Mean ergodicity of power-bounded operators in countably order complete Banach lattices
type: Item
instance of: scholarly article
title: Mean ergodicity of power-bounded operators in countably order complete Banach lattices
zbMATH Open document ID: 0575.47027
DOI: 10.1007/BF01162022
author: R. Zaharopol
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173717
review text: Our goal is to prove that given a countably order complete Banach lattice E, if every power-bounded positive operator \(T: E\to E\) is mean ergodic (i.e., the sequence (1/n\(\sum^{n-1}_{i=0}T^ ix)_{n\in {\mathbb{N}}}\) converges in E for every \(x\in E)\), then E is reflexive. The proof is in a certain sense a constructive one; for every non- reflexive countably order complete Banach lattice E we construct a positive power-bounded operator \(T: E\to E\) which is not mean ergodic. The result obtained in this article is related to the following still open problem suggested by Louis Sucheston in 1975: Let E be a Banach space and let us assume that every contraction of E is mean ergodic. Decide whether E is reflexive or not.
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 3919150
zbMATH Keywords: countably order complete Banach lattice
zbMATH Keywords: power-bounded positive operator
zbMATH Keywords: mean ergodic
zbMATH Keywords: reflexive Banach lattices
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3919150