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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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  • [1] Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101–121. MR 0470754
  • [2] J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier, Grenoble 5 (1953–54), 305–370 (1955) (French). MR 0074787
  • [3] Roe W. Goodman, One-parameter groups generated by operators in an enveloping algebra., J. Functional Analysis 6 (1970), 218–236. MR 0268330
  • [4] Leonard Gross, Existence and uniqueness of physical ground states, J. Functional Analysis 10 (1972), 52–109. MR 0339722
  • [5] SigurÄ‘ur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
  • [6] P. E. T. Jórgensen, Extensions of symmetric operators and unbounded derivations, J. Math. Anal. Appl. 73 (1980), 115-133.
  • [7] Palle E. T. Jørgensen, Partial differential operators and discrete subgroups of a Lie group, Math. Ann. 247 (1980), no. 2, 101–110. MR 568200,
  • [8] -, Spectral theory of finite volume domains in $ {{\mathbf{R}}^n}$, Adv. in Math, (to appear).
  • [9] Hubert Kalf, On the characterization of the Friedrichs extension of ordinary or elliptic differential operators with a strongly singular potential, J. Functional Analysis 10 (1972), 230–250. MR 0348256
  • [10] M. Krein, The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 431–495 (Russian, with English summary). MR 0024574
  • [11] Peter D. Lax and Ralph S. Phillips, Scattering theory for automorphic functions, Princeton Univ. Press, Princeton, N.J., 1976. Annals of Mathematics Studies, No. 87. MR 0562288
  • [12] Edward Nelson and W. Forrest Stinespring, Representation of elliptic operators in an enveloping algebra, Amer. J. Math. 81 (1959), 547–560. MR 0110024,
  • [13] F. Rellich, Ein Satz über mittlere Konvergenz, Nachr. Acad. Wiss. Göttingen Math.-Phys. Kl.II (1930), 30-35.

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

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