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.
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- © 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