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

Recently, J. H'michane et al. 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.