The Corona Property in Nevanlinna quotient algebras and Interpolating sequences

TL;DR

This paper characterizes when Bezout equations in Nevanlinna quotient algebras can be solved, showing it occurs precisely when the zeros of the inner function form a finite union of Nevanlinna interpolating sequences, unlike in bounded analytic function algebras.

Contribution

It establishes a necessary and sufficient condition for solving Bezout equations in Nevanlinna quotient algebras based on zero sets being finite unions of interpolating sequences.

Findings

Bezout equations solvable iff zeros form finite union of Nevanlinna interpolating sequences

Contrast with bounded analytic functions where the condition is only sufficient

Characterization of Blaschke products with zeros as finite unions of interpolating sequences

Abstract

Let $I$ be an inner function in the unit disk $\mathbb D$ and let $\mathcal N$ denote the Nevanlinna class. We prove that under natural assumptions, Bezout equations in the quotient algebra $\mathcal N/I\mathcal N$ can be solved if and only if the zeros of $I$ form a finite union of Nevanlinna interpolating sequences. This is in contrast with the situation in the algebra of bounded analytic functions, where being a finite union of interpolating sequences is a sufficient but not necessary condition. An analogous result in the Smirnov class is proved as well as several equivalent descriptions of Blaschke products whose zeros form a finite union of interpolating sequences in the Nevanlinna class.