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Accretive (*)-extensions and Realization Problems

  • Yury ArlinskiĭEmail author
  • Sergey Belyi
  • Eduard Tsekanovskiĭ
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)

Abstract

We present a solution of the extended Phillips-Kato extension problem about existence and parametrization of all accretive (*)-extensions (with the exit into triplets of rigged Hilbert spaces) of a densely defined non-negative operator.I n particular, the analogs of the von Neumann and Friedrichs theorems for existence of non-negative self-adjoint (*)-extensions are obtained. Relying on these results we introduce the extremal classes of Stieltjes and inverse Stieltjes functions and show that each function from these classes can be realized as the impedance function of an L-system.I t is proved that in this case the realizing L-system contains an accretive operator and, in case of Stieltjes functions, an accretive (*)-extension.M oreover, we establish the connection between the above-mentioned classes and the Friedrichs and Kre”n-von Neumann extremal non-negative extensions.

Keywords

Accretive operator L-system realization 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Yury Arlinskiĭ
    • 1
    Email author
  • Sergey Belyi
    • 2
  • Eduard Tsekanovskiĭ
    • 3
  1. 1.Department of MathematicsEast Ukrainian National UniversityLuganskUkraine
  2. 2.Department of MathematicsTroy UniversityTroyUSA
  3. 3.Department of MathematicsNiagara UniversityNew YorkUSA

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