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 $\Psi_{*}$ in the set $A_{k}^{n,m}$ such that \int_{\mathbb{T}}{\rm trace}(\Phi(\zeta)\Psi_{*}(\zeta))dm(\zeta)=\sup_{\Psi\in A_{k}^{n,m}}|\int_{\mathbb{T}}{\rm trace}(\Phi(\zeta)\Psi(\zeta))dm(\zeta)|? The set $A_{k}^{n,m}$ is defined by A_{k}^{n,m}={\Psi\in H_{0}^{1}: \|\Psi\|_{L^{1}}\leq 1, {\rm rank}\Psi(\zeta)\leq k{a.e.}\zeta\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}(\Phi(\zeta)\Psi(\zeta))dm(\zeta) 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.