Niels Laustsen (Lancaster University)

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

Abstract: I shall report on joint work with Sanne ter Horst (North-West University, South Africa), in which we study two relations for bounded operators on Banach spaces: equivalence after extension (EAE) and Schur coupling (SC). They originate in the study of integral equations and have found numerous applications, often relying on a proof that they coincide in the case at hand. More precisely, it has been known for 30 years that SC implies EAE, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC. Motivated by this result, we investigate when EAE and SC coincide for Fredholm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer n and every pair of Banach spaces (X,Y), either no pair of Fredholm operators of index n acting on X and Y, respectively, is SC, or every pair of this kind which is EAE is also SC. Consequently, whether EAE and SC coincide for Fredholm operators of index n depends only on the geometry of the underlying Banach spaces X and Y, not on the properties of the operators themselves. I intend to make the talk accessible to a broad audience of analysts; in particular, I shall not assume any prior knowledge of EAE and SC or specialist knowledge of Banach space theory.