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 A n,m k such that Z T trace(Φ(ζ)Ψ * (ζ))dm(ζ) = sup Ψ∈A n,m k ˛ ˛ ˛ ˛ Z T trace(Φ(ζ)Ψ(ζ))dm(ζ) ˛ ˛ ˛ ˛ ? The set A n,m k is defined by A n,m k def = n Ψ ∈ H 1 0 (Mn,m) : Ψ L 1 (Mn,m) ≤ 1, rank Ψ(ζ) ≤ k a.e. ζ ∈ T o. 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 Z 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 ∈ A n,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.