Very badly approximable matrix functions

2003/03/15 by V. V. Peller, S. R. Treil, Peller, V. V. +1
Mathematics · #30D50 #30D55 #30E10 #42B30 #46E40 #47B35 #Category Theory (math.CT) #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #math.CA #math.CO #math.CT #math.CV #math.FA #msc:30D50 #msc:30D55 #msc:30E10 #msc:42B30 #msc:46E40 #msc:47B35

paper · pdf · doi:10.48550/arxiv.math/0303186

27 pages

arxiv created 2003/03/15 · arxiv updated 2016/09/07

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 \citePY and \citeAP. It involves analyticity of certain families of subspaces defined in terms of Schmidt vectors of the matrices Φ(\z), \z∈\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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