Parametrization of contractive block operator matrices and passive discrete-time systems
This paper presents some new proofs and new formulas concerning passive discrete-time systems \(\tau = \left\{A,B,C,D: \mathcal{H},\mathcal{M},\mathcal{N}\right\}\) and their transfer function \(\Theta_{\tau}(\lambda)=D+\lambda C(I-\lambda A)^{-1}B\). Concerning the realization of operator-valued Schur functions \(\Theta(\lambda)\) as transfer functions of passive systems \(\tau\), some new results are obtained. A characterization of controllable, observable and simple systems is given as the main operator \(A\) belongs to \( C_{0.}\), \(C_{.0}\) and \( C_{00}\), respectively. The paper has an attractive form, presenting the topic in the first section and the Sz.-Nagy and Foiaş model in the second section. In the following two sections, an identity for contraction and contractive block operators are analysed. The fifth section is dedicated to the analysis of the transfer functions of passive systems, and in the sixth one, using especially some results of Sz.-Nagy and Foiaş concerning factorization theorems, isometric, co-isometric and conservative systems are studied. Inner, co-inner and bi-inner dilations and characterizations for strongly stable, strongly co-stable systems are obtained in the seventh section of the paper. In the last section, operators of the class \(C(\alpha)\) and corresponding passive systems are analysed.
scientific article; zbMATH DE number 5178718