Order continuous extensions of positive compact operators on Banach lattices

TL;DR

This paper proves that order continuous positive compact operators on certain sublattices of Banach lattices can be extended to the whole lattice while preserving compactness, positivity, and order continuity.

Contribution

It establishes conditions under which positive compact operators on sublattices extend to the entire Banach lattice with preserved properties.

Findings

Extensions preserve compactness and positivity.

Unique extension for positive orthomorphisms.

Applicable to order dense sublattices.

Abstract

Let $E$ and $F$ be Banach lattices. Let $G$ be a vector sublattice of $E$ and $T: G\rightarrow F$ be an order continuous positive compact (resp. weakly compact) operators. We show that if $G$ is an ideal or an order dense sublattice of $E$, then $T$ has a norm preserving compact (resp. weakly compact) positive extension to $E$ which is likewise order continuous on $E$. In particular, we prove that every compact positive orthomorphism on an order dense sublattice of $E$ extends uniquely to a compact positive orthomorphism on $E$.