Disjointly non-singular operators on Banach lattices

TL;DR

This paper studies disjointly non-singular operators on Banach lattices, providing new characterizations and revealing different behaviors in $L_p$ spaces for $p=2$ and $p eq 2$, with applications to subspace topology.

Contribution

It introduces a perturbative characterization of $DN$-$S$ operators and analyzes their behavior specifically in $L_p$ spaces, highlighting differences at $p=2$.

Findings

$DN$-$S$ operators have a perturbative characterization.

Behavior of $DN$-$S$ operators differs between $p=2$ and $p eq 2$ in $L_p$ spaces.

Strongly embedded subspaces of $L_p$ form an open set among all closed subspaces.

Abstract

An operator $T$ from a Banach lattice $E$ into a Banach space is disjointly non-singular ($DN$-$S$, for short) if no restriction of $T$ to a subspace generated by a disjoint sequence is strictly singular. We obtain several results for $DN$-$S$ operators, including a perturbative characterization. For $E=L_p$ ($1< p<\infty$) we improve the results, and we show that the $DN$-$S$ operators have a different behavior in the cases $p=2$ and $p\neq 2$. As an application we prove that the strongly embedded subspaces of $L_p$ form an open subset in the set of all closed subspaces.