Commuting operators over Pontryagin spaces with applications to system theory

TL;DR

This paper extends vessel theory and overdetermined 2D systems to Pontryagin spaces, developing realization theorems and an indefinite de Branges-Rovnyak theory with applications to system theory on Riemann surfaces.

Contribution

It introduces an indefinite version of vessel theory and de Branges-Rovnyak spaces over Riemann surfaces, expanding the mathematical framework for system analysis in Pontryagin spaces.

Findings

Developed realization theorems for characteristic functions in Pontryagin spaces.

Formulated an indefinite de Branges-Rovnyak theory on Riemann surfaces.

Proved a Beurling type theorem for indefinite Hardy spaces.

Abstract

In this paper we extend vessel theory, or equivalently, the theory of overdetermined $2D$ systems to the Pontryagin space setting. We focus on realization theorems of the various characteristic functions associated to such vessels. In particular, we develop an indefinite version of de Branges-Rovnyak theory over real compact Riemann surfaces. To do so, we use the theory of contractions in Pontryagin spaces and the theory of analytic kernels with a finite number of negative squares. Finally, we utilize the indefinite de Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beurling type theorem on indefinite Hardy spaces on finite bordered Riemann surfaces.