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

TL;DR

This paper derives explicit formulas for Wiener-Hopf indices of rational matrix functions using realization theory, even when the functions are not unitary on the unit circle, by employing factorization techniques.

Contribution

It extends previous results by removing the unitarity requirement on the matrix function and provides explicit formulas through realization-based factorizations.

Findings

Explicit formulas for Wiener-Hopf indices are obtained.

A factorization approach using bi-inner functions is developed.

The method applies to non-unitary rational matrix functions on the unit circle.

Abstract

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 $m\times m$ matrix function $R(z)$ that has no poles and no zeros on the unit circle $\mathbb{T}$ but, in contrast with that paper, 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 $R(z)=\Xi(z)\Psi(z)$, where $\Xi(z)$ and $\Psi(z)$ are rational $m\times m$ matrix functions, $\Xi(z)$ is unitary on the unit circle and $\Psi(z)$ is an invertible outer function. Furthermore, the fact that $\Xi(z)$ is unitary on the unit circle allows us to factor as $\Xi(z) =V(z)W^*(z)$ 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 $\Xi(z)$ which are given in the last two sections.