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

DOI:
https://doi.org/10.1090/S0002-9939-1991-1027100-4

MathSciNet review:
1027100

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove two convexity results:

(1) Let be a family of selfadjoint operators in a Krein space with separated spectrum so that the maximum of the spectrum of negative type of is an isolated simple eigenvalue. Then is convex.

(2) Let be the left distinguished eigenvalue of the generalized eigenvalue problem where and are real diagonal matrices and is an irreducible essentially nonnegative matrix. Then is convex.

In both cases is strictly convex unless it is linear.

**[1]**J. Bognar,*Indefinite inner product spaces*, Springer-Verlag, Berlin, Heidelberg, and New York, 1974. MR**0467261 (57:7125)****[2]**J. E. Cohen,*Convexity of the dominant eigenvalue of an essentially nonnegative matrix*, Proc. Amer. Math. Soc.**81**(1981), 657-658. MR**601750 (82a:15016)****[3]**T. Kato,*A short introduction to the perturbation theory of linear operators*, Springer-Verlag, Berline, Heidelberg, and New York, 1982. MR**678094 (83m:47015)****[4]**-,*Superconvexity of the spectral radius and convexity of the spectral bound and the type*, Math. Z.**180**(1982), 265-273. MR**661703 (84a:47049)****[5]**B. Najman,*Eigenvalues of the Klein-Gordon equation*, Proc. Edinburgh Math. Soc.**26**(1983), 181-190. MR**705262 (84k:35106)****[6]**K. Veselić,*A spectral theory for the Klein-Gordon equation with an external electrostatic potential*, Nuclear Phys. A.**147**(1970), 215-224. MR**0269218 (42:4114)**

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1027100-4

Article copyright:
© Copyright 1991
American Mathematical Society