Wiener-Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators

We provide new formulas for the Wiener-Hopf factorization indices of a rational matrix function R which has neither poles nor zeros on the unit circle. In addition, we recover recent results on the Fredholm characteristics of the Toeplitz operator with symbol R via the method of matricial coupling. Furthermore, we present an alternative formula for the index in terms of the Fourier coefficients of R.