Domination by positive weak* Dunford-Pettis operators on Banach lattices

Recently, J. H'michane et al. [‘On the class of limited operators’, Acta Math. Sci. (submitted)] introduced the class of weak\(^*\) Dunford-Pettis operators on Banach spaces, that is, operators which send weakly compact sets onto limited sets. In this paper the domination problem for weak\(^*\) Dunford-Pettis operators is considered. Let \(S, T:E\rightarrow F\) be two positive operators between Banach lattices \(E\) and \(F\) such that \(0\leq S\leq T\). We show that if \(T\) is a weak\(^{*}\) Dunford-Pettis operator and \(F\) is \(\sigma\)-Dedekind complete, then \(S\) itself is weak\(^*\) Dunford-Pettis. DOI: 10.1017/S000497271400032