On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices

Document Type: Original Article

Authors

1 Ibno Tofail University

2 Moulay Ismail University

3 Universite Ibn Tofail

Abstract

We characterize Banach lattices on which each positive weak* Dunford--Pettis operator is weakly (resp., M-weakly, resp., order weakly) compact. More precisely, we prove that if $F$ is a Banach lattice with order continuous norm, then each positive weak* Dunford--Pettis operator $T : E\longrightarrow F$ is weakly compact if, and only if, the norm of $E^{\prime}$ is order continuous or $F$ is reflexive. On the other hand, when the Banach lattice $F$ is Dedekind $\sigma$-complete, we show that every positive weak* Dunford--Pettis operator $T: E\longrightarrow F$ is M-weakly compact if, and only if, the norms of $E^{\prime}$ and $F$ are order continuous or $E$ is finite-dimensional.

Keywords