Advances in Operator Theory

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

El Fahri Kamal, H'michane Jawad, El Kaddouri Abdelmonim, and Aboutafail Moulay Othmane

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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.

Article information

Adv. Oper. Theory, Volume 2, Number 3 (2017), 192-200.

Received: 12 December 2016
Accepted: 17 March 2017
First available in Project Euclid: 4 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B42: Banach lattices [See also 46A40, 46B40]
Secondary: 47B60: Operators on ordered spaces 47B65: Positive operators and order-bounded operators

weak* Dunford–Pettis operator weakly compact operator M-weakly compact operator order weakly compact operator DP* property


Kamal, El Fahri; Jawad, H'michane; Abdelmonim, El Kaddouri; Moulay Othmane, Aboutafail. On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices. Adv. Oper. Theory 2 (2017), no. 3, 192--200. doi:10.22034/aot.1612-1078.

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