Canonical L-systems with Contractive and Accretive Operators

  • Yuri Arlinskii
  • Sergey Belyi
  • Eduard Tsekanovskii
Part of the Operator Theory: Advances and Applications book series (OT, volume 217)


In this chapter the Kreın classical theorem is extended to the case of quasiself-adjoint contractive extensions of symmetric contractions, and their complete parametrization is given. On its basis we present the solution of the Phillips-Kato restricted extension problem on existence and description of all proper maximal accretive and sectorial extensions of a densely-defined non-negative symmetric operator. The criterion in terms of the impedance function of an L-system for the state-space operator to be a contraction (or so-called θ-co-sectorial contraction) is obtained. We establish the conditions for a given Stieltjes and inverse Stieltjes function to be realized as an impedance function of some L-system. We also verify when the state-space operator of an L-system is maximal accretive or sectorial. The connections between the Friedrichs and Kre?in-von Neumann extensions and impedance function of L-systems are provided.


Hilbert Space Symmetric Operator Impedance Function Accretive Operator Contractive Extension 
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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Yuri Arlinskii
    • 1
  • Sergey Belyi
    • 2
  • Eduard Tsekanovskii
    • 3
  1. 1.Department of MathematicsEast Ukrainian National UniversityLuganskUkraine
  2. 2.Department of MathematicsTroy UniversityTroyUSA
  3. 3.Department of MathematicsNiagara UniversityLewistonUSA

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