Very badly approximable matrix functions

• V.V. Peller ^[1] ; S.R. Treil ^[2]
  1. [1] Michigan State University

     Michigan State University

     City of East Lansing, Estados Unidos

     
  2. [2] Brown University

     Brown University

     City of Providence, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 11, Nº. 1, 2005, págs. 127-154
• Idioma: inglés
• DOI: 10.1007/s00029-005-0001-1
• Enlaces
  • Texto completo (pdf)
• Resumen

  • 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 [PY1] and [AP]. It involves analyticity of certain families of subspaces defined in terms of Schmidt vectors of the matrices Φ(ζ),ζ∈T. This characterization can be extended to the wider class of admissible functions, i.e., the class of matrix functions Φ such that the essential norm ||HΦ||e of the Hankel operator HΦ 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.