The Bézout Equation on the Right Half-plane in a Wiener Space Setting

Abstract

This paper deals with the Bézout equation ${G}(s){X}(s) = {I}_{m}, \mathfrak{R}{s} \leq {0}$, in the Wiener space of analytic matrix-valued functions on the right halfplane. In particular, G is an m × p matrix-valued analytic Wiener function, where p ≥ m, and the solution X is required to be an analyticWiener function of size p × m. The set of all solutions is described explicitly in terms of a p × p matrix-valued analyticWiener function Y , which has an inverse in the analytic Wiener space, and an associated inner function Θ defined by Y and the value of G at infinity. Among the solutions, one is identified that minimizes the H ²- norm. A Wiener space version of Tolokonnikov’s lemma plays an important role in the proofs. The results presented are natural analogs of those obtained for the discrete case in [11].

Mathematics Subject Classification (2010). Primary 47A56; Secondary 47A57, 47B35, 46E40, 46E15.