The Bezout equation on the right half plane in a Wiener space setting

This paper deals with the Bezout equation \(G(s)X(s)=I_m\), \(\Re s \geq 0\), in the Wiener space of analytic matrix-valued functions on the right half plane. In particular, \(G\) is an \(m\times p\) matrix-valued analytic Wiener function, where \(p\geq m\), and the solution \(X\) is required to be an analytic Wiener function of size \(p\times m\). The set of all solutions is described explicitly in terms of a \(p\times p\) matrix-valued analytic Wiener function \(Y\), which has an inverse in the analytic Wiener space, and an associated inner function \(\Theta\) 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].