Weak Unbounded Norm Topology and Dounford-Pettis Operators

TL;DR

This paper explores the properties of the weakly unbounded norm topology on Banach lattices, compares it with existing topologies, and introduces $wun$-Dunford-Pettis operators, analyzing their characteristics and relationships with other operators.

Contribution

It introduces the $wun$-topology on Banach lattices and studies $wun$-Dunford-Pettis operators, providing new insights into their properties and connections with known operator classes.

Findings

$E^* = ext{ud}(E)$ if $E^*$ has order continuous norm

The $wun$-topology is distinct from weak and $uaw$-topologies

Properties and relationships of $wun$-Dunford-Pettis operators are characterized

Abstract

In this paper, we study $un$-dual (in symbol, $\ud{E}$) of Banach lattice $E$ and compare it with topological dual $E^*$. If $E^*$ has order continuous norm, then $E^* = \ud{E}$. We introduce and study weakly unbounded norm topology ($wun$-topology) on Banach lattices and compare it with weak topology and $uaw$-topology. In the final, we introduce and study $wun$-Dunford-Pettis opertors from a Banach lattice $E$ into Banach space $X$ and we investigate some of its properties and its relationships with others known operators.