Ideal membership in $H^\infty$: Toeplitz corona approach

TL;DR

This paper investigates the ideal membership problem in the algebra of bounded analytic functions on the unit disk, providing a new proof of a sharp sufficient condition using Toeplitz corona techniques.

Contribution

It offers a novel proof of Treil's theorem on ideal membership in $H^$, utilizing the Toeplitz corona approach and solving a related problem in $H^2$.

Findings

Provided the sharpest known sufficient condition for ideal membership.

Connected the problem in $H^$ to a Hilbert space $H^2$ problem via the Nevanlinna-Pick property.

Enhanced understanding of ideal membership criteria in $H^$.

Abstract

We study the ideal membership problem in $H^\infty$ on the unit disc. Thus, given functions $f,f_1,\ldots,f_n$ in $H^\infty$, we seek sufficient conditions on the size of $f$ in order for $f$ to belong to the ideal of $H^\infty$ generated by $f_1,\ldots,f_n$. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space $H^2$, which is equivalent to the ideal membership problem by the Nevanlinna-Pick property of $H^2$.