Abstract: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: $(\star )$ \hskip 1em\relax $U(t)[K]\subseteq K$ $ (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.
Keywords: semigroups of positive operators, quasimonotonicity
AMS Subject Classification: 47D06