Convergence of iterates of averages of certain operator representations and of convolution powers
Let G be a locally compact \(\sigma\)-compact group and let \(\mu\) be a probability measure on G that is adapted and strictly aperiodic. Moreover let T be a strongly continuous representation of G by isometries on a uniformly convex Banach space B. Then the strong integral \(P_{\mu}:=\int_{G}T(t)\mu (dt)\) yields a contraction on B. By uniform convexity the mean ergodic theorem applies to \(P_{\mu}\). The paper is concerned with the strong convergence of the sequence \((P^ n_{\mu})_{n\geq 1}\) (to the projection onto the subspace of \(P_{\mu}\)-invariant vectors). Whereas weak convergence of \((P^ n_{\mu})_{n\geq 1}\) always holds (Theorem 2.6) strong convergence is proved only under additional conditions. If G is compact then \((P^ n_{\mu})_{n\geq 1}\) converges strongly to \(P_ m\) in view of the Ito-Kawada theorem (where m denotes Haar measure on G). If \(\mu\) is normal then \((P^ n_{\mu})_{n\geq 1}\) strongly converges (Theorem 2.4). The main result of this paper (Theorem 2.8) is the strong convergence of \((P^ n_{\mu})_{n\geq 1}\) if \(\mu\) is spread-out (i.e. some convolution power \(\mu^ k\) is not singular). This property is well known from the theory of random walks on groups. These results are applied to the concentration function \(C(\mu,K):=\sup \{\mu (sK):\) \(s\in G\}\) of \(\mu\) where K runs through the compact subsets of G: If G is not compact and if \(\mu\) is normal or spread-out then \(\lim_{n}C(\mu^ n,K)=0\) (Theorems 3.3 and 3.5). There already exist profound investigations on the convergence of \((C(\mu^ n,K))_{n\geq 1}\) to 0 and on the rate of this convergence. Let us only mention the paper by \textit{K. Hofmann} and \textit{A. Mukherjea} [Math. Ann. 256, 535-548 (1981; Zbl 0471.60015)] where it is proved that \(\lim_ nC(\mu^ n,K)=0\) for all irreducible probability measures \(\mu\) on G provided G belongs to a large class of locally compact non-compact groups including the discrete, the Abelian and the Lie groups. Thus this result can in general not be compared with the results in the paper under review.
locally compact \(\sigma \) -compact group
scientific article; zbMATH DE number 4171246