Very badly approximable matrix functions

We study in this paper very badly approximable matrix functions on the unit circle $\text{T}$, i.e., matrix functions $\mathrm{\Phi }$ such that the zero function is a superoptimal approximation of $\mathrm{\Phi }$. 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 $\mathrm{\Phi }(\text{z})$, $\text{z}\in \text{T}$. This characterization can be extended to the wider class of {\em admissible} functions, i.e., the class of matrix functions $\mathrm{\Phi }$ such that the essential norm $\mathrm{\parallel }{H}_{\mathrm{\Phi }}{\mathrm{\parallel }}_{\mathrm{e}}$ of the Hankel operator $H_{\mathrm{\Phi }}$ is less than the smallest nonzero superoptimal singular value of $\mathrm{\Phi }$. In the final section we obtain a similar characterization of badly approximable matrix functions.