Quasi-reflexive Fréchet spaces and mean ergodicity
type: Item
instance of: scholarly article
title: Quasi-reflexive Fréchet spaces and mean ergodicity
zbMATH Open document ID: 1190.46003
author: Krzysztof Piszczek
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-11-05T00:00:00Z
review text: Several results showing the relation between the properties of a Banach space \(X\) with a Schauder basis and the existence of power bounded non mean ergodic operators on \(X\) were proved in the important paper by \textit{V.\,P.\thinspace Fonf, M.\,Lin} and \textit{P.\,Wojtaszczyk} [J.~Funct.\ Anal.\ 187, No.\,1, 146--162 (2001; Zbl 1017.46007)]. The proof of their main results is based on classical results of Pełczyński and of Zippin, connecting bases with reflexivity. Their work was continued in the setting of Fréchet spaces by \textit{A.\,A.\thinspace Albanese, W.\,J.\thinspace Ricker} and the reviewer [Ann.\ Acad.\ Sci.\ Fenn., Math.\ 34, No.\,2, 401--436 (2009; Zbl 1194.47012)]. The results of Pełczyński and of Zippin were there extended to Fréchet spaces, thus solving an open problem of Kalton. The aim of the present paper is to complement the results of the aforementioned papers by obtaining a characterization of nonreflexive Fréchet spaces \(E\) with a basis which are quasi-reflexive of order one (i.e., \(E\) has codimension one in its bidual \(E''\)) in terms of the existence of power bounded operators which are not mean ergodic in the space \(E\) or in the dual \(E'\). To do this, the author shows that certain results concerning bases in Banach spaces due to Singer in 1963 can be extended to Fréchet spaces. This extension requires new ideas and techniques.
reviewed by: José Bonet
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A25
Mathematics Subject Classification ID: 46A35
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 5625400
zbMATH Keywords: Fréchet spaces
zbMATH Keywords: Schauder basis
zbMATH Keywords: mean ergodic operator
zbMATH Keywords: power bounded operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.08.060
OpenAlex ID: W2046942358
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DOI: 10.1016/J.JMAA.2009.08.060
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description: scientific article; zbMATH DE number 5625400