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Following the concept of L–limited sets in dual Banach spaces introduced by Salimi and Moshtaghioun, we introduce the concepts of L–Dunford–Pettis and almost L–Dunford–Pettis sets in dual Banach lattices and then by a class of operators on Banach lattices, so called disjoint Dunford–Pettis completely continuous operators, we characterize Banach lattices in which almost L–Dunford–Pettis subsets of their dual, coincide with L–Dunford–Pettis sets.

L-Dunford–Pettis and Almost L-Dunford–Pettis Sets in Dual Banach Lattices