A note on homotopies of rational matrix inner functions

TL;DR

This paper proves that the space of matrix-valued rational inner functions with more rows than columns in the disk is connected through continuous paths, enhancing understanding of their topological structure.

Contribution

It establishes the path connectedness of the space of m×n rational inner functions for m>n, a previously unexplored topological property.

Findings

The space of m×n rational inner functions is path connected for m>n.

Provides new insights into the topology of matrix-valued inner functions.

Advances understanding of the structure of rational inner functions in complex analysis.

Abstract

We show that when $m>n$, the space of $m\times n$-matrix-valued rational inner functions in the disk is path connected.