Uniform convergence of operators on \(L^{\infty}\) and similar spaces
type: Item
instance of: scholarly article
title: Uniform convergence of operators on \(L^{\infty}\) and similar spaces
zbMATH Open document ID: 0623.47033
DOI: 10.1007/BF01160459
author: Heinrich P. Lotz
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173619
review text: Let E be a Banach space. It is said to have the Dunford-Pettis property if every weakly compact operator from E into any Banach space maps weakly compact sets into norm compact sets. It is called a Grothendieck space if every weak* convergent sequence in the dual \(E^*\) converges also for the weak topology of \(E^*.\) The paper is concerned with the uniform convergence of a sequence of bounded linear operators on a Grothendieck space with the Dunford-Pettis property. Sequences which are relatively compact for the strong or the weak operator topology are considered and their convergence in one of these topologies is discussed. It is shown that uniform convergence is possible under some additional hypotheses of algebraic, topological or spectral theoretic nature. The discussion is classified into (1) operators on C(X), (2) semi-groups of operators, (3) ergodic theorems, and (4) Tauberian theorems. Among the many interesting results is one which solves a problem of \textit{A. Kishimoto} and \textit{D. W. Robinson} [J. Austr. Math. Soc., Ser. A 31, 59-76 (1981; Zbl 0491.47021), p. 75], namely, any strongly continuous one-parametric semi-group of bounded linear operators on a Grothendieck space with Dunford-Pettis property is necessarily uniformly continuous.
reviewed by: Srinivasa Swaminathan
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 4011258
zbMATH Keywords: uniform convergence of a sequence of bounded linear operators on a Grothendieck space with the Dunford-Pettis property
zbMATH Keywords: semi-groups of operators
zbMATH Keywords: ergodic theorems
zbMATH Keywords: Tauberian theorems
MaRDI profile type: Publication
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OpenAlex ID: W2319894550
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description: scientific article; zbMATH DE number 4011258