# On Disjointly singular centralizers

## TL;DR

This paper explores various notions of disjoint singularity in quasi-linear maps within K"othe function spaces, revealing their relationships and distinctions, especially in superreflexive and non-atomic contexts.

## Contribution

It introduces and analyzes disjoint versions of triviality and singularity for centralizers, establishing key equivalences and non-existence results in specific Banach space settings.

## Key findings

- Trivial and disjoint trivial notions coincide on reflexive spaces.
- No singular centralizers exist on non-atomic superreflexive K"othe spaces.
- Super disjointly singular centralizers include Kalton-Peck centralizers.

## Abstract

We study ``disjoint" versions of the notions of trivial, locally trivial, strictly singular and super-strictly singular quasi-linear maps in the context of K\"othe function spaces. Among other results, we show: i) (locally) trivial and (locally) disjointly trivial notions coincide on reflexive spaces; ii) On non-atomic superreflexive K\"othe spaces, no centralizer is singular, although most are disjointly singular. iii) No super singular quasi-linear maps exist between superreflexive spaces although Kalton-Peck centralizers are super disjointly singular; iv) Disjoint singularity does not imply super disjoint singularity.