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semigroups of positive operators; quasimonotonicity
In a Banach space $E$, let $U(t)$ $\,(t>0)$ be a $C_0$-semigroup with generating operator $A$. For a cone $K\subseteq E$ with non-empty interior we show: $(\star)$ \quad $U(t)[K]\subseteq K$ $\,(t>0)$ holds if and only if $A$ is quasimonotone increasing with respect to $K$. On the other hand, if $A$ is not continuous, then there exists a regular cone $K\subseteq E$ such that $A$ is quasimonotone increasing, but $(\star)$ does not hold.
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