On the sum of superoptimal singular values

Abstract

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 k n , m such that ∫ T trace ( Φ ( ζ ) Ψ ∗ ( ζ ) ) d m ( ζ ) = sup Ψ ∈ A k n , m | ∫ T trace ( Φ ( ζ ) Ψ ( ζ ) ) d m ( ζ ) | ? The set A k n , m is defined by A k n , m = def { Ψ ∈ H 0 1 ( M n , m ) : ‖ Ψ ‖ L 1 ( M n , m ) ⩽ 1 , rank Ψ ( 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 ) ) d m ( 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 k n , m . Moreover, we provide a representation of any such function Ψ when Φ is an admissible very badly approximable unitary-valued n × n matrix function.