Monotone thematic factorizations of matrix functions

TL;DR

This paper investigates the existence and properties of monotone non-decreasing thematic factorizations of admissible very badly approximable matrix functions, extending previous work on monotone non-increasing factorizations and their uniqueness.

Contribution

It proves the existence of monotone non-decreasing factorizations and explores their non-uniqueness, expanding understanding of thematic factorizations in matrix function approximation.

Findings

Existence of monotone non-decreasing thematic factorizations.

Thematic indices in such factorizations are not uniquely determined.

Monotone non-increasing factorizations generate many other factorizations.

Abstract

We continue the study of the so-called thematic factorizations of admissible very badly approximable matrix functions. These factorizations were introduced by V.V. Peller and N.J. Young for studying superoptimal approximation by bounded analytic matrix functions. Even though thematic indices associated with a thematic factorization of an admissible very badly approximable matrix function are not uniquely determined by the function itself, R.B. Alexeev and V.V. Peller showed that the thematic indices of any monotone non-increasing thematic factorization of an admissible very badly approximable matrix function are uniquely determined. In this paper, we prove the existence of monotone non-decreasing thematic factorizations for admissible very badly approximable matrix functions. It is also shown that the thematic indices appearing in a monotone non-decreasing thematic factorization are not uniquely determined by the matrix function itself. Furthermore, we show that the monotone non-increasing thematic factorization gives rise to a great number of other thematic factorizations.