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

We extend the classical construction of operator colligations and characteristic functions. Consider the group G of finitary block unitary matrices of order $\alpha+\infty+\dots+\infty$ ($m$ times) and its subgroup $K \cong U(\infty)$, which consists of block diagonal unitary matrices with the identity block of order $\alpha$ and a matrix $u \in U(\infty)$ repeated $m$ times. It turns out that there is a natural multiplication on the space $G$//$K$ of conjugacy classes. We construct “spectral data” of conjugacy classes, which visualize the multiplication and are sufficient for reconstructing a conjugacy class.