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.
Citation
El Fahri Kamal. H'michane Jawad. El Kaddouri Abdelmonim. Aboutafail Moulay Othmane. "On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices." Adv. Oper. Theory 2 (3) 192 - 200, Summer 2017. https://doi.org/10.22034/aot.1612-1078
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