关于超优奇异值的和
On the sum of superoptimal singular values
摘要：
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 $Φ$ on the unit circle $\mathbb{T}$ , when is there a matrix function $Ψ_{*}$ 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 $Φ$ for some $Ψ\in A_{k}^{n,m}$ . Moreover, we provide a representation of any such function $Ψ$ when $Φ$ is an admissible very badly approximable unitary-valued $n\times n$ matrix function.
英文摘要：
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 $Φ$ on the unit circle $\mathbb{T}$, when is there a matrix function $Ψ_{*}$ 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 $Φ$ for some $Ψ\in A_{k}^{n,m}$. Moreover, we provide a representation of any such function $Ψ$ when $Φ$ is an admissible very badly approximable unitary-valued $n\times n$ matrix function.
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