Abstract
In the present paper, the notion of almost p-(V) sets is defined to characterize order bounded almost p-convergent operators from a Banach lattice E into another Banach lattice F. Also, for 1≤p≤q≤∞ we introduce the concept of almost q-(V) property of order p on Banach lattices in order to find an operator theoretic characterization of q-weakly compactness in terms of almost p-(V) sets. Moreover, some characterizations of those Banach lattices with the disjoint Dunford-Pettis property of order p are investigated. Finally, we introduce the class of almost weak p-convergent operators and we give a characterization of these operators in terms of weakly compact operators and the adjoint of almost p-convergent operators.
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