ON SUPEROPTIMAL APPROXIMATION BY ANALYTIC AND MEROMORPHIC MATRIX-VALUED FUNCTIONS

It is a well-known Fact that for any continuous scalar-valued function phi on the unit circle there is a unique best approximation in the Ha rdy class H infinity of bounded analytic functions. However, in the ma trix-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 a re seeking F is an element of H infinity (matrix-valued) to minimize n ot only sup{parallel to Phi(zeta)-F(zeta)parallel to:\zeta\=1} =sup(s( 0)(Phi(zeta)-F(zeta)):\zeta\=1) but also the suprema of further singul ar values s(j)(Phi(zeta)-F(zeta)), j greater than or equal to 1. It wa s proved by V.V. Peller and N.J. Young that for matrix-valued function Phi in H infinity + C such superoptimal solution is unique. In this p aper an alternative geometric approach to the problem is presented. It allows us to obtain another proof of the uniqueness and some new resu lts: uniqueness of the superoptimal approximation by meromorphic funct ions (a generalization of the Adamyan-Arov-Krein result on approximati on of scalar-valued functions), inequalities between s-numbers of a Ha nkel operator, and superoptimal singular values. Our approach works as well for operator-valued functions as for matrix-valued ones. (C) 199 5 Academic Press, Inc.