Convergence of iterates of averages of certain operator representations and of convolution powers
type: Item
instance of: scholarly article
title: Convergence of iterates of averages of certain operator representations and of convolution powers
zbMATH Open document ID: 0712.22008
DOI: 10.1016/0022-1236(89)90047-5
author: Yves Derriennic
author: Michael Lin
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 43A05
zbMATH DE Number: 4171246
zbMATH Keywords: spread-out probability measure
zbMATH Keywords: locally compact \(\sigma \) -compact group
zbMATH Keywords: strongly continuous representation
zbMATH Keywords: uniformly convex Banach space
zbMATH Keywords: strong integral
zbMATH Keywords: uniform convexity
zbMATH Keywords: mean ergodic theorem
zbMATH Keywords: strong convergence
zbMATH Keywords: random walks on groups
zbMATH Keywords: concentration function
reviewed by: Eberhard Siebert
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90047-5
OpenAlex ID: W1981821445
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description: scientific article; zbMATH DE number 4171246