Matrix-valued regular functions of a quaternionic variable - a decomposition and some approximation results

Abstract

A quaternionic matrix-valued regular function is a map $F:\Omega \to M_n(\mathbb{H})$ whose entries are (left) regular functions of a quaternion variable, where $\Omega$ is a domain in $\mathbb{H}$. Given any $n\times n$ matrix-valued regular function $F(q)$ over $\mathbb{H}$ having norm at most $1$, we derive a decomposition of $F(q)$ in the form $U\star \left[\begin{array}{cc}||F(q_0)||& 0\\ 0& G(q)\end{array}\right]V$ for some constant unitary matrices $U,V$, some vector $q_0$ and an $(n-1)\times (n-1)$ matrix-valued regular function $G(q)$, where $\star$ denotes the star product over $\mathbb{H}$. We use this to deduce that any matrix-valued regular function $F(q)$ having norm at most $1$ can be uniformly approximated by a convex combination of rational unitary functions.