Inner functions of matrix argument and conjugacy classes in unitary groups

TL;DR

This paper explores the structure of inner holomorphic functions between matrix balls and their relation to conjugacy classes in unitary groups, revealing closure properties under natural operations and explicit descriptions of these operations.

Contribution

It introduces a new framework linking inner functions with conjugacy classes in unitary groups, showing closure properties and providing explicit descriptions of operations.

Findings

Inner functions form a closed class under natural operations.

Characteristic functions encode conjugacy classes in unitary groups.

Explicit formulas for operations on conjugacy classes are provided.

Abstract

Denote by $B_n$ the set of complex square matrices of order $n$, whose Euclidean operator norms are $<1$. Its Shilov boundary is the set $U(n)$ of all unitary matrices. A holomorphic map $B_m\to B_n$ is inner if it sends $U(m)$ to $U(n)$. On the other hand we consider a group $U(n+mj)$ and its subgroup $U(j)$ embedded to $U(n+mj)$ in a block-diagonal way ($m$ blocks $U(j)$ and a unit block of size $n$). For any conjugacy class of $U(n+mj)$ with respect to $U(j)$ we assign a 'characteristic function', which is a rational inner map $B_m\to B_n$. We show that the class of inner functions, which can be obtained as 'characteristic functions', is closed with respect to natural operations as pointwise direct sums, pointwise products, compositions, substitutions to finite-dimensional representations of general linear groups, etc. We also describe explicitly the corresponding operations on conjugacy classes.