Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

S. ter Horst, M. Messerschmidt, A. C.M. Ran, M. Roelands

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In 1994, H. Bart and V. É. Tsekanovskii posed the question whether the Banach space operator relations matricial coupling (MC), equivalence after extension (EAE) and Schur coupling (SC) coincide, leaving only the implication EAE/MC (Formula presented.) SC open. Despite several affirmative results, in this paper we show that the answer in general is no. This follows from a complete description of EAE and SC for the case that the operators act on essentially incomparable Banach spaces, which also leads to a new characterisation of the notion of essential incomparability. Concretely, the forward shift operators (Formula presented.) on (Formula presented.) and (Formula presented.) on (Formula presented.), for (Formula presented.), (Formula presented.), are EAE but not SC. As a corollary, SC is not transitive. Under mild assumptions, given (Formula presented.) and (Formula presented.) that are Atkinson or generalised invertible and EAE, we give a concrete operator (Formula presented.) that is SC to both (Formula presented.) and (Formula presented.), even if (Formula presented.) and (Formula presented.) are not SC themselves. Some further affirmative results for the case where the Banach spaces are isomorphic are also obtained.

Original languageEnglish
Pages (from-to)1005-1014
Number of pages10
JournalBulletin of the London Mathematical Society
Issue number6
Early online date18 Oct 2019
Publication statusPublished - Dec 2019


  • 47A53 (secondary)
  • 47A62 (primary)


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