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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
MathSciNet review: 601731
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Abstract: Let $ \tilde H$ denote the Friedrichs extension of a given semibounded operator $ H$ in a Hilbert space. Assume $ \lambda I \leqslant H$, and $ \lambda \in \sigma (\tilde H)$. If for a finite-dimensional projection $ P$ in the Hubert space we have $ I - P \leqslant $ Const. $ (H - \lambda I)$, then it follows that $ \lambda $ is an eigenvalue of $ \tilde H$, and the corresponding eigenspace is contained in the range of $ P$. 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 $ {U_g}(1) = 1$ for all $ g \in G$ for the following problem.

$ U$ is a unitary representation of a Lie group $ G$, and acts on the Hilbert space $ {L^2}(\Omega )$ for some Nikodym-domain $ \Omega \subset G$. Moreover $ U$ is obtained as a certain normalized integral for the left-$ G$-in variant vector fields on $ \Omega $, that is, for each such vector field $ X$, the skew-adjoint operator $ dU(X)$ is an extension of $ X$ when regarded as a skew-symmetric operator in $ {L^2}(\Omega )$ with domain $ C_0^\infty (\Omega )$.

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Keywords: Estimates for operators, extensions, eigenvalues
Article copyright: © Copyright 1981 American Mathematical Society

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