Classes of realizable Herglotz-Nevanlinna functions

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


In this chapter we are going to introduce three distinct subclasses N0(R), N1(R), and N01(R) of the class of functions N(R) realizable as impedance functions of L-systems, that was studied in Chapter 6. We give complete proofs of direct and inverse realization theorems in each subclass. We show that each subclass is characterized by a different property of the state-space operator in the corresponding realizing L-system. Based on this partition of the class N(R), we introduce the corresponding structure of subclasses Ω0(R, J), Ω1(R, J), and Ω01(R, J) on the class Ω (R, J) consisting of functions realizable as transfer functions of L-systems. Multiplication and coupling theorems are then proved for each subclass of Ω (R, J).


Hilbert Space Function Versus Symmetric Operator Impedance Function Boundary Triplet 
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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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