Quasi-compactness and uniform stabilization on general Banach spaces under $\theta$-subordinate perturbations of semigroup generators

Abstract

We prove that the quasi-compactness of an analytic semigroup is preserved under $\theta$-subordinate perturbations of its generator on general Banach spaces, provided the perturbations are compact along trajectories. This allows us to prove the permanence of uniform exponential stability for an analytic semigroup under similar perturbations of its generator, provided the perturbed generator generates a strongly stable semigroup. We then show how the analyticity assumption can be relaxed to the class of Crandall-Pazy semigroups under a smaller class of $\theta$-subordinate perturbations, and to immediately differentiable semigroups under bounded perturbations. As an application of the stabilization result, we consider the one-dimensional Neumann Laplacian on a non-reflexive Banach space. Finally, we present counterexamples that demonstrate the lack of uniform stabilization under the larger class of $A$-bounded perturbations.