On the sum of superoptimal singular values

Abstract

We discuss the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an $m\times n$ matrix function $\Phi$ on the unit circle $\mathbb{T}$, when is there a matrix function ${\Psi }_{\ast }$ in the set $A_k^{n,m}$ such that \int_{\mathbb{T}}{\rm trace}(Φ(ζ)Ψ_{*}(ζ))dm(ζ)=\sup_{Ψ\in A_{k}^{n,m}}|\int_{\mathbb{T}}{\rm trace}(Φ(ζ)Ψ(ζ))dm(ζ)|? The set $A_k^{n,m}$ is defined by A_{k}^{n,m}={Ψ\in H_{0}^{1}: \|Ψ\|_{L^{1}}\leq 1, {\rm rank}Ψ(ζ)\leq k{a.e.}ζ\in T}. 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. We also characterize the smallest number $k$ for which \int_{\mathbb{T}}{\rm trace}(Φ(ζ)Ψ(ζ))dm(ζ) equals the sum of all the superoptimal singular values of an admissible matrix function $\Phi$ for some $\Psi \in A_k^{n,m}$. Moreover, we provide a representation of any such function $\Psi$ when $\Phi$ is an admissible very badly approximable unitary-valued $n\times n$ matrix function.

24 pages