Some properties on the class of $\sigma$-un-Dunford-Pettis operators in Banach lattice

Abstract

In this work, we studied more properties concerning the class of $\sigma$-un-Dunford-Pettis operators on Banach lattices. Precisely, we present a characterization of $\sigma$-un-Dunford-Pettis operators, it's an extension of Proposition 3.2 obtained by N. Hafidi et al in their article, and another characterization of Banach lattices for which each $\sigma$-un-Dunford-Pettis operator $T:E\to E$ is $\sigma$-un-compact (respectively, $\sigma$-uaw-compact) and $T^{2}$ is $\sigma$-un-compact. Additionally, we give sufficient conditions under which $T$ is a $\sigma$-un-Dunford-Pettis operator if and only if $|T|$ is a $\sigma$-un-Dunford-Pettis operator. Furthermore, we introduce a new property that generalizes the Dunford-Pettis property, which we call the $\sigma$-un-Dunford-Pettis property. After that, we investigate proprieties about this new property. On the other side, we examined its connections with the other classes of operators as u-M-weakly compact and AM-$\sigma$-un-compact. Finally, we presented a necessary and sufficient condition under which for each $\sigma$-un-Dunford-Pettis operator is u-M-weakly compact and under which for each AM-$\sigma$-un-compact is $\sigma$-un-Dunford-Pettis operator.