Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology

TL;DR

This paper studies disjointly non-singular (DNS) operators on order continuous Banach lattices, showing their openness and characterizing them via topology, and draws parallels with semi-Fredholm operators in Banach space theory.

Contribution

It proves DNS operators form an open subset in order continuous Banach lattices and characterizes them through topological complementarity, extending understanding of their structure.

Findings

DNS operators form an open subset in $L(F,E)$ when $F$ is order continuous

A DNS operator is characterized by complementing the unbounded norm topology

Parallel results are established for semi-Fredholm operators and the weak topology

Abstract

In this article we investigate the disjointly non-singular (DNS) operators. Following [8] we say that an operator $T$ from a Banach lattice $F$ into a Banach space $E$ is DNS, if no restriction of $T$ to a subspace generated by a disjoint sequence is strictly singular. We partially answer a question from [8] by showing that this class of operators forms an open subset of $L\left(F,E\right)$ as soon as $F$ is order continuous. Moreover, we show that in this case $T$ is DNS if and only if the norm topology is the minimal topology which is simultaneously stronger than the unbounded norm topology and the topology generated by $T$ as a map (we say that $T$ "complements" the unbounded norm topology in $F$). Since the class of DNS operators plays a similar role in the category of Banach lattices as the upper semi-Fredholm operators play in the category of Banach spaces, we investigate and indeed uncover a similar characterization of the latter class of operators, but this time they have to complement the weak topology.