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almost weak$^{\star}$ Dunford-Pettis operator; almost Dunford-Pettis operator; weak Dunford-Pettis$^{\star}$ property; positive Schur property; order continuous norm
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.
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