Wiener-Hopf indices of unimodular functions on the imaginary axis

TL;DR

This paper investigates Wiener-Hopf indices of unimodular rational matrix functions on the imaginary axis, providing formulas based on factorization matrices and employing operator theory and Cayley transform techniques.

Contribution

It introduces new formulas for Wiener-Hopf indices using realizations from Douglas-Shapiro-Shields factorizations and presents two analytical approaches.

Findings

Formulas for Wiener-Hopf indices in terms of factorization matrices

Two methods: operator theoretic approach and Cayley transform approach

Connections established between unimodular functions on the imaginary axis and Toeplitz operators

Abstract

This paper is concerned with the Wiener-Hopf indices of unimodular rational matrix functions on the imaginary axis. These indices play a role in the Fredholm theory for Wiener-Hopf integral operators. Our main result gives formulas for the Wiener-Hopf indices in terms of the matrices appearing in realizations of the factors in a Douglas-Shapiro-Shields factorization of the unimodular function. Two approaches to this problem are presented: one direct approach using operator theoretic methods, and a second approach using the Cayley transform which allows to use results for an analogous problem regarding unimodular functions on the unit circle and corresponding Toeplitz operators.