Uniform convergence of operators and Grothendieck spaces with the Dunford-Pettis property
type: Item
instance of: scholarly article
title: Uniform convergence of operators and Grothendieck spaces with the Dunford-Pettis property
zbMATH Open document ID: 0618.46023
DOI: 10.1007/BF01161628
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183756
review text: A theorem of D. W. Dean states that all Grothendieck spaces with the Dunford-Pettis property fail to admit Schauder decompositions. Similar results of H. P. Lotz concern the uniform convergence of ergodic operators and semigroups of operators. This paper considers extensions of these results to wider classes of Banach spaces. We say that a Banach space E has the surjective Dunford-Pettis property if every operator from E onto a reflexive Banach space maps weakly compact sets onto norm compact sets. The main result can now be stated. Theorem. Let \(E\) be a Grothendieck space with the surjective Dunford- Pettis property, then (1) The space \(E\) admits no Schauder decompositions; (2) Every strongly ergodic operator \(T\) on \(E\) with \(\| T^ n/n\| \to 0\) is uniformly ergodic; and (3) Every strongly continuous semi-group of operators \((T_ t)_{t\geq 0}\) on E is uniformly continuous. Examples are presented to distinguish the surjective Dunford-Pettis property from the closely related Dunford- Pettis property.
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 4002803
zbMATH Keywords: Grothendieck spaces with the Dunford-Pettis property
zbMATH Keywords: Schauder decompositions
zbMATH Keywords: uniform convergence of ergodic operators
zbMATH Keywords: semigroups of operators
zbMATH Keywords: surjective Dunford-Pettis property
author: Denny H. Leung
MaRDI profile type: Publication
cites work: Schauder Decompositions in (m)
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cites work: On subspaces of L\(^p\)
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description: scientific article; zbMATH DE number 4002803