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

Research output: Contribution to JournalArticleAcademicpeer-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 languageEnglish
Pages (from-to)1350-1361
Number of pages12
JournalIndagationes Mathematicae
Volume29
Issue number5
Early online date9 Jul 2018
DOIs
Publication statusPublished - Oct 2018

Keywords

  • Compact operators
  • Equivalence after extension
  • Fredholm operators
  • Inessential operators
  • Schur coupling

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