On unitary interpolants and Fredholm infinite block Toeplitz matrices

Abstract

Conditions for the existence and uniqueness of unitary n × n matrix valued functions f on the unit circle with prescribed Fourier coefficients f_j for j ⩾ 0 are given (in terms of infinite block Hankel matrices based on the prescribed coefficients f₀,f₁,⋯ ) for a natural class of functions. A unitary function belongs to this class if and only if it admits a generalized factorization (in a sense which will be made precise in the paper) or equivalently if and only if any one (and hence both) of the two Toeplitz operators defined by the function are Fredholm. In particular this class includes all continuous unitary n × n matrix valued functions. It is shown that the nonnegative factorization indices of every such unitary f are uniquely determined by f₀,f₁,⋯ and formulas for them are given.