Characterization of dense minimum-attaining operator pairs

Characterize all pairs of Banach spaces $(X,Y)$ for which the set of minimum-attaining bounded linear operators from $X$ into $Y$ is dense in the space of bounded linear operators $L(X,Y)$.

Background

The paper discusses minimum-attaining operators as the linear analogue of strongly minimum-attaining Lipschitz maps. While existing results relate density of minimum-attaining operators to geometric properties such as the Radon–Nikodým property, they do not provide a complete characterization for individual domain–range pairs.

References

A complete characterization of the individual pairs $(X,Y)$ for which $\MA(X,Y)$ is dense in $L(X,Y)$, however, is still unknown.

— A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps  (2609.11080 - Choi, 10 Sep 2026) in Section 1, Introduction