On the weak compactness of weak\(^*\) Dunford-Pettis operators on Banach lattices. (English) Zbl 1380.46012

El Fahri, Kamal; H’Michane, Jawad; El Kaddouri, Abdelmonim; Aboutafail, Othmane

Adv. Oper. Theory 2, No. 3, 192-200 (2017).

Summary: 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 \(T : E\to F\) is weakly compact if, and only if, the norm of \(E'\) 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 \(T: E\to F\) is M-weakly compact if, and only if, the norms of \(E'\) and \(F\) are order continuous or \(E\) is finite-dimensional.

MSC:

46B42	Banach lattices
47B60	Linear operators on ordered spaces
47B65	Positive linear operators and order-bounded operators

Keywords:

weak\(^*\) Dunford-Pettis operator; weakly compact operator; \(M\)-weakly compact operator; order weakly compact operator; DP\(^*\) property