Minimal passive realizations of generalized Schur functions in Pontryagin spaces

TL;DR

This paper studies minimal passive system realizations in Pontryagin spaces, proving their existence and uniqueness conditions for generalized Schur functions, and introduces a new approach based on passive system theory.

Contribution

It establishes the existence of optimal minimal realizations for generalized Schur functions and generalizes criteria for their unitary similarity, using a novel passive system approach.

Findings

Proves existence of optimal minimal realizations for generalized Schur functions.

Generalizes criteria for unitary similarity of passive realizations.

Introduces a new definition for defects functions in this context.

Abstract

Passive discrete-time systems in Pontryagin space setting are investigated. In this case the transfer functions of passive systems, or characteristic functions of contractive operator colligations, are generalized Schur functions. The existence of optimal and *-optimal minimal realizations for generalized Schur functions are proved. By using those realizations, a new definition, which covers the case of generalized Schur functions, is given for defects functions. A criterion due to D.Z. Arov and M.A. Nudelman, when all minimal passive realizations of the same Schur function are unitarily similar, is generalized to the class of generalized Schur functions. The approach used here is new; it relies completely on the theory of passive systems.