# $C_0$-semigroups

In the setting of partially ordered banach spaces me and Prof. Mikhail Igorevich Kamenskii have proven necessary and sufficient conditions for a $C_0$-semigroup of the form $e^{(-\Gamma+M)t}$ to be stable. The generator $-\Gamma+M$ is a (possibly) unbounded linear operator $-\Gamma$ perturbed by a bounded, positive (in the sense of the partial order structure) linear operator $M$. Our main requrements are that the unperturbed semigroup $e^{-\Gamma t}$ is stable and the product $\Gamma^{-1}M$ is contracting.