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 α+∞+···+∞ (m times) and its subgroup K ? U(∞), which consists of block diagonal unitary matrices with the identity block of order α and a matrix u ∈ U(∞) 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.