Badly approximable matrix functions and canonical factorizations

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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 Φ and the notion of a canonical factorization of a very badly approximable matrix function Φ. 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 Φ 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 Φ∈ X implies that all factors in a canonical factorization of Φ belong to the same space X. In the last section we characterize the very badly approximable unitary-valued functions U that satisfy the condition Hᵤₑ<1.