Inner functions as strongly extreme points: stability properties

Abstract

Given a Banach space \(\mathcal X\), let \(x\) be a point in ball\((\mathcal X)\), the closed unit ball of \(\mathcal X\). We say that \(x\) is a strongly extreme point of ball\((\mathcal X)\) if it has the following property: for every \(\varepsilon>0\) there is \(\delta>0\) such that the inequalities \(\|x\pm y\|<1+\delta\) imply, for \(y\in\mathcal X\), that \(\|y\|<\varepsilon\). We are concerned with certain subspaces of \(H^\infty\), the space of bounded holomorphic functions on the disk, that arise upon imposing finitely many linear constraints and can be viewed as small perturbations of \(H^\infty\). It is well known that the strongly extreme points of ball\((H^\infty)\) are precisely the inner functions, while the (usual) extreme points of this ball are the unit-norm functions \(f\in H^\infty\) with \(\log(1-|f|)\) non-integrable over the circle. Here we show that similar characterizations remain valid for our perturbed \(H^\infty\)-type spaces. Also, we investigate to what extent a non-inner function can differ from a strongly extreme point.