Continuity properties of best analytic approximation
Vladimir Peller and Nicholas J. Young
Let ${\mathcal A}$ 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 ${\mathcal A}$ with respect to various norms. Our main result is that, for a class of norms satifying certain natural axioms, ${\mathcal A}$ 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 ${\mathcal A}$ at point and a sufficient condition for the continuity of superoptimal singular values.