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, Volume 29, Issue 2–4, pp 113–118 | Cite as

On the spectral kernel of symmetric extensions

  • Winfried Kaballo
  • Albert Schneider


In a Hilbert space H we consider closed and symmetric operators A and à with closed ranges such that A⊂Ã. We prove a necessary and sufficient condition for the existence of a closed and symmetric operator B with A⊂B⊂à the range of which is not closed. We show that this condition can be fulfilled and, by the way, we get a counter example to the assertion that the continuous part of the spectral kernel of a symmetric operator is contained in the corresponding part of a symmetric extension, as is claimed in the books of Achieser-Glasmann [1], Neumark [2] and Smirnow [3].


Hilbert Space Number Theory Algebraic Geometry Topological Group Symmetric Operator 
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  1. 1.
    Achieser, N.I., Glasmann, I.M.: Theorie der linearen Operatoren im Hilbertraum. Akademie-Verlag, Berlin 1968Google Scholar
  2. 2.
    Neumark, N.A.: Lineare Differentialoperatoren. Akademie-Verlag, Berlin 1967Google Scholar
  3. 3.
    Smirnow, W.I.: Lehrgang der höheren Mathematik V. Deutscher Verlag der Wissenschaften, Berlin 1967Google Scholar
  4. 4.
    Triebel, H.: Höhere Analysis. Deutscher Verlag der Wissenschaften, Berlin 1972Google Scholar
  5. 5.
    Weidmann, J.: Lineare Operatoren in Hilberträumen. Teubner, Stuttgart 1976Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Winfried Kaballo
    • 1
  • Albert Schneider
    • 1
  1. 1.Abteilung MathematikUniversität DortmundDortmund 50

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