On structured finite–rank perturbations of positive operator semigroups

In this note, we give a Weiss–Staffans type perturbation result for the generator $ A $ of a positive semigroup on a Banach lattice $ X $. Assuming that the perturbation $ P:Z\to X_{-1} $ can be factorized as $ P = BC $ for positive operators $ C:Z\to\mathbb{R}^N $ and $ B:\mathbb{R}^N\to X_{-1} $, we show that the admissibility and invertibility conditions for the associated input-output map $ {\mathcal{F}}_\infty $ follow from the spectral condition $ \mathrm{r}(C R(\lambda,A_{-1})B)<1 $ for some $ \lambda>\omega_0(A) $. The abstract results are applied to domain perturbations of generators and perturbations of the first derivative.