Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument

TL;DR

This paper generalizes classical operator colligation theory by defining a natural multiplication on conjugacy classes of infinite block unitary matrices, introducing spectral data that captures this structure and allows for class reconstruction.

Contribution

It extends the theory of operator colligations and characteristic functions to a new setting involving conjugacy classes of infinite block matrices with a novel multiplication operation.

Findings

Defined a natural multiplication on conjugacy classes G//K.

Constructed spectral data that visualizes the multiplication.

Provided a method for reconstructing conjugacy classes from spectral data.

Abstract

We extend the classical construction of operator colligations and characteristic functions. Consider the group $G$ of finite block unitary matrices of size $\alpha+\infty+...+\infty$ ($k$ times). Consider the subgroup $K=U(\infty)$, which consists of block diagonal unitary matrices (with a block 1 of size $\alpha$ and a matrix $u\in U(\infty)$ repeated $k$ times). It appears that there is a natural multiplication on the conjugacy classes $G//K$. We construct 'spectral data' of conjugacy classes, which visualize the multiplication and are sufficient for a reconstruction of a conjugacy class.