Geodesic
Inequality for second characteristic values of positive operators of certain classes
Matematičeskie zametki, Tome 9 (1971) no. 1, pp. 27-33.

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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 $u_0: Au_0=\lambda_0u_0$ in $K$. It is proved that if $AK\subset K_{u_0,\rho}$ for some $\rho\geqslant1$, then any other characteristic value $\lambda$ of $A$ satisfies the inequality $|\lambda|(\rho-1)/(\rho+1)\lambda_0$. This is the best possible upper bound in the class of operators considered. 
@article{MZM_1971_9_1_a3,
     author = {Yu. V. Pokornyi},
     title = {Inequality for second characteristic values of positive operators of certain classes},
     journal = {Matemati\v{c}eskie zametki},
     pages = {27--33},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_9_1_a3/}
}
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Yu. V. Pokornyi. Inequality for second characteristic values of positive operators of certain classes. Matematičeskie zametki, Tome 9 (1971) no. 1, pp. 27-33. http://geodesic.mathdoc.fr/item/MZM_1971_9_1_a3/