Aqzzouz, Belmesnaoui; H’michane, Jawad b-weak compactness of weak Dunford-Pettis operators. (English) Zbl 1279.46011 Oper. Matrices 7, No. 1, 219-224 (2013). The aim of this paper is to study when weak Dunford-Pettis operators are b-weakly compact. Recall that an operator from a Banach lattice \(E\) into a Banach space \(X\) is called weakly Dunford-Pettis if for each weakly null sequence \((x_n)\) in \(E\) and weakly null sequence \((y_n')\) in \(X'\), we have \(y'_n(Tx_n)\rightarrow 0\). An operator \(T:E\rightarrow X\) is called b-weakly compact if for each disjoint sequence in \(E\) which is order bounded in \(E''\), the bidual of \(E\), we have that \(||T(x_n)||\rightarrow 0\).The main results of the paper are as follows:Theorem. Suppose that the Banach lattice \(E\) has order continuous norm. Each positive weak Dunford-Pettis operator \(E\rightarrow F\) is b-weakly compact if and only if one of \(E\) or \(F\) is a KB-space.Theorem. Let \(E,F\) be Banach lattices such that \(F\) is Dedekind \(\sigma\)-complete. If each weak Dunford-Pettis operator \(T:E\rightarrow F\) is b-weakly compact, then \(E\) is a KB-space or \(F\) has order continuous norm. Reviewer: Şafak Alpay (Ankara) Cited in 1 Document MSC: 46B42 Banach lattices 47B07 Linear operators defined by compactness properties 47B60 Linear operators on ordered spaces Keywords:b-weakly compact operator; weak Dunford-Pettis operator; Banach lattice; (b)-property; order continuous norm; KB-space PDFBibTeX XMLCite \textit{B. Aqzzouz} and \textit{J. H'michane}, Oper. Matrices 7, No. 1, 219--224 (2013; Zbl 1279.46011) Full Text: DOI Link