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Complex Analysis and Operator Theory

, Volume 3, Issue 1, pp 19–56 | Cite as

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

  • Yury M. Arlinskiĭ
  • Seppo HassiEmail author
  • Henk S. V. de Snoo
Article
  • 44 Downloads

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.

Mathematics Subject Classification (2000).

Primary 47A45, 47A48, 47A56 Secondary 93B15, 93B28 

Keywords.

Passive system transfer function quasi-selfadjoint contraction Q-function 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Yury M. Arlinskiĭ
    • 1
  • Seppo Hassi
    • 2
    Email author
  • Henk S. V. de Snoo
    • 3
  1. 1.Department of Mathematical AnalysisEast Ukrainian National UniversityLuganskUkraine
  2. 2.Department of Mathematics and StatisticsUniversity of VaasaVaasaFinland
  3. 3.Department of Mathematics and Computing ScienceUniversity of GroningenGroningenNederland

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