Canonical Wiener-Hopf factorization on the unit circle: matching subspaces versus Riccati equations

Abstract

Wiener-Hopf factorization is an important tool in the study of block Toeplitz and block Wiener-Hopf operators, and many applications involving these operators. In this paper we compare two approaches to Wiener-Hopf factorization, namely, the more classical approach based on matching invariant subspaces and a more recent approach based on solutions to a non-symmetric Riccati equation. The latter approach is extended to the case of Hilbert space operator-valued functions that are analytic on a neighborhood of the unit disc $\BT$, but need not be rational. In both approaches, existence of canonical right Wiener-Hopf factorization is characterized by existence of a stabilizing solution to a Riccati equation, however, the Riccati equations are not the same. We analyse the solution sets of the Riccati equations and show that they indeed are not the same, but they do have the same stabilizing solution.