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 $S,T:E\to F$ be two positive operators between Banach lattices $E$ and $F$ such that $0\leq S\leq T$\mathrm{.}Weshowthatif$T$isaweak$^{}$Dunford-Pettisoperatorand$F$is$\sigma$-Dedekindcomplete,then$S$ itself is weak Dunford-Pettis.