On Livšic-Brodskii nodes with strongly regular \(J\)-inner characteristic matrix functions in the Hardy class. (English) Zbl 1002.47005
Kérchy, László (ed.) et al., Recent advances in operator theory and related topics. The Béla Szőkefalvi-Nagy memorial volume. Proceedings of the memorial conference, Szeged, Hungary, August 2-6, 1999. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 127, 83-96 (2001).
The Livšic-Brodskii nodes were introduced and investigated in the paper [D. Z. Arov and H. Dym, Integral Equations Oper. Theory 29, No. 4, 373-454 (1997; Zbl 0902.30026)]. The author considers the case when the Livšic-Brodskii node \(\Sigma\) has the characteristic matrix function \(\Theta=\Theta_\Sigma\) from the class of strongly regular \(J\)-inner matrix functions such that \((z+i)^{-1}\Theta(z)\in H^{m\times m}_2\) (the Hardy class). A full characterization of such nodes is obtained. In particular, it is proved that the basic operator \(A\) of such a node is similar to a dissipative operator and that the evolution semigroup \(T(t)=e^{iAt}\) is bistable. Also some equivalent conditions for this class of nodes in terms of the theory of linear time-invariant systems are presented.
For the entire collection see [Zbl 0971.00017].
For the entire collection see [Zbl 0971.00017].
Reviewer: Andrzej Sołtysiak (Poznań)
MSC:
| 47A48 | Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. |
| 47A45 | Canonical models for contractions and nonselfadjoint linear operators |
| 93B07 | Observability |
| 93C99 | Model systems in control theory |
| 93B05 | Controllability |
