On the sum of superoptimal singular values

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 $m\times n$matrixfunction$\Phi$ontheunitcircle$$\mathbb{T}$$, when is there a matrix function ${\mathrm{\Psi }}_{\ast }$ 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 $\mathrm{\Phi }$ for some $\mathrm{\Psi }\in A_k^{n,m}$. Moreover, we provide a representation of any such function $\mathrm{\Psi }$ when $\mathrm{\Phi }$ is an admissible very badly approximable unitary-valued $n\times n$ matrix function.