Spaces of $u\tau$-Dunford-Pettis and $u\tau$-Compact Operators on Locally Solid Vector Lattices

TL;DR

This paper explores the structure of bounded operators on locally solid vector lattices, generalizes concepts of $uaw$-Dunford-Pettis and $uaw$-compact operators, and examines their topological and lattice relations.

Contribution

It introduces conditions under which classes of bounded operators form locally solid vector lattices and extends operator notions from Banach lattices to more general settings.

Findings

Characterization of spaces where operator classes coincide

Conditions for bounded operators to form locally solid vector lattices

Relations between topological and lattice structures of operators

Abstract

Suppose $X$ is a locally solid vector lattice. It is known that there are several non-equivalent spaces of bounded operators on $X$. In this paper, we consider some situations under which these classes of bounded operators form locally solid vector lattices. In addition, we generalize some notions of $uaw$-Dunford-Pettis operators and $uaw$-compact operators defined on a Banach lattice to general theme of locally solid vector lattices. With the aid of appropriate topologies, we investigate some relations between topological and lattice structures of these operators. In particular, we characterize those spaces for which these concepts of operators and the corresponding classes of bounded ones coincide.