The \(b\)-weak compactness of weak Banach-Saks operators. (Q2856914)

Let \(T\:X\to Y\) be a bounded operator between Banach spaces. \(T\) is called weakly Banach-Saks if, for each weakly null sequence \((x_k)\) in \(X\), the sequence \((Tx_k)\) has a Cesàro convergent subsequence. If \(X\) is, moreover, a Banach lattice, then \(T\) is called \(b\)-weakly compact if it maps each \(b\)-order bounded set in \(X\) (i.e., a subset of \(X\) which is order-bounded in the bidual) to a relatively weakly compact set in \(Y\).NEWLINENEWLINEThe main result is as follows. Let \(E\) and \(F\) be Banach lattices such that the norm on \(E\) is order continuous. Then the following assertions are equivalent.NEWLINENEWLINE(1) Each operator \(T\: E\to F\) is \(b\)-weakly compact.NEWLINENEWLINE(2) Each positive weak Banach-Saks operator \(T:E\to F\) is \(b\)-weakly compact.NEWLINENEWLINE(3) Either \(E\) or \(F\) is a KB-space.NEWLINENEWLINEReviewer's remark: In the summary, the authors claim that they give a characterization of Banach lattices on which every Banach-Saks operator is \(b\)-weakly compact. Yet in fact they provide such a characterization only within lattices with order continuous norm.