Almost band preservers

TL;DR

This paper investigates the stability of band preserving operators on Banach lattices by introducing psilon-band preserving mappings, showing they are close to true band preserving operators under general conditions, with some exceptions and continuity results.

Contribution

It introduces psilon-band preserving mappings, demonstrating their proximity to band preserving operators and exploring automatic continuity, advancing understanding of operator stability in Banach lattices.

Findings

psilon-band preserving operators are close to band preserving operators under general assumptions.

Counterexamples exist where psilon-band preserving operators are not close to band preserving operators.

Automatic continuity results for psilon-band preserving maps are established.

Abstract

We study the stability of band preserving operators on Banach lattices. To this end the notion of $\varepsilon$-band preserving mapping is introduced. It is shown that, under quite general assumptions, a $\varepsilon$-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of $\varepsilon$-band preserving maps are also obtained.