SUPEROPTIMAL SINGULAR-VALUES AND INDEXES OF INFINITE MATRIX FUNCTIONS

We study the problem of finding a best approximation of a given bounde d operator function Phi, on the unit circle T by bounded operator func tions Q analytic in the unit disk. We minimize not only esssup(zeta is an element of T)\\Phi(zeta) - Q(zeta)\\ but also the L infinity norms of further singular values of Phi(zeta) - Q(zeta). We show that if th e Hankel operator H-Phi is compact, then such an approximation (which is called superoptimal) exists and uniqe. We study related factorizati ons (thematic factorizations), establish invariance of their indices a nd prove an inequality between the singular values of the Hankel opera tor H-Phi and the superoptimal singular values esssup(zeta is an eleme nt of T)s(j)(Phi(zeta) - Q(zeta)).