2023/01/26 by Konstantin M. Dyakonov, Dyakonov, Konstantin M.
Mathematics · #30H05 #30H10 #30J05 #46A55 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2301.11162
openalex publication_date 2023/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Banach space \mathcal X, let x be a point in ball(\mathcal X), the closed unit ball of \mathcal X. One says that x is a strongly extreme point of ball(\mathcal X) if it has the following property: for every ε>0 there is δ>0 such that the inequalities ‖x± y‖<1+δ imply, for y∈\mathcal X, that ‖y‖<ε. We are concerned with certain subspaces of H^∞, 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^∞. It is well known that the strongly extreme points of ball(H^∞) are precisely the inner functions, while the (usual) extreme points of this ball are the unit-norm functions f∈ H^∞ with log(1-|f|) non-integrable over the circle. Here we show that similar characterizations remain valid for our perturbed H^∞-type spaces. Also, we investigate to what extent a non-inner function can differ from a strongly extreme point.