Passive systems with a normal main operator and quasi-selfadjoint systems

TL;DR

This paper studies passive systems with a normal main operator, characterizes the subclass of quasi-selfadjoint systems, and explores their spectral properties, transfer functions, and minimal realizations.

Contribution

It introduces and characterizes the class of quasi-selfadjoint passive systems and analyzes their spectral and transfer function properties.

Findings

Characterization of the subclass $S^{qs}$ of the Schur class.

Spectral theoretic conditions for unitary similarity of systems.

Connection between transfer functions in $S^{qs}$ and $Q$-functions.

Abstract

Passive systems $\tau={T,M,N,H}$ with $M$ and $N$ as an input and output space and $H$ as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system $\tau$ with $M=N$ is said to be quasi-selfadjoint if $ran(T-T^*)\subset N$. The subclass $S^{qs}$ of the Schur class $S$ is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass $S^{qs}$ is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass $S^{qs}$ and the $Q$-function of $T$ is given.