Wiener–Hopf factorization indices of rational matrix functions with respect to the unit circle in terms of realization

As in the paper Groenewald et al. (2017) our aim is to obtain explicitly the Wiener–Hopf indices of a rational m×m matrix function R(z) that has no poles and no zeros on the unit circle T but, in contrast with Groenewald et al. (2017), the function R(z) is not required to be unitary on the unit circle. On the other hand, using a Douglas–Shapiro–Shields type of factorization, we show that R(z) factors as R(z)=Ξ(z)Ψ(z), where Ξ(z) and Ψ(z) are rational m×m matrix functions, Ξ(z) is unitary on the unit circle and Ψ(z) is an invertible outer function. Furthermore, the fact that Ξ(z) is unitary on the unit circle allows us to factor as Ξ(z)=V(z)W^∗(z) where V(z) and W(z) are rational bi-inner m×m matrix functions. The latter allows us to solve the Wiener–Hopf indices problem. To derive explicit formulas for the functions V(z) and W(z) requires additional realization properties of the function Ξ(z) which are given in the last two sections.