Gelfand-Phillips-type properties in Banach lattices

TL;DR

This paper explores properties related to $p$-limited sets in Banach lattices, examining their connections with compactness and weak compactness, and characterizes KB-spaces via these properties.

Contribution

It introduces conditions under which various $p$-Gelfand-Phillips properties coincide and characterizes KB-spaces through $p$-limited set properties.

Findings

Characterization of KB-spaces via $p$-limited sets.

Conditions for the equivalence of $p$-Gelfand-Phillips properties.

Connections between $p$-limited sets and compactness in Banach lattices.

Abstract

We study $p$-limited and almost $p$-limited sets in Banach lattices and their connections with relatively $p$-compact and relatively compact sets. We investigate the weak and the strong Gelfand-Phillips property of order $p$, as well as the $p$-GP property introduced by Delgado and Pi\~neiro, providing conditions under which these properties may coincide. Additionally, we prove that a Banach lattice $E$ is a KB-space if and only if every almost $p$-limited set in $E$ is relatively weakly compact if and only if every the adjoint of a weakly compact taking values on $E$ is disjoint $p$-summing.