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 nonreal 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 Kreln-Rutman theorems, and we conclude the paper by describing applications to ordinary and partial differential equations and to tridiagonal matrices.

Additional Metadata
Keywords Hermitian maps, Krein space, Perron-FrÖbenius
Persistent URL dx.doi.org/10.1090/S0002-9939-1994-1197541-6
Journal Proceedings of the American Mathematical Society
Citation
Mingarelli, A. (1994). A class of maps in an algebra with indefinite metric. Proceedings of the American Mathematical Society, 121(4), 1177–1183. doi:10.1090/S0002-9939-1994-1197541-6