Very badly approximable matrix functions

V. V. Peller; S. R. Treil

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Very badly approximable matrix functions

V. V. Peller·S. R. Treil

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Abstract

We study in this paper very badly approximable matrix functions on the unit circle $\T$, i.e., matrix functions $Φ$ such that the zero function is a superoptimal approximation of $Φ$. The purpose of this paper is to obtain a characterization of the continuous very badly approximable functions. Our characterization is more geometric than algebraic characterizations earlier obtained in \cite{PY} and \cite{AP}. It involves analyticity of certain families of subspaces defined in terms of Schmidt vectors of the matrices $Φ(\z)$, $\z\in\T$. This characterization can be extended to the wider class of {\em admissible} functions, i.e., the class of matrix functions $Φ$ such that the essential norm $\|H_Φ\|_{\rm e}$ of the Hankel operator $H_Φ$ is less than the smallest nonzero superoptimal singular value of $Φ$. In the final section we obtain a similar characterization of badly approximable matrix functions.

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BibTeX RIS

V. V. Peller, S. R. Treil. 2003-03-15. Very badly approximable matrix functions. https://arxiv.org/abs/math/0303186

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