Maximum principles for matrix-valued regular functions of a quaternionic variable

Abstract

A quaternionic matrix-valued regular function is a map $F: \Omega \rightarrow M_n(\mathbb{H})$ whose entries are (left) regular functions of a quaternion variable, where $\Omega$ is a domain in $\mathbb{H}$. Our aim is to bring out some maximum norm principles for such functions. We derive an SVD type decomposition theorem for such functions, using the notion of maximizing vectors. Some maximum principles for singular values of matrix-valued regular function are brought out next. We then proceed to prove a Fisher type approximation theorem for regular functions $f: \mathbb{B} \rightarrow \overline{\mathbb{B}}$ that are continuous on $\partial \mathbb{B}$, in terms of convex combinations of finite Blaschke products over $\mathbb{H}$ ($\mathbb{B}$ being the quaternionic unit ball). This in turn yields a Fisher type approximation theorem for an $n \times n$ matrix-valued regular function on the quaternionic unit ball, where each entry of the matrix satisfies the same condition as above.