Topological characterisation of weakly compact operators
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).
scientific article; zbMATH DE number 5078869
scientific article