Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Abstract

Passive linear systems τ = {A, B, C, D; H, M, N} have their transfer function Θτ (λ) = D + λC(I − λA) −1 B in the Schur class S(M, N). Using a parametrization of contractive block operators the transfer function Θτ (λ) is connected to the Sz.-Nagy-Foiaş characteristic function ΦA(λ) of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions Θτ (λ) and ΦA(λ). The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Mathematics Subject Classification (2000). Primary 47A45, 47A48, 47A56, 47N70; Secondary 93B15.

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

“…Then in (3.9) one has ϕ Θ (λ) = 0 and ψ Θ (λ) = 0. Thus by [5,Theorem 1.1] τ is conservative (in fact, this conclusion can be derived also from the proof of Theorem 5.8 above by applying Corollary 2.6). As Θ(λ) is nonconstant, A is nonisometric, K is isometric, and X appearing in (4.1) is unitary in D K * .…”

“…Hence, the system η = {T; V, V, H} is conservative. Since A ∈ C 00 , the system η is minimal; see Corollary 2.4 and, e.g., [5,Proposition 5.2]. In addition it is easy to see that the transfer function of η coincides with Θ(λ) and therefore Θ(λ) ∈ S qs (V) by Proposition 4.5.…”

“…In addition it is easy to see that the transfer function of η coincides with Θ(λ) and therefore Θ(λ) ∈ S qs (V) by Proposition 4.5. Since η is minimal and conservative, Θ(λ) ∈ S(V) is a bi-inner dilation of Θ(λ) ∈ S(N) in view of [5,Corollary 5.3]. …”

Minimal Passive Realizations of Generalized Schur Functions in Pontryagin Spaces

“…This was done by the author in [27] by using the Kreȋn-Langer factorizations. With the definition given therein, the main results of [4] were generalized to the Pontryagin state space setting. The main subjects of [27] include some continuation of the study of products of systems and the stability properties of passive systems, subjects treated earlier by Saprikin et al in [11].…”

“…The key idea here is to use optimal minimal passive realizations and conservative embeddings. By using such a definition, it is shown that one can generalize and improve some of the main results from [4], using different proofs than those given in [4] or [27], see Theorem 4.8. Furthermore, in Theorem 4.10, the main results from [9,10] concerning the criterion when all the minimal realizations of a Schur function are unitarily similar, is generalized to the present indefinite setting.…”