Characteristic function of M. Liv\v{s}ic and some developments

TL;DR

This paper explores M. Livšic's characteristic matrix functions, discussing their properties, factorization, inverse problems, and applications across various mathematical fields.

Contribution

It reviews and connects multiple developments related to Livšic's characteristic functions, including factorization, inverse problems, and applications in prediction theory and Riemann-Hilbert problems.

Findings

Factorization of transfer operator functions

Solutions to inverse problems

Applications in Wiener--Masani prediction and Riemann-Hilbert problems

Abstract

The area related to M. Liv\v{s}ic's characteristic matrix functions is too vast to be discussed in one paper and we selected for this article the problems which are close to our scientific interests. We discuss M.Liv\v{s}ic's results connected with characteristic matrix functions and various important developments including factorization of the transfer operator function, inverse problems, triangular models, reduction of the operators to the simplest form as well as applications to Wiener--Masani prediction theory, to random matrices, and to Riemann--Hilbert problems.