Abstract
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].
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References
Achieser, N.I., Glasmann, I.M.: Theorie der linearen Operatoren im Hilbertraum. Akademie-Verlag, Berlin 1968
Neumark, N.A.: Lineare Differentialoperatoren. Akademie-Verlag, Berlin 1967
Smirnow, W.I.: Lehrgang der höheren Mathematik V. Deutscher Verlag der Wissenschaften, Berlin 1967
Triebel, H.: Höhere Analysis. Deutscher Verlag der Wissenschaften, Berlin 1972
Weidmann, J.: Lineare Operatoren in Hilberträumen. Teubner, Stuttgart 1976
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Kaballo, W., Schneider, A. On the spectral kernel of symmetric extensions. Manuscripta Math 29, 113–118 (1979). https://doi.org/10.1007/BF01303622
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DOI: https://doi.org/10.1007/BF01303622