On the class of positive almost weak$^{\star}$ Dunford-Pettis operators

summary:In this paper, we introduce and study the class of almost weak$^{\star}$ Dunford-Pettis operators. As consequences, we derive the following interesting results: the domination property of this class of operators and characterizations of the wDP$^{\star}$ property. Next, we characterize pairs of Banach lattices for which each positive almost weak$^{\star}$ Dunford-Pettis operator is almost Dunford-Pettis