On the sum of superoptimal singular values

TL;DR

This paper investigates the extremal problem related to the sum of superoptimal singular values of matrix functions, characterizing solutions via Hankel-type operators and identifying the minimal rank for optimality.

Contribution

It introduces Hankel-type operators on matrix function spaces and characterizes when extremal solutions exist, also determining the minimal rank for achieving the sum of superoptimal singular values.

Findings

Solution existence tied to Hankel-type operator having a maximizing vector

Characterization of the minimal rank k for superoptimal singular value sum

Representation of solutions for badly approximable unitary-valued matrices

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.