Equivalence after extension and Schur coupling for relatively regular operators

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Authors

Ter Horst, Sanne
Messerschmidt, Miek
Ran, A.C.M.

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Springer

Abstract

It was recently shown in Ter Horst et al. (Bull Lond Math Soc 51:1005–1014, 2019) 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 spaces are not essentially incomparable.

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Keywords

Equivalence after extension, Fredholm operators, Generalized invertible operators, Relatively regular operators, Schur coupling

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Citation

Ter Horst, S., Messerschmidt, M. & Ran, A.C.M. Equivalence After Extension and Schur Coupling for Relatively Regular Operators. Integral Equations and Operator Theory 92, 40 (2020). https://doi.org/10.1007/s00020-020-02597-2.