Characterization of strongly extreme points in H^∞/E

Develop a characterization of the strongly extreme points of the unit ball in the quotient space H^∞/E, where E is a proper weak-star closed subspace of H^∞. Here, a unit-norm element x in a Banach space X is strongly extreme if for every ε>0 there exists δ>0 such that ||x±y||<1+ε implies ||y||<δ for all y∈X.

Background

The paper proves a characterization of extreme points of the unit ball in H∞/E when E is a proper weak-star closed subspace of H∞: a unit-norm coset f+E is extreme if and only if it has a unique norm-attaining representative h∈H∞ with log(1−|h|) not integrable. This is Theorem 1.3 in the introduction.

The authors raise the question of extending such results to strongly extreme points in the same quotient spaces. In H∞ (without taking quotients), strongly extreme points are precisely the inner functions, but an analogous description for H∞/E is not known.

References

Finally, we mention an open question that comes to mind in light of Theorem \ref{thm:hinfty}. Namely, we wonder whether that theorem could be adjusted to yield a description of the {\it strongly extreme} points of $\text{\rm ball}\,(H\infty/E)$, with $E$ as above.

— Extreme points in quotients of Hardy spaces  (2603.29103 - Dyakonov, 31 Mar 2026) in Final paragraph, Section 1 (Introduction and statement of results)