Topological characterisation of weakly compact operators
type: Item
instance of: scholarly article
title: Topological characterisation of weakly compact operators
zbMATH Open document ID: 1117.47014
author: Antonio M. Peralta
author: Ignacio Villanueva
author: J. D. Maitland Wright
author: K. Ylinen
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: Let \(X\) be Banach a space. The main result of this paper identifies a locally convex topology \(\tau_R\) on \(X\) which has the property that a linear map \(T: X \to Y\) is weakly compact if and only if \(T\) is continuous \((X,\tau_R) \to Y\), where the Banach space \(Y\) is equipped with the norm topology. Here, \(\tau_R\) is the topology induced on \(X\) by the Mackey topology of the dual pair \((X^{**},X^*)\), that is, the topology on \(X^{**}\) defined by uniform convergence on absolutely convex, \(w\)-compact subsets \(K\subset X^*\), see also \textit{J.--H.\thinspace Qiu} [Proc.\ Am.\ Math.\ Soc.\ 129, No.\,5, 1419--1425 (2001; Zbl 0967.46002)]. The authors also discuss the class of operators \(T:X\to Y\), called pseudo weakly compact, for which \(T\) is sequentially continuous \((X,\tau_R)\to Y\). They observe that any pseudo weakly compact operator is unconditionally convergent and that the class of Banach spaces \(X\) for which every pseudo weakly compact operator \(X\to Y\) is weakly compact includes the \(C^*\)-algebras and the spaces having Peł{c}zynski's property (V).
reviewed by: Hans-Olav Tylli
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47L05
Mathematics Subject Classification ID: 46B99
zbMATH DE Number: 5078869
zbMATH Keywords: weakly compact operators
zbMATH Keywords: Mackey topology
zbMATH Keywords: right topology
Wikidata QID: Q62470999
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.02.066
OpenAlex ID: W2033109060
cites work: \(JB^*\)-triples have Pełczynski's property V
cites work: Q3329946
cites work: Q5569793
cites work: Local completeness and dual local quasi-completeness
cites work: Q5664372
cites work: Theory of operator algebras I.
cites work: Multilinear maps on products of operator algebras
DOI: 10.1016/J.JMAA.2006.02.066
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description: scientific article; zbMATH DE number 5078869