Geodesic
On the positivity of semigroups of operators
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 3, pp. 483-489.

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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.
Classification : 47B65, 47D06
Keywords: semigroups of positive operators; quasimonotonicity 
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Lemmert, Roland; Volkmann, Peter. On the positivity of semigroups of operators. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 3, pp. 483-489. http://geodesic.mathdoc.fr/item/CMUC_1998__39_3_a4/