Abstract
A homogeneous additive operator A, positive on a cone K of a Banach space E partially ordered by K, is investigated. It is assumed that K is a reproducing cone in E and that A has a characteristic vector u0: Au0 = λ0u0 in K. It is proved that if\(AK \subset K_{ u_0 ,\rho }\) for some ρ≥1, then any other characteristic value λof A satisfies the inequality |λ|<(ρ-1)/(ρ+1) λ0. This is the best possible upper bound in the class of operators considered.
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Translated from Matematicheskie Zametki, Vol. 19,No. 1, pp. 27–33, January, 1971.
The author takes this opportunity to thank M. A. Krasnosel'skii for his interest in this work.
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Pokornyi, Y.V. Inequality for second characteristic values of positive operators of certain classes. Mathematical Notes of the Academy of Sciences of the USSR 9, 17–20 (1971). https://doi.org/10.1007/BF01405044
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DOI: https://doi.org/10.1007/BF01405044