NDLI: Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions

Content Provider	Springer Nature Link
Author	Arov, Damir Z. Staffans, Olof J.
Copyright Year	2008
Abstract	Let S(U; Y) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping one separable Hilbert space U into another separable Hilbert space Y , and which are defined on a domain $$\Omega \subset {\mathbb{C}}$$ , which is either the open unit disk $${\mathbb{D}}$$ or the open right half-plane $${\mathbb{C}}^+$$ . In the development of the Darlington method for passive linear time-invariant input/state/output systems (by Arov, Dewilde, Douglas and Helton) the following question arose: do there exist simple necessary and sufficient conditions under which a function $$\theta \in S(U; Y )$$ has a bi-inner dilation $$\Theta = \left[ \begin{array}{ll} \theta _{11}&\theta \\ \theta _{21}&\theta _{22} \end{array} \right] $$ mapping $$U_1 \bigoplus U$$ into $$Y \bigoplus Y1$$ ; here U 1 and Y 1 are two more separable Hilbert spaces, and the requirement that Θ is bi-inner means that Θ is analytic and contractive on Ω and has unitary nontangential limits a.e. on ∂Ω. There is an obvious well-known necessary condition: there must exist two functions $$\psi_r \in S(U; Y_{1})$$ and $$\psi_l \in S(U_{1}; Y)$$ (namely $$\psi_r = \theta_{22}$$ and $$\psi_l = \theta_{11}$$ ) satisfying $$\psi_{r}^{}(z)\psi_{r}(z) = I - \theta^{}(z)\theta(z)$$ and $$\psi_{l}(z)\psi_{l}^(z) = I - \theta(z)\theta^{}(z)$$ for almost all $$z \in \partial\Omega$$ . We prove that this necessary condition is also sufficient. Our proof is based on the following facts. 1) A solution ψ r of the first factorization problem mentioned above exists if and only if the minimal optimal passive realization of θ is strongly stable. 2) A solution ψ l of the second factorization problem exists if and only if the minimal *-optimal passive realization of θ is strongly co-stable (the adjoint is strongly stable). 3) The full problem has a solution if and only if the balanced minimal passive realization of θ is strongly bi-stable (both strongly stable and strongly co-stable). This result seems to be new even in the case where θ is scalar-valued.
Ending Page	42
Page Count	14
Starting Page	29
File Format	PDF
ISSN	0378620X
e-ISSN	14208989
Journal	Integral Equations and Operator Theory
Issue Number	1
Volume Number	62
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2008-09-17
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	optimal passive realization Analytic circuit theory Realizations from input-output data Analysis *-optimal passive realization Darlington method balanced passive realization Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Analysis

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