Inversion of the oscillatory property of focusing operators

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.