SAMS Subject Classification Number: 11

The relations between two bounded linear Banach space operators known as Equivalence After Extension (EAE) and Schur Coupling (SC) originated from the study of holomorphic operator functions and integral operators and provide information about the relative spectral properties of the operators near the origin, e.g., Fredholm properties, level of compactness, etc. In the application of these relations it is essential that they coincide, i.e., the operators in question are EAE if and only if they are SC. It was observed in the 1990s that at the level of bounded linear Banach space operators SC implies EAE, while the converse implication was posed as an open question that was only resolved in the negative in 2019 by considering Fredholm operators on Banach spaces that are essentially incomparable. In this talk we investigate the question whether EAE and SC coincide for Fredholm operators without the essential incomparability assumption. Surprisingly, some of the observations from the case of essentially incomparable Banach spaces remain in the general case, while, on the other hand, we show that the results for essentially incomparable Banach spaces do not extend in general when weakening the assumption to projective incomparability.

The talk is based on joint work with N.J. Laustsen.