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A class of maps in an algebra with indefinite metric

Author: Angelo B. Mingarelli
Journal: Proc. Amer. Math. Soc. 121 (1994), 1177-1183
MSC: Primary 47B50; Secondary 34A12, 46C20, 46H99, 47N20
MathSciNet review: 1197541
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Abstract: We study a class of hermitian maps on an algebra endowed with an indefinite inner product. We show that, in particular, the existence of a non-real eigenvalue is incompatible with the existence of a real eigenvalue having a right-invertible eigenvector. It also follows that for this class of maps the existence of an appropriate extremal for an indefinite Rayleigh quotient implies the nonexistence of nonreal eigenvalues. These results are intended to complement the Perron-Fröbenius and Kreĭn-Rutman theorems, and we conclude the paper by describing applications to ordinary and partial differential equations and to tridiagonal matrices.

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

  • [1] W. Allegretto and A. B. Mingarelli, Boundary problems of the second order with an indefinite weight-function, J. Reine Angew. Math. 398 (1989), 1–24. MR 998469
  • [2] -, On the non-existence of positive solutions for a Schrödinger equation with an indefinite weight function, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 69-73.
  • [3] F. V. Atkinson and A. B. Mingarelli, Asymptotics of the number of zeros and of the eigenvalues of general weighted Sturm-Liouville problems, J. Reine Angew. Math. 375/376 (1987), 380–393. MR 882305
  • [4] János Bognár, Indefinite inner product spaces, Springer-Verlag, New York-Heidelberg, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78. MR 0467261
  • [5] M. A. Krasnosel′skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. MR 0181881

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Keywords: Kreĭn space, hermitian maps, Perron-Fröbenius
Article copyright: © Copyright 1994 American Mathematical Society