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

Autor: Miek Messerschmidt, Sanne ter Horst, André C. M. Ran, Mark Roelands
Přispěvatelé: 24116327 - Ter Horst, Sanne, 20000212 - Ran, Andreas Cornelis Maria, Mathematics
Jazyk: angličtina
Rok vydání: 2019
Předmět:
Zdroj: Horst, S T, Messerschmidt, M, Ran, A C M & Roelands, M 2019, ' Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces ', Bulletin of the London Mathematical Society, vol. 51, no. 6, pp. 1005-1014 . https://doi.org/10.1112/blms.12292
Bulletin of the London Mathematical Society, 51(6), 1005-1014. Oxford University Press
ISSN: 0024-6093
Popis: In 1994 H. Bart and V.\'{E}. 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 $\Rightarrow$ 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 characterization of the notion of essential incomparability. Concretely, the forward shift operators $U$ on $\ell^p$ and $V$ on $\ell^q$, for $1\leq p,q\leq \infty$, $p\neq q$, are EAE but not SC. As a corollary, SC is not transitive. Under mild assumptions, given $U$ and $V$ that are Atkinson or generalized invertible and EAE, we give a concrete operator $W$ that is SC to both $U$ and $V$, even if $U$ and $V$ are not SC themselves. Some further affirmative results for the case where the Banach spaces are isomorphic are also obtained.
Comment: 10 pages
Databáze: OpenAIRE
Externí odkaz: