On modulated ergodic theorems for Dunford-Schwartz operators
type: Item
instance of: scholarly article
title: On modulated ergodic theorems for Dunford-Schwartz operators
zbMATH Open document ID: 0939.47008
author: Michael Lin
author: James Olsen
author: A. A. Tempelman
published in: Illinois Journal of Mathematics
publication date: 2000-07-05T00:00:00Z
review text: Let \((\Omega,{\mathcal F},\mu)\) be a \(\sigma\)-finite measure space. A linear operator in \(L_1\) is called contraction if \(\|T\|\leq 1\). It is called Dunford-Schwartz operator of it is, in addition, a contraction in \(L_\infty\). Let \(a= (a_k)\) be a sequence of complex numbers, and, for \(f\in L_1\), \[ L_nf= L_n(a, T)f= n^{-1} \sum^n_{k=1} a_k T^kf. \] The authors study sequences \(a\) for which \(L_nf\) converges in norm for every weakly almost periodic linear operator in a Banach space. For Dunford-Schwartz operators on probability spaces the a.e.-convergence for \(f\in L_p\) is studied. The limit is identified in some special cases. Also necessary and sufficient conditions for a sequence \(a\) are derived for the norm convergence of \(L_nf\) for every mean ergodic power bounded \(T\).
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 1345978
zbMATH Keywords: contraction
zbMATH Keywords: Dunford-Schwartz operator
zbMATH Keywords: weakly almost periodic linear operator
zbMATH Keywords: mean ergodic power bounded
reviewed by: Ulrich Krengel
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1345978