Abstract
For a semibounded below self-adjoint operatorA in a Hilbert spaceH and a singular operatorV acting in theA-scale of Hilbert spaces, the notion of generalized sumA∓V is introduced. Conditions are found forA∓V to be self-adjoint in ℋ. In particular, it is shown that if a symmetric operatorV is semibounded or has a spectral gap, then there exists an α such that the generalized sumA∓αV is a self-adjoint operator inH. For a symmetric restrictionA = A‖D, D C D(A), with deficiency indices (1, 1), it is proved that each self-adjoint extension à of A admits representation as a generalized sum Ã=A∓V.
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Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 671–681, November, 1999.
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Karataeva, T.V., Koshmanenko, V.D. Generalized sum of operators. Math Notes 66, 556–564 (1999). https://doi.org/10.1007/BF02674196
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DOI: https://doi.org/10.1007/BF02674196