Geodesic
Inversion of the oscillatory property of focusing operators
Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 753-760.

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Suppose that $E$ is a real Banach space with a cone $K$ and suppose that the homogeneous additive operator $A$ that is positive on $K$ is focusing, i.e., $AK\subset K_{u_0\rho}$ for certain $u_0\in K$ and $\rho\ge1$. Then, as is well known, the operator $A$ uniformly reduces the oscillation (osc) between the elements of $K$. In this paper we show that only the focusing operators have this property. 
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     author = {Yu. V. Pokornyi and S. V. Smitskikh},
     title = {Inversion of the oscillatory property of focusing operators},
     journal = {Matemati\v{c}eskie zametki},
     pages = {753--760},
     publisher = {mathdoc},
     volume = {20},
     number = {5},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a15/}
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Yu. V. Pokornyi; S. V. Smitskikh. Inversion of the oscillatory property of focusing operators. Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 753-760. http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a15/