Time regularity for random walks on locally compact groups (Q863487)

From the author's abstract: ``Let \(G\) be a compactly generated, locally compact group, and let \(T\) be the operator of convolution with a probability measure \(\mu\) on \(G\). Our main results give sufficient conditions on \(\mu\) for the operator \(T\) to be analytic in \(L^p(G)\), \(1< p<\infty\), where analyticity means that one has an estimate of form \(\|(I- T)T^n\|< cn^{-1}\) for all \(n= 1,2,\dots\) in \(L^p\) operator norm.'' Sufficient for analyticity is, e.g., that \(\mu\) satisfies a moment condition and dominates a multiple of the unit mass at the identity \(e\) or a multiple of a left invariant Haar measure on a neighborhood of \(e\). Counterexamples show that analyticity may not hold if some of the conditions are not met. E.g., \(T\) may not be analytic if \(\mu\) is not spread-out or if its support does not generate the whole group \(G\).