The strong Dunford-Pettis relatively compact property of order $p$

Abstract

We introduce and study Banach lattices with the strong Dunford-Pettis relatively compact property of order $p$ ($1 \le p <\infty$); that is, spaces in which every weakly $p$-compact and almost Dunford-Pettis set is relatively compact. We also introduce the notion of the weak Dunford-Pettis property of order $p$ and then characterize this property
in terms of sequences.
In particular, in terms of disjoint weakly compact operators into $c_0$, an operator characterization of those Banach lattices with the weak Dunford-Pettis property of order $p$ is given.
Moreover, some results about Banach lattices with the positive Dunford-Pettis relatively compact property of order $p$ are presented.