The Krengel's theorem for compact operators between locally solid vector lattices

TL;DR

This paper extends Krengel's theorem to various types of compact operators on locally solid vector lattices, exploring the $AM$-property and its implications.

Contribution

It introduces a variant of Krengel's theorem applicable to different compact operator types in locally solid vector lattices.

Findings

Established a new version of Krengel's theorem for locally solid vector lattices.

Analyzed the $AM$-property in the context of compact operators.

Connected the $AM$-property with compactness notions in vector lattices.

Abstract

Suppose $X$ is a locally solid vector lattice. It is known that there are several non-equivalent notions for compact operators on $X$. Furthermore, notion of the $AM$-property in $X$ as an extension for the $AM$-spaces in Banach lattices has been considered, recently. In this paper, we establish a variant of the known Krengel's theorem for different types of compact operators between locally solid vector lattices.