2001/01/26 by R. B. Alexeev, V. V. Peller, Alexeev, R. B. +2
Mathematics · #30D55 #46E15 #47B35 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.CA #math.CV #math.FA #msc:30D55 #msc:46E15 #msc:47B35
paper · pdf · doi:10.48550/arxiv.math/0101225
36 pages
arxiv created 2001/01/26 · openalex publication_date 2001/01/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ‖HU‖\text e<1.