Unitary Interpolants and Factorization Indices of Matrix Functions

For an n×n bounded matrix function Φ we study unitary interpolants U, i.e., unitary-valued functions U such that U(j)=Φ(j), j<0. We are looking for unitary interpolants U for which the Toeplitz operator TU is Fredholm. We give a new approach based on superoptimal singular values and thematic factorizations. We describe Wiener–Hopf factorization indices of U in terms of superoptimal singular values of Φ and thematic indices of Φ−F, where F is a superoptimal approximation of Φ by bounded analytic matrix functions. The approach essentially relies on the notion of a monotone thematic factorization introduced in [AP]. In the last section we discuss hereditary properties of unitary interpolants. In particular, for matrix functions Φ of class H∞+C we study unitary interpolants U of class QC.