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

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.

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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

Information

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

zbMATH: 1380.46012
MathSciNet: MR3730048
Digital Object Identifier: 10.22034/aot.1612-1078

Subjects:
Primary: 46B42
Secondary: 47B60 , 47B65

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

Rights: Copyright © 2017 Tusi Mathematical Research Group

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Vol.2 • No. 3 • Summer 2017
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