## Advances in Operator Theory

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

### 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

#### 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.

#### Article information

**Source**

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

**Dates**

Received: 12 December 2016

Accepted: 17 March 2017

First available in Project Euclid: 4 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1512431670

**Digital Object Identifier**

doi:10.22034/aot.1612-1078

**Mathematical Reviews number (MathSciNet)**

MR3730048

**Zentralblatt MATH identifier**

1380.46012

**Subjects**

Primary: 46B42: Banach lattices [See also 46A40, 46B40]

Secondary: 47B60: Operators on ordered spaces 47B65: Positive operators and order-bounded operators

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aot/1512431670