On Superoptimal Approximation by Analytic and Meromorphic Matrix-Valued Functions

Abstract

It is a well-known fact that for any continuous scalar-valued function φ on the unit circle there is a unique best approximation in the Hardy class H ∞ of bounded analytic functions. However, in the matrix-valued case a best approximation by bounded analytic functions is almost never unique. To make it unique, one has to impose additional assumptions. In 1986 N. J. Young introduced the so-called superoptimal solution of the Nehari problem. The word superoptimal means that we are seeking F ∈ H ∞ (matrix-valued) to minimize not only sup {|| Φ (ζ) − F(ζ) || : |ζ| = 1} = sup{ s 0(Φ(ζ) − F (ζ)) : |ζ| = 1} but also the suprema of further singular values s j (Φ(ζ) − F(ζ)), j ≥ 1. It was proved by V. V. Peller and N. J. Young that for matrix-valued function Φ in H ∞ + C such superoptiminal solution is unique. In this paper an alternative geometric approach to the problem is presented. It allows us to obtain another proof of the uniqueness and some new results: uniqueness of the superoptimal approximation by meromorphic functions (a generalization of the Adamyan-Arov-Krein result on approximation of scalar-valued functions), inequalities between s-numbers of a Hankel operator, and superoptimal singular values. Our approach works well for operator-valued functions as for matrix-valued ones.