Convexity properties of distinguished eigenvalues of certain classes of operators

Najman, B.

doi:10.1090/S0002-9939-1991-1027100-4

Convexity properties of distinguished eigenvalues of certain classes of operators
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Proc. Amer. Math. Soc. 111 (1991), 221-225 Request permission

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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