Resumen de Very badly approximable matrix functions

V.V. Peller, S.R. Treil

• 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.