Invariance Properties of Thematic Factorizations of Matrix Functions

We study the problem of invariance of indices of thematic factorizations. Such factorizations were introduced by V. V. Peller and N. J. Young (1994, J. Funct. Anal.120, 300–343) for studying superoptimal approximation by bounded analytic matrix functions. As shown by Peller and Young, the indices may depend on the choice of a thematic factorization. We introduce the notion of a monotone thematic factorization. The main result shows that under natural assumptions a matrix function that admits a thematic factorization also admits a monotone thematic factorization and the indices of a monotone thematic factorization are uniquely determined by the matrix function itself. We obtain similar results for so-called partial thematic factorizations.