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

TL;DR

This paper characterizes when quotient algebras of bounded analytic functions satisfy a strong invertibility property, linking it to the growth of inner functions and their geometric zero set properties.

Contribution

It establishes the equivalence between the Strong Invertibility Property and the maximal asymptotic growth of inner functions away from their zeros, providing geometric and functional characterizations.

Findings

SIP is equivalent to maximal asymptotic growth of inner functions.

Inner functions satisfying SIP are characterized by narrow sublevel sets.

Finite entropy sets relate to divisors of inner functions with SIP.

Abstract

We consider inner functions $\Theta$ with the zero set $\mathcal Z(\Theta)$ such that the quotient algebra $H^\infty / \Theta H^\infty$ satisfies the Strong Invertibility Property (SIP), that is for every $\varepsilon>0$ there exists $\delta>0$ such that the conditions $f \in H^\infty$, $\|[f]\|_{H^\infty/ \Theta H^\infty}=1$, $\inf_{\mathcal Z(\Theta)} |f| \ge 1-\delta$ imply that $[f]$ is invertible in $H^\infty / \Theta H^\infty$ and $\| 1/ [f] \|_{H^\infty/ \Theta H^\infty}\le 1+\varepsilon$. We prove that the SIP is equivalent to the maximal asymptotic growth of $\Theta $ 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.