Approximation by analytic operator functions. Factorizations and very badly approximable functions

摘要

This is a continuation of our earlier paper \cite{PT3}. We consider hereoperator-valued functions (or infinite matrix functions) on the unit circle$\T$ and study the problem of approximation by bounded analytic operatorfunctions. We discuss thematic and canonical factorizations of operatorfunctions and study badly approximable and very badly approximable operatorfunctions. We obtain algebraic and geometric characterizations of badly approximable andvery badly approximable operator functions. Note that there is an importantdifference between the case of finite matrix functions and the case of operatorfunctions. Our criteria for a function to be very badly approximable in thecase of finite matrix functions also guarantee that the zero function is theonly superoptimal approximant. However in the case of operator functions thisis not true.