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A class of maps in an algebra with indefinite metric. (English) Zbl 0806.46022

Summary: 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- Frobenius and Krejn-Rutman theorems, and we conclude the paper by describing applications to ordinary and partial differential equations and to tridiagonal matrices.

MSC:

46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
47B50 Linear operators on spaces with an indefinite metric
46K15 Hilbert algebras
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B24 Sturm-Liouville theory
35P05 General topics in linear spectral theory for PDEs
39A70 Difference operators
15A18 Eigenvalues, singular values, and eigenvectors
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References:

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