ is a continuous operator semigroup on the space C(G) with the infinitesimal generator N. The generating functional A of this semigroup is defined by Af:=limt01t(Tμtf(e)f(e)). We consider the problem of construction of a subspace C(2)(G) of C(G) such that the generating functional A on C(2)(G) exists. This result will be used later to show that Lvy-Khinchine formula holds for Hilbert-Lie groups.