Ter Horst, SanneMesserschmidt, MiekRan, A.C.M.Roelands, MarkWortel, Marten2018-10-102018-10Ter Horst, S., Messerschmidt, M., Ran, A.C.M. et al. 2018, 'Equivalence after extension and Schur coupling coincide for inessential operators', Indagationes Mathematicae, vol. 29, no. 5, pp. 1350-1361.0019-357710.1016/j.indag.2018.07.001http://hdl.handle.net/2263/66836In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators that can in norm be approximated by invertible operators. In this paper we prove that the implication EAE=SC also holds for inessential Banach space operators. The inessential operators were introduced as a generalization of the compact operators, and include, besides the compact operators, also the strictly singular and strictly co-singular operators; in fact they form the largest ideal such that the invertible elements in the associated quotient algebra coincide with (the equivalence classes of) the Fredholm operators.en© 2018 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier B.V. All rights reserved. Notice : this is the author’s version of a work that was accepted for publication in Indagationes Mathematicae. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. A definitive version was subsequently published in Indagationes Mathematicae, vol. 29, no. 5, pp. 1350-1361, 2018. doi : 10.1016/j.indag.2018.07.001.Compact operatorsEquivalence after extension (EAE)Fredholm operatorsInessential operatorsSchur coupling (SC)Postprint Article