Banach lattices with the positive Dunford-Pettis relatively compact property

Abstract

The paper is devoted to such Banach lattices E that every Dunford-Pettis and weakly null sequence (x_n) ⊂ E with disjoint terms is norm null (the positive Dunford-Pettis relatively compact property). It is established that a Banach lattice E has the positive Dunford-Pettis relatively compact property if and only if its almost Dunford-Pettis subsets are L-weakly compact. Consequently, we derive the following result: Banach lattices with the property that their almost Dunford-Pettis subsets are relatively compact, are precisely the discrete KB-spaces.