On Disjointly singular centralizers

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