Equivalence after extension and Schur coupling for Fredholm operators on Banach spaces

Sanne ter Horst, Niels Jakob Laustsen

Published 2022-10-30Version 1

In 2019 a pair of bounded Banach space operators was constructed which are equivalent after extension (EAE), but not Schur coupled (SC), thereby answering a question posed by Bart and Tsekanovskii in the early 1990s. The operators in question are Fredholm operators defined on a pair of Banach spaces which are essentially incomparable. For context, we remark that SC always implies EAE, and that the two notions coincide for Fredholm operators defined on isomorphic Banach spaces. In the present paper we investigate EAE and SC for Fredholm operators without (in)comparability conditions on the underlying Banach spaces. Fredholm operators that are EAE must have the same Fredholm index, and hence the same is true for SC. Surprisingly, the set of pairs $(U,V)$ of SC Fredholm operators of a fixed index $k$, where $U$ acts on the Banach space $X$ and $V$ on $Y$, is either empty or equal to the set of pairs of Fredholm operators of index $k$ acting on $X$ and $Y$, respectively, that are EAE. This dichotomy implies that the question whether EAE and SC coincide for Fredholm operators of index $k$ depends only on the geometry of the underlying Banach spaces $X$ and $Y$, not on the properties of the operators themselves. We analyse the question whether EAE and SC coincide for Fredholm operators further and define two numerical indices, $\operatorname{eae}(X,Y)$ and $\operatorname{sc}(X,Y)$, which capture the coincidence of EAE and SC for pairs $(U,V)$ of Fredholm operators acting on $X$ and $Y$, respectively. Finally, we provide a number of examples illustrating the possible values of the indices $\operatorname{eae}(X,Y)$ and $\operatorname{sc}(X,Y)$ and their interplay with the geometry of the Banach spaces $X$ and $Y$. These examples are based on certain ``exotic'' Banach spaces constructed by Gowers and Maurey, and also use ideas from subsequent work of Aiena, Gonzalez and Ferenczi.