On Structured Perturbations of Positive Semigroups

In this thesis we study perturbation results for the generator A of a positive C_0--semigroup on a Banach lattice X. We generalize results that hold for an AM- or AL-space X and extend them to an arbitrary Banach lattice X in case the perturbation can be factorized as P=BC via an AM- or AL-space U. This allows to deal with applications on (infinite dimensional) reflexive Banach spaces, which are a priori excluded if X is an AM- or an AL-space. First, B and C are assumed to be positive operators satisfying the spectral condition r(CR(\lambda,A_{-1})B)<1 for some \lambda>\omega_0(A). We then prove that the generation results obtained in this case still hold when B and C are not necessarily positive, but only dominated by positive operators. These abstract results are applied to domain perturbations of generators, a heat equation with boundary feedback and perturbations of the first derivative. Combining the previous results we then deal with "doubly unbounded" perturbations P:Z\to X_{-1}, provided they can be factorized as P=BC via U=\mathbb{R}^N. In this context, we prove a Staffans--Weiss type perturbation result. We show that the classically assumed admissibility and invertibility conditions for the associated input--output map \mathcal{F}_\infty follow from the (much simpler to verify) spectral condition mentioned above. The abstract results are applied to domain perturbations and finite rank perturbations of the first derivative.