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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1227–1311 | Cite as

Perturbations of Donoghue Classes and Inverse Problems for L-Systems

  • S. BelyiEmail author
  • E. Tsekanovskiĭ
Article
  • 22 Downloads

Abstract

We study linear perturbations of Donoghue classes of scalar Herglotz–Nevanlinna functions by a real parameter Q and their representations as impedance of conservative L-systems. Perturbation classes \({{\mathfrak {M}}}^Q\), \({{\mathfrak {M}}}^Q_\kappa \), \({{\mathfrak {M}}}^{-1,Q}_\kappa \) are introduced and for each class the realization theorem is stated and proved. We use a new approach that leads to explicit new formulas describing the von Neumann parameter of the main operator of a realizing L-system and the unimodular one corresponding to a self-adjoint extension of the symmetric part of the main operator. The dynamics of the presented formulas as functions of Q is obtained. As a result, we substantially enhance the existing realization theorem for scalar Herglotz–Nevanlinna functions. In addition, we solve the inverse problem (with uniqueness condition) of recovering the perturbed L-system knowing the perturbation parameter Q and the corresponding non-perturbed L-system. Resolvent formulas describing the resolvents of main operators of perturbed L-systems are presented. A concept of a unimodular transformation as well as conditions of transformability of one perturbed L-system into another one are discussed. Examples that illustrate the obtained results are presented.

Keywords

L-system Transfer function Impedance function Herglotz–Nevanlinna function Weyl–Titchmarsh function Livšic function Characteristic function Donoghue class Symmetric operator Dissipative extension Von Neumann parameter Unimodular transformation 

Mathematics Subject Classification

Primary 81Q10 Secondary 35P20 47N50 

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsTroy State UniversityTroyUSA
  2. 2.Department of MathematicsNiagara UniversityLewistonUSA

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