Invariance properties of thematic factorizations of matrix functions

TL;DR

This paper investigates the invariance of indices in thematic factorizations of matrix functions, introducing monotone factorizations to ensure unique, matrix-dependent indices, advancing understanding in superoptimal approximation theory.

Contribution

It introduces the concept of monotone thematic factorizations, proving their existence and uniqueness of indices under natural conditions, improving the theoretical framework of matrix function factorizations.

Findings

Monotone thematic factorizations have uniquely determined indices.

Indices depend only on the matrix function in monotone factorizations.

Results extend to partial thematic factorizations.

Abstract

We study the problem of invariance of indices of thematic factorizations. Such factorizations were introduced in [PY1] for studying superoptimal approximation by bounded analytic matrix functions. As shown in [PY1], 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.