Point-spectrum of semibounded operator extensions

Author:
Palle E. T. Jørgensen

Journal:
Proc. Amer. Math. Soc. **81** (1981), 565-569

MSC:
Primary 47A70; Secondary 47D10

DOI:
https://doi.org/10.1090/S0002-9939-1981-0601731-9

MathSciNet review:
601731

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Abstract: Let denote the Friedrichs extension of a given semibounded operator in a Hilbert space. Assume , and . If for a finite-dimensional projection in the Hubert space we have Const. , then it follows that is an eigenvalue of , and the corresponding eigenspace is contained in the range of . Using this, together with the known order structure on the family of selfadjoint extensions, with given lower bound 0, of minus the Laplace-Beltrami operator, we establish the identity for all for the following problem.

is a unitary representation of a Lie group , and acts on the Hilbert space for some Nikodym-domain . Moreover is obtained as a certain normalized integral for the left--in variant vector fields on , that is, for each such vector field , the skew-adjoint operator is an extension of when regarded as a skew-symmetric operator in with domain .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1981-0601731-9

Keywords:
Estimates for operators,
extensions,
eigenvalues

Article copyright:
© Copyright 1981
American Mathematical Society