Banach lattice versions of strict singularity

TL;DR

This paper investigates the relationship between two types of strictly singular operators in Banach lattices, providing new results that clarify when these classes coincide.

Contribution

It introduces new results that clarify the conditions under which disjointly strictly singular and lattice strictly singular operators are equivalent.

Findings

Disjointly strictly singular operators coincide with lattice strictly singular operators under certain conditions.

New criteria are established for the equivalence of these two classes of operators.

The paper advances understanding of operator invertibility in Banach lattices.

Abstract

We explore the relation between lattice versions of strict singularity for operators from a Banach lattice to a Banach space. In particular, we study when the class of disjointly strictly singular operators, those not invertible on the span of any disjoint sequence, coincides with that of lattice strictly singular operators, i.e. those not invertible on any (infinite dimensional) sublattice. New results are given which help to clarify the existing relation between these two classes.