Convergence of arithmetic means of operators in Fréchet spaces
type: Item
instance of: scholarly article
title: Convergence of arithmetic means of operators in Fréchet spaces
zbMATH Open document ID: 1283.47017
author: José Bonet
author: Angela A. Albanese
author: Werner J. Ricker
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-03-12T00:00:00Z
review text: A well-known result of \textit{M. Lin} [Proc. Am. Math. Soc. 46, 217--225 (1974; Zbl 0291.47006)] states that, if \(T\) is a bounded operator on a Banach space \(X\) which satisfies the assumption \(\lim_{n\to+\infty}\frac{||T^{n}||}{n}=0\), then its arithmetic means \[ T_{[n]}=\frac{1}{n}\sum_{j=1}^{n}T^{j} \] converge uniformly to a bounded projection \(P\) on \(X\) (\(T\) is then called uniformly mean ergodic) if and only if \(I-T\) has closed range, in which case \(X\) can be decomposed as the direct sum \(X=\ker(I-T)\oplus (I-T)X\). The purpose of the paper under review is to study in a systematic way how far this result may be extended to Fréchet spaces, given that it has been observed in [\textit{A. Albanese, J. Bonet} and \textit{W. Ricker}, Ann. Acad. Sci. Fenn., Math. 34, No. 2, 401--436 (2009; Zbl 1194.47012)] that it fails on the space \(s\) of rapidly decreasing sequences. It is shown that on every Montel Köthe space not isomorphic to a countable product of copies of the scalar field, there exists a continuous power-bounded linear operator \(T\) which is uniformly mean ergodic but such that \(I-T\) does not have closed range. On the other hand, it is shown that Lin's result can be extended to a special class of Fréchet spaces, called prequojections. The arithmetic means of the Cesàro operator on some classical sequence spaces are then studied, and finally Browder's equality, which states that for a power-bounded operator \(T\) on a reflexive Banach space \(X\), \[ (I-T)X=\left\{x\in X : \sup_{n\geq 1}\left\|\sum_{j=1}^{n}T^{j}x\right\|<+\infty\right\} \] is investigated in the Fréchet space setting.
reviewed by: Sophie Grivaux
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46A45
zbMATH DE Number: 6143657
zbMATH Keywords: Fréchet space
zbMATH Keywords: Köthe echelon space
zbMATH Keywords: quojection
zbMATH Keywords: prequojection
zbMATH Keywords: power bounded operator
zbMATH Keywords: uniformly mean ergodic operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.11.060
OpenAlex ID: W2057768672
DOI: 10.1016/J.JMAA.2012.11.060
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description: scientific article; zbMATH DE number 6143657