Continuity properties of best analytic approximation

Abstract

Let $\A$ be the operator which assigns to each $m \times n$ matrix-valued function on the unit circle with entries in $H^\infty + C$ its unique superoptimal approximant in the space of bounded analytic $m \times n$ matrix-valued functions in the open unit disc. We study the continuity of $\A$ with respect to various norms. Our main result is that, for a class of norms satifying certain natural axioms, $\A$ is continuous at any function whose superoptimal singular values are non-zero and is such that certain associated integer indices are equal to 1. We also obtain necessary conditions for continuity of $\A$ at point and a sufficient condition for the continuity of superoptimal singular values.