Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions |
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Authors: | Damir Z Arov Olof J Staffans |
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Institution: | (1) Division of Mathematical Analysis, South-Ukrainian Pedagogical University, 65020 Odessa, Ukraine;(2) Department of Mathematics, Ǻbo Akademi University, Biskopsgatan 8, Axelia 3 vǻn, FIN-20500 Ǻbo, Finland |
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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 , which is either the open unit disk or the open right half-plane . 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 has a bi-inner dilation mapping into ; 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 and (namely and ) satisfying and for almost all . 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.
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Keywords: | " target="_blank"> Darlington method optimal passive realization *-optimal passive realization balanced passive realization |
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