The Bezout-corona problem revisited: Wiener space setting

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.