Superoptimal singular values and indices of matrix functions (Q1341060)

Let \(G\) be a matrix-valued function on the unit circle which is the sum of a continuous function and an \(H^\infty\) 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 of \(G\) 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 function \(G-Q\) with \(Q \in H^\infty\). 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 by the authors in J. Funct. Anal. 120, No. 2, 300-343 (1994; Zbl 0808.47011)].