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 → X−1 can be factorized as P = BC for positive operators C : Z → RN and B : R^N → X−1 , we show that the admissibility and invertibility conditions for the associated input-output map F_∞ follow from the spectral condition r(CR(λ, A_−1 )B) < 1 for some λ > ω_0 (A). The abstract results are applied to domain perturbations of generators and perturbations of the first derivative.