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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Convexity properties of distinguished eigenvalues of certain classes of operators


Author: B. Najman
Journal: Proc. Amer. Math. Soc. 111 (1991), 221-225
MSC: Primary 47A55; Secondary 15A18, 47A70, 47B15, 47B50
MathSciNet review: 1027100
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Abstract: We prove two convexity results:

(1) Let $ A(\varepsilon ) = {A_0} + \varepsilon {A_1}$ be a family of selfadjoint operators in a Krein space with separated spectrum so that the maximum $ \lambda \_(\varepsilon )$ of the spectrum of negative type of $ A(\varepsilon )$ is an isolated simple eigenvalue. Then $ \lambda \_(\varepsilon )$ is convex.

(2) Let $ \lambda \_(\varepsilon )$ be the left distinguished eigenvalue of the generalized eigenvalue problem $ ({A_0} + \varepsilon {A_1})x = \lambda Bx$ where $ {A_1}$ and $ B$ are real diagonal matrices and $ {A_0}$ is an irreducible essentially nonnegative matrix. Then $ \lambda \_(\varepsilon )$ is convex.

In both cases $ \lambda \_(\varepsilon )$ is strictly convex unless it is linear.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1027100-4
PII: S 0002-9939(1991)1027100-4
Article copyright: © Copyright 1991 American Mathematical Society