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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A comparison theorem for selfadjoint operators


Author: Amin Boumenir
Journal: Proc. Amer. Math. Soc. 111 (1991), 161-175
MSC: Primary 47B25; Secondary 34L40, 47A70, 47E05
MathSciNet review: 1021896
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Abstract: In this work we shall establish a result concerning the spectral theory of differential operators. Let $ {L_1}$ and $ {L_2}$ be two self-adjoint operators acting in two different Hubert spaces. Then under some conditions we shall prove that

$\displaystyle (d{\Gamma _1}/d{\Gamma _2})({L_2}) = \overline V V',$

where $ {\Gamma _1}(\lambda )$ and $ {\Gamma _2}(\lambda )$ are the spectral functions associated with $ {L_1}$ and $ {L_2}$ respectively. $ V$ is the shift operator mapping the set of generalized eigenfunctions of $ {L_1}$ into the set of generalized eigenfunctions of $ {L_2}$, that is

$\displaystyle y = V\varphi ,$

where $ {L_2}y = \lambda y$ and $ {L_1}\varphi = \lambda \varphi $.

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

  • [1] Aleksandrjian, Spectral decomposition of arbitrary self-adjoint operators into eigenfunctionals, Soviet Mat. 5 (1985), 607-611.
  • [2] W. N. Everitt and A. Zettl, On a class of integral inequalities, J. London Math. Soc. (2) 17 (1978), no. 2, 291–303. MR 0477234 (57 #16775)
  • [3] I. M. Gel′fand and A. G. Kostyučenko, Expansion in eigenfunctions of differential and other operators, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 349–352 (Russian). MR 0073136 (17,388g)
  • [4] I. M. Gel′fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2) 1 (1955), 253–304. MR 0073805 (17,489c)
  • [5] I.M. Gelfand and G. E. Shilov, Generalized functions, vols. 2-4, Academic Press, New York, 1961. (English transl.)
  • [6] Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0200692 (34 #580)
  • [7] K. Maurin, General eigenfunctions expansions, Polska. Akad. Nauk. 48 (1969).
  • [9] N. Naimark, Linear differential operators, Part 2, Ungar, New York, 1968. (English transl.)
  • [10] A. I. Plesner and V. A. Rohlin, Spectral theory of linear operators, Amer. Math. Soc. Transl. (2) 62 (1946), 24-101.
  • [11] J. Weidman, Spectral theory of differential operators, Lecture Notes in Math., vol. 1258, Springer-Verlag.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1021896-3
PII: S 0002-9939(1991)1021896-3
Article copyright: © Copyright 1991 American Mathematical Society