On norm-attaining positive operators between Banach lattices

TL;DR

This paper investigates conditions under which positive operators between Banach lattices attain their norm, establishing equivalences with compactness and applying these results to positive weak maximizing properties.

Contribution

It introduces an absolute version of James boundaries to characterize norm-attainment of positive operators between specific Banach lattices, linking it to compactness.

Findings

Positive operators from reflexive Banach lattices with a basis are compact iff they attain their norm.

Analogous results hold when the order in the codomain is continuous and given by a basis.

Results are applied to analyze a positive version of the weak maximizing property.

Abstract

In this paper we study the norm-attainment of positive operators between Banach lattices. By considering an absolute version of James boundaries, we prove that: If $E$ is a reflexive Banach lattice whose order is given by a basis and $F$ is a Dedekind complete Banach lattice, then every positive operator from $E$ to $F$ is compact if and only if every positive operator from $E$ to $F$ attains its norm. An analogue result considering that $E$ is reflexive and the order in $F$ is continuous and given by a basis was proven. We applied our result to study a positive version of the weak maximizing property.