Very Badly Approximable Matrix Functions

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 H Φ 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.