Equivalence after extension and Schur coupling coincide for inessential operators

S. ter Horst; M. Messerschmidt; A.C.M. Ran; M. Roelands; M. Wortel

doi:10.1016/j.indag.2018.07.001

Equivalence after extension and Schur coupling coincide for inessential operators

S. ter Horst^*
, M. Messerschmidt
, A.C.M. Ran
, M. Roelands
, M. Wortel

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

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Abstract

In 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.

Original language	English
Pages (from-to)	1350-1361
Number of pages	12
Journal	Indagationes Mathematicae
Volume	29
Issue number	5
Early online date	9 Jul 2018
DOIs	https://doi.org/10.1016/j.indag.2018.07.001
Publication status	Published - Oct 2018

Funding

This work is based on research supported in part by the National Research Foundation of South Africa and the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard. Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the CoE-MaSS. This work is based on research supported in part by the National Research Foundation of South Africa and the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard. Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the CoE-MaSS.

Funders
DST-NRF Centre of Excellence in Mathematical and Statistical Sciences
National Research Foundation of South Africa
National Research Foundation

Keywords

Compact operators
Equivalence after extension
Fredholm operators
Inessential operators
Schur coupling

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10.1016/j.indag.2018.07.001

Equivalence_after_extension_and_Schur_coupling_coincide_for_inessential_operatorsFinal published version, 271 KBLicence: Article 25fa Dutch Copyright Act

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AU - ter Horst, S.

AU - Messerschmidt, M.

AU - Ran, A.C.M.

AU - Roelands, M.

AU - Wortel, M.

PY - 2018/10

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Equivalence after extension and Schur coupling coincide for inessential operators

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