Inner functions as strongly extreme points: stability properties

Abstract

Given a Banach space $\mathcal{X}$, let $x$ be a point in $\text{ball}(\mathcal{X})$, the closed unit ball of $\mathcal{X}$. One says that $x$ is a strongly extreme point of $\text{ball}(\mathcal{X})$ if it has the following property: for every $\epsilon >0$ there is $\delta >0$ such that the inequalities $\Vert x\pm y\Vert <1+\delta$ imply, for $y\in \mathcal{X}$, that $\Vert y\Vert <\epsilon$. 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 finite-dimensional perturbations of $H^{\infty }$. It is well known that the strongly extreme points of $\text{ball}(H^{\infty })$ 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.

10 pages

References in corpus (3)

• Lacunary polynomials in $L^1$: geometry of the unit sphere
• Nearly outer functions as extreme points in punctured Hardy spaces
• A Rudin--de Leeuw type theorem for functions with spectral gaps

Cited by in corpus (1)

• Extremal problems in BMO and VMO involving the Garsia norm