Equivalence after extension and Schur coupling do not coincide, on essentially incomparable Banach spaces
| Autor: | ter Horst, Sanne, Messerschmidt, Miek, Ran, Andre C. M., Roelands, Mark |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1112/blms.12292 |
| 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: | arXiv |
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