On the sum of superoptimal singular values

In this paper, we study the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an m × n matrix function Φ, when is there a matrix function Ψ∗ in the set An,m k such that T trace Φ(ζ )Ψ∗(ζ ) dm(ζ ) = sup Ψ∈An,m k T trace Φ(ζ)Ψ(ζ) dm(ζ ) ? The set An,m k is defined by An,m k def = Ψ ∈ H1 0 (Mn,m): Ψ L1(Mn,m) 1, rankΨ(ζ) k a.e. ζ ∈ T . To address this extremal problem, we introduce Hankel-type operators on spaces of matrix functions and prove that this problem has a solution if and only if the corresponding Hankel-type operator has a maximizing vector. The main result of this paper is a characterization of the smallest number k for which T trace Φ(ζ)Ψ(ζ) dm(ζ ) equals the sum of all the superoptimal singular values of an admissible matrix function Φ (e.g. a continuous matrix function) for some function Ψ ∈ An,m k . Moreover, we provide a representation of any such function Ψ when Φ is an admissible very badly approximable unitary-valued n×n matrix function. © 2009 Elsevier Inc. All rights reserved