Inner functions of matrix argument and conjugacy classes in unitary groups

Abstract

Let $\mathrm{B}_n$ denote the set of complex square matrices of order $n$ whose Euclidean operator norms are less than one. Its Shilov boundary is the set $\operatorname{U}(n)$ of all unitary matrices. A holomorphic map $\mathrm{B}_m\to\mathrm{B}_n$ is inner if it sends $\operatorname{U}(m)$ to $\operatorname{U}(n)$. On the other hand we consider the group $\operatorname{U}(n+mj)$ and its subgroup $\operatorname{U}(j)$ that is embedded in $\operatorname{U}(n+mj)$ in the block-diagonal way ($m$ blocks $\operatorname{U}(j)$ and a unit block of size $n$). To any conjugacy class of $\operatorname{U}(n+mj)$ with respect to $\operatorname{U}(j)$ we assign a ‘characteristic function’, which is a rational inner map $\mathrm{B}_m\to\mathrm{B}_n$. We show that the class of inner functions that can be obtained as ‘characteristic functions’ is closed with respect to such natural operations as pointwise direct sums, pointwise products, compositions, substitutions into finite-dimensional representations of general linear groups and so on. We also describe explicitly the corresponding operations on conjugacy classes. Bibliography: 24 titles.