Wiener-Hopf indices of unimodular functions on the imaginary axis

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.