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Local completeness for eigenfunctions of regular maximal ordinary differential operators


Author: Robert Carlson
Journal: Proc. Amer. Math. Soc. 79 (1980), 400-404
MSC: Primary 47E05; Secondary 34B25
DOI: https://doi.org/10.1090/S0002-9939-1980-0567980-2
MathSciNet review: 567980
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Abstract: Let $ L = \sum\nolimits_{m = 0}^n {{A_m}(x){D^m}} $ be a differential operator on $ { \oplus ^k}{L^2}[0,1]$ with infinitely differentiable $ k \times k$ matrix valued coefficients. Assume that $ \det \,{A_n}(x) \ne 0$ for $ x \in [0,1]$. The domain of L is the set of k-vector valued functions f such that $ f \in {C^{n - 1}}([0,1]),{f^{(n - 1)}}$ is absolutely continuous on [0, 1] and $ Lf \in { \oplus ^k}{L^2}[0,1]$. Let $ {x_0} \in (0,1)$. Then there is a neighborhood $ U({x_0})$ containing $ {x_0}$ such that the restrictions of the eigenfunctions of L to $ U({x_0})$ have dense span in $ { \oplus ^k}{L^2}[U({x_0})]$. The example $ L = {e^{ - ix}}d/dx$ shows that this is the best possible abstract result.


References [Enhancements On Off] (What's this?)

  • [1] R. Carlson, Expansions associated with non-self-adjoint boundary-value problems, Proc. Amer. Math. Soc. 73 (1979), 173-179. MR 516459 (80e:47040)
  • [2] E. A. Coddington and A. Dijksma, Adjoint subspaces in Banach spaces, with applications to ordinary differential subspaces, Ann. Mat. Pura Appl. (to appear). MR 533601 (81k:47036)
  • [3] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monographs, vol. 18, Amer. Math. Soc., Providence, R. I., 1969. MR 0246142 (39:7447)
  • [4] S. Goldberg, Unbounded linear operators, McGraw-Hill, New York, 1966. MR 0200692 (34:580)
  • [5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford Univ. Press, Oxford, 1956.
  • [6] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [7] J. Ringrose, Compact non-self-adjoint operators, Van Nostrand Reinhold, London, 1971.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0567980-2
Article copyright: © Copyright 1980 American Mathematical Society

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