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

As in the paper [G. Groenewald, M.A. Kaashoek, A.C.M. Ran, Wiener-Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators. \emph{Indag. Math.} 28 (2017) 694–710] our aim is to obtain explicitly the Wiener-Hopf indices of a rational $m\times m$matrixfunction$R(z)$thathasnopolesandnozerosontheunitcircle$$\mathbb{T}$$but,incontrastwiththatpaper,thefunction$R(z)$ 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 $R(z)=\mathrm{\Xi }(z)\mathrm{\Psi }(z)$, where $\mathrm{\Xi }(z)$ and $\mathrm{\Psi }(z)$ are rational $m\times m$ matrix functions, $\mathrm{\Xi }(z)$ is unitary on the unit circle and $\mathrm{\Psi }(z)$ is an invertible outer function. Furthermore, the fact that $\mathrm{\Xi }(z)$ is unitary on the unit circle allows us to factor as $\mathrm{\Xi }(z)=V(z)W^{\ast }(z)$ where $V(z)$ and $W(z)$ are rational bi-inner $m\times m$ 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 $\mathrm{\Xi }(z)$ which are given in the last two sections.