Kamal, E., Jawad, H., Abdelmonim, E., Moulay Othmane, A. (2017). On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices. Advances in Operator Theory, 2(3), 192-200. doi: 10.22034/aot.1612-1078
El Fahri Kamal; H'michane Jawad; El Kaddouri Abdelmonim; Aboutafail Moulay Othmane. "On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices". Advances in Operator Theory, 2, 3, 2017, 192-200. doi: 10.22034/aot.1612-1078
Kamal, E., Jawad, H., Abdelmonim, E., Moulay Othmane, A. (2017). 'On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices', Advances in Operator Theory, 2(3), pp. 192-200. doi: 10.22034/aot.1612-1078
Kamal, E., Jawad, H., Abdelmonim, E., Moulay Othmane, A. On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices. Advances in Operator Theory, 2017; 2(3): 192-200. doi: 10.22034/aot.1612-1078
On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices
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.