Characterization of norm-attaining operator pairs

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

Background

The paper places its Lipschitz minimum-attainment problem in the broader context of the Bishop–Phelps theory for norm-attaining operators. Although norm-attaining functionals can be densely approximated, norm-attaining operators need not be dense in general. A complete geometric characterization of the domain–range pairs for which density holds remains unresolved.

References

Nevertheless, a complete characterization of the pairs $(X,Y)$ for which $\NA(X,Y)$ is dense in $L(X,Y)$ remains unknown.

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