Equivalence after extension and Schur coupling for relatively regular operators

TL;DR

This paper investigates the relations between Equivalence After Extension and Schur Coupling for relatively regular operators, revealing new cases of their equivalence and non-equivalence on Banach spaces.

Contribution

It extends the analysis of EAE and SC relations to relatively regular operators, providing new examples and conditions for their coincidence or difference.

Findings

EAE and SC coincide in new classes of relatively regular operators.

A novel example shows EAE and SC do not always coincide on Banach spaces.

The paper reduces the problem to an equivalent Banach space operator problem.

Abstract

It was recently shown in [24] that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regular operators, the latter being operators with complementable kernels and ranges. In this paper we analyse the relations EAE and SC for the class of relatively regular operators, leading to an equivalent Banach space operator problem from which we derive new cases where EAE and SC coincide and provide a new example for which EAE and SC do not coincide and where the Banach space are not essentially incomparable.