Superoptimal singular values and indices of matrix functions

Abstract

LetG be a matrix-valued function on the unit circle which is the sum of a continuous function and anH ∞ function. We establish an inequality between corresponding terms of the sequence of singular values of the Hankel operator {s j (H G )} and a sequence formed from the superoptimal singular values ofG with repetitions. The number of times each superoptimal singular value is repeated is a positive integer index which is the winding number of a related scalar function on the circle and gives information about the superoptimal error functionG-Q withQ∈H ∞ . In the second part of the paper we establish a property of invariance of the sum of the indices corresponding to a particular superoptimal singular value. This establishes the truth of two conjectures made in [PeY].