On the sum of superoptimal singular values (Q2391257)

The present paper is based on the author's PhD dissertation. Its aim is to give a characterization of the smallest number for which the sum of the so-called superoptimal singular values of an admissible matrix function is evaluated. The paper is organized in five sections, the first two of which containing the main definitions, notations, and some principles of functional analysis expressed in matrix language. The notions of superoptimal approximation and very badly approximation by bounded analytic matrix functions are explained. In particular, explicit formulations are given for the space \(L^p(S_q^{m,n})\), where \(S_q^{m,n}\) is the space of the \(m\times n\) matrices equipped with the Schatten-von Neumann norm. In the third section, Hankel-type operators on spaces of matrix functions are introduced, and the approximation is given in terms of Hankel-type operators. In the fourth section, the main result of the paper on the best approximation is obtained. The last section of the paper is concerned with the class of admissible unitary valued very badly approximable \(n\times n\) matrix functions.