The Bezout-corona problem revisited: Wiener space setting

TL;DR

This paper provides an explicit description of solutions to the matrix-valued Bezout-corona problem within Wiener space, extending some results to the $H^inity$ setting and clarifying the structure of solutions.

Contribution

It offers a new explicit parametrization of solutions to the Bezout-corona problem in Wiener space and explores the extent of these results in the $H^inity$ setting.

Findings

All Wiener solutions can be explicitly described using two matrices and a Wiener function Y.

Some results extend to the $H^inity$ setting, with Y being an $H^2$ function.

Explicit solution formulas are provided for Wiener and $H^2$ solutions.

Abstract

The matrix-valued {Bezout-corona} problem $G(z)X(z)=I_m$, $|z|<1$, is studied in a Wiener space setting, that is, the given function $G$ is an analytic matrix function on the unit {disc} whose Taylor coefficients are absolutely summable and the same is required for the solutions $X$. It turns out that all Wiener solutions can be described explicitly in terms of two matrices and a square analytic Wiener function $Y$ satisfying $\det Y(z)\not =0$ for all $|z|\leq 1$. It is also shown that some of the results hold in the $H^\infty$ {setting, but} not all. In fact, if $G$ is an $H^\infty$ function, then $Y$ is just an $H^2$ function. Nevertheless, in this case, using the two matrices and the function $Y$, all $H^2$ solutions to the Bezout-corona problem can be described explicitly in a form analogous to the one appearing in the Wiener setting.