Mean ergodicity on Banach lattices and Banach spaces
type: Item
instance of: scholarly article
title: Mean ergodicity on Banach lattices and Banach spaces
zbMATH Open document ID: 0938.47006
author: Eduard Yu. Emel'yanov
author: Manfred P. H. Wolff
published in: Archiv der Mathematik
publication date: 2000-06-09T00:00:00Z
review text: Let \(E\) be a Banach space. The linear operator \(T:E\rightarrow E\) is called power-bounded if \(\sup\{||T^n||:n\in \mathbb N\}<\infty\). \(T\) is mean ergodic whenever the sequence \((\frac{1}{n+1}\sum_{k=0}^nT^kx)\) converges in \(E\) for every \(x\in E\). Now let \(E\) be a Banach lattice. \(T\) is called power-order-bounded if, for every \(x\in E^+\), there exists \(z\in E^+\) such that \(T^n([-x,x])\subseteqq [-z,z]\) for all \(n\in \mathbb N\). A contractive operator \(T\) possesses a quasi order bounded attractor if there exist \(z>0\) and \(\eta\in [0,1]\) such that \(\lim_{n\to\infty}\inf\{\|T^nx-y\|: y\in [-z,z]+B(0,\eta)\}=0\) where \(B(0,\eta)\) denotes the closed ball around \(0\) with radius \(\eta\). In the paper under review the following facts are established: (i) The Banach lattice \(E\) has order continuous norm iff every power-order-bounded operator \(T\) is mean ergodic. (Theorem 1.1) (ii) A countably order complete Banach lattice is a KB-space iff every positive operator which possesses a quasi order bounded attractor is mean ergodic. (Theorem 2.2) (iii) A Banach space does not contain \(c_0\) if every power bounded Fredholm operator with index \(0\) is mean ergodic. (Theorem 3.1.).
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 1295493
zbMATH Keywords: mean ergodic and power bounded operators
zbMATH Keywords: KB-spaces
zbMATH Keywords: order continuous norm
zbMATH Keywords: countably order complete Banach lattice
zbMATH Keywords: positive operator
zbMATH Keywords: quasi-order bounded attractor
zbMATH Keywords: power bounded Fredholm operator
reviewed by: Alexander A. Mekler
MaRDI profile type: Publication
full work available at URL: s000130050324
OpenAlex ID: W2034245828
DOI: 10.1007/S000130050324
description: scientific article; zbMATH DE number 1295493