Almost p-Right sets in Banach lattices and classes of operators

Abstract

The paper is devoted to those subsets in the dual of a Banach lattice E for which every almost Dunford-Pettis weakly p-summable sequence (x_n ) in E converges uniformly on them (the almost Right sets of order p). As an application, Banach lattices with the strong relatively compact Dunford-Pettis property of order p and weak Dunford-Pettis property of order p are characterized, with respect to almost Dunford-Pettis p-convergent operators. After that, the classes of disjoint Dunford-Pettis p-convergent operators and positively unconditionally p-converging operators are studied. It is established that positively unconditionally p-converging operators are exactly the disjoint p-convergent operators for all p ≥ 2.