Equality of quasi-norm-attaining and norm-attaining operators

Determine whether there exists an infinite-dimensional Banach space X for which the class of quasi-norm-attaining operators equals the class of norm-attaining operators, that is, whether \(QNA(X,X)=NA(X,X)\) can hold.

Background

Quasi-norm attainment requires the closure of the image of the unit ball to meet the sphere of radius equal to the operator norm, whereas norm attainment requires an actual unit vector at which the norm is attained. The paper notes that it is unknown whether these two classes can coincide for all operators on some infinite-dimensional Banach space.

The authors' renorming theorem shows that every infinite-dimensional Banach space admits an equivalent norm for which quasi-norm-attaining operators do not exhaust all bounded operators. Thus, even if the equality holds for some particular normed structure, it is not preserved under all equivalent renormings.

References

To the best of the authors' knowledge, it is unknown whether there exists an infinite-dimensional Banach space $X$ such that $QNA(X,X)=NA(X,X)$ (see Problem 7.11).

— On operators whose adjoints or second adjoints attain their norms  (2609.00794 - Dantas et al., 1 Sep 2026) in Section 6, Section 6.1? final Remark following Theorem \ref{Ostrovskii-for-NA1-and-NA2}