\(C_{0}\)-semigroups and mean ergodic operators in a class of Fréchet spaces
type: Item
instance of: scholarly article
title: \(C_{0}\)-semigroups and mean ergodic operators in a class of Fréchet spaces
zbMATH Open document ID: 1192.47041
author: Angela A. Albanese
author: José Bonet
author: Werner J. Ricker
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-02-19T00:00:00Z
review text: The authors prove that there is an important class of Fréchet spaces, namely the quojections, in which important results concerning \({\mathcal C}_0\)-semigroups on Banach spaces continue to hold. They prove that if \(X\) is a quojection Fréchet space and \((T(t))_{t\geq 0}\) is an exponentially equicontinuous, uniformly continuous semigroup on \(X\), then the infinitesimal generator \(A\) of \((T(t))_{t\geq 0}\) is everywhere defined, i.e., \(D(A)=X\) and hence, \(A\in {\mathcal L}(X)\). They also prove that, if \(X\) is a quojection GDP-Fréchet space and \((T(t))_{t\geq 0}\) is an exponentially equicontinuous \({\mathcal C}_0\)-semigroup on \(X\), then \((T(t))_{t\geq 0}\) is uniformly continuous and its infinitesimal generator belongs to \({\mathcal L}(X)\) (a GDP-space is a Grothendieck locally convex Hausdorff space with the Dunford-Pettis property). A result concerning any mean ergodic operator satisfying \({\frac{1}{n}}T^n\rightarrow 0\) in \({\mathcal L}_b(X)\) and defined in a GDP-space is presented. One proves that there exists a projection \(P\in {\mathcal L}(X)\) such that the sequence of squares \((T_{[n]})^2\rightarrow P\) in \({\mathcal L}_b(X)\) as \(n\rightarrow \infty\) (here, \(T_{[n]}:={\frac{1}{n}}\sum_{m=1}^nT^m, \; n\in {\mathbb N}\)). If, in addition, \(X\) is a quojection Fréchet space, then \(\lim_{n\rightarrow \infty}T_{[n]}= P\) in \({\mathcal L}_b(X)\), i.e., \(T\) is uniformly mean ergodic. For GDP-Banach spaces, one recovers a classical result of H.\,P.\thinspace Lotz.
reviewed by: Silvia-Otilia Corduneanu
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 5673263
zbMATH Keywords: \(C_{0}\)-semigroup
zbMATH Keywords: quojection space
zbMATH Keywords: prequojection space
zbMATH Keywords: Grothendieck space
zbMATH Keywords: Dunford-Pettis property
zbMATH Keywords: Köthe space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.10.014
OpenAlex ID: W2015983029
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DOI: 10.1016/J.JMAA.2009.10.014
description: scientific article; zbMATH DE number 5673263