CONTINUITY PROPERTIES OF BEST ANALYTIC APPROXIMATION

Let A be the operator which assigns to each mxn matrix-valued function on the unit circle with entries in H-infinity+C its unique superoptim al approximant in the space of bounded analytic mxn matrix-valued func tions 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 satif ying certain natural axioms, A is continuous at any function whose sup eroptimal singular values are non-zero and is such that certain associ ated integer indices are equal to 1. We also obtain necessary conditio ns for continuity of A at a point and a sufficient condition for the c ontinuity of superoptimal singular values.