Badly approximable matrix functions and canonical factorizations

We continue studying the problem of analytic approximation of matrix functions. We introduce the notion of a partial canonical factorization of a badly approximable matrix function $\Phi$ and the notion of a canonical factorization of a very badly approximable matrix function $\Phi$. Such factorizations are defined in terms of so-called balanced unitary-valued functions which have many remarkable properties. Unlike the case of thematic factorizations studied earlier in [PY1], [PY2], [PT], [AP1], the factors in canonical factorizations (as well as partial canonical factorizations) are uniquely determined by the matrix function $\Phi$ up to constant unitary factors. We study many properties of canonical factorizations. In particular we show that under certain natural assumptions on a function space $X$ the condition $\Phi\in X$ implies that all factors in a canonical factorization of $\Phi$ belong to the same space $X$. In the last section we characterize the very badly approximable unitary-valued functions $U$ that satisfy the condition $\|H_U\|_{\text e}<1$.