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.
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.
@article{CMUC_1998_39_3_a4,
author = {Lemmert, Roland and Volkmann, Peter},
title = {On the positivity of semigroups of operators},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {483--489},
year = {1998},
volume = {39},
number = {3},
mrnumber = {1666770},
zbl = {0970.47026},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_1998_39_3_a4/}
}
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/