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

Authors:	Yury M Arlinski? Seppo Hassi Henk S V de Snoo

Institution:	(1) Department of Mathematical Analysis, East Ukrainian National University, Kvartal Molodyozhny 20-A, UA-91034 Lugansk, Ukraine;(2) Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland;(3) Department of Mathematics and Computing Science, University of Groningen, P.O. Box 407, NL-9700 AK Groningen, Nederland

Abstract:	Passive systems $\tau = \{T,{\mathfrak{M}},{\mathfrak{N}},{\mathfrak{H}}\}$ with ${\mathfrak{M}}$ and ${\mathfrak{N}}$ as an input and output space and ${\mathfrak{H}}$ as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system $\tau$ with ${\mathfrak{M}} = {\mathfrak{N}}$ is said to be quasi-selfadjoint if ran $(T - T^*) \subset {\mathfrak{N}}$ . The subclass ${\bf S}^{qs}({\mathfrak{N}})$ of the Schur class ${\bf S}({\mathfrak{N}})$ is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass ${\bf S}^{qs}({\mathfrak{N}})$ is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass ${\bf S}^{qs}({\mathfrak{N}})$ and the Q-function of T is given. Received: December 16, 2007., Accepted: March 4, 2008.

Keywords:	Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 47A45 47A48 47A56 Secondary 93B15 93B28
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