The Bezout-Corona problem revisited: Wiener space setting

Abstract

The matrix-valued Bezout-corona problem G(z)X(z)=Im, |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 detY(z)≠0 for all |z|≤1. It is also shown that some of the results hold in the H∞ setting, but not all. In fact, if G is an H∞ function, then Y is just an H2 function. Nevertheless, in this case, using the two matrices and the function Y, all H2 solutions to the Bezout-corona problem can be described explicitly in a form analogous to the one appearing in the Wiener setting