Sharp Invertibility in Quotient Algebras of $H^\infty$

TL;DR

The paper investigates Sharp Invertibility in quotient algebras $H^\infty/\Theta H^\infty$ by linking SIP to maximal asymptotic growth (MAG) of $|\Theta|$ away from its zero set via $\eta_\Theta$ and $\varkappa_\Theta$. It proves that SIP is equivalent to MAG and explores stability under finite products, then characterizes SIP through narrow sublevel sets and connects it with the Weak Embedding Property (WEP), Carleson-Newman (CN) Blaschke products, and Frostman shifts. It further relates a Beurling-Carleson entropy condition on boundary sets to divisors of SIP inner functions and provides extensive examples, including thin Blaschke products, Stolz-angle zeros, and perturbation stability, to map the SIP landscape within $H^\infty$. The results illuminate how invertibility in $H^\infty/\Theta H^\infty$ is governed by geometric and boundary-measure properties of $\Theta$, with implications for divisors, level-set geometry, and entropy-type criteria. Overall, the work clarifies when SIP holds and how it interacts with WEP, CN structure, and zero-set geometry, enriching the theory of function algebras on the disc.

Abstract

We consider inner functions $Θ$ with the zero set $\mathcal Z(Θ)$ such that the quotient algebra $H^\infty / ΘH^\infty$ satisfies the Strong Invertibility Property (SIP), that is for every $\varepsilon>0$ there exists $δ>0$ such that the conditions $f \in H^\infty$, $\|[f]\|_{H^\infty/ ΘH^\infty}=1$, $\inf_{\mathcal Z(Θ)} |f| \ge 1-δ$ imply that $[f]$ is invertible in $H^\infty / ΘH^\infty$ and $\| 1/ [f] \|_{H^\infty/ ΘH^\infty}\le 1+\varepsilon$. We prove that the SIP is equivalent to the maximal asymptotic growth of $Θ$ away from its zero set. We also describe inner functions satisfying the SIP in terms of the narrowness of their sublevel sets and relate the SIP to the Weak Embedding Property introduced by P.Gorkin, R.Mortini, and N.Nikolski as well as to inner functions whose Frostman shifts are Carleson--Newman Blaschke products. We finally study divisors of inner functions satisfying the SIP. We describe geometrically the zero set of inner functions such that all its divisors satisfy the SIP. We also prove that a closed subset $E$ of the unit circle is of finite entropy if and only if any singular inner function associated to a singular measure supported on $E$ is a divisor of an inner function satisfying the SIP.

Theorems & Definitions (55)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1
• Proposition 2.2