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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A class of maps in an algebra with indefinite metric
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by Angelo B. Mingarelli PDF
Proc. Amer. Math. Soc. 121 (1994), 1177-1183 Request permission

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
  • 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, DOI 10.1515/crll.1989.398.1
    • —, 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.
  • 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
  • János Bognár, Indefinite inner product spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78, Springer-Verlag, New York-Heidelberg, 1974. MR 0467261
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 121 (1994), 1177-1183
  • MSC: Primary 47B50; Secondary 34A12, 46C20, 46H99, 47N20
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1197541-6
  • MathSciNet review: 1197541