Positive Desch-Schappacher perturbations of bi-continuous semigroups on AM-spaces

We consider positive Desch–Schappacher perturbations of bi-continuous semigroups on AM-spaces with an additional property concerning the additional locally convex topology. As an examples, we discuss perturbations of the left-translation semigroup on the space of bounded continuous function on the real line and perturbations of the implemented semigroup on the space of bounded linear operators. Dynamical processes occurring, e.g., in population models, quantum mechanics, or the financial world, are frequently expressed by a particular class of partial differential equations, the so-called evolution equations. A general operator theoretical method for dealing with those equations is the one using abstract Cauchy problems on a Banach space. In some cases, it is possible to write a given operator (A,D(A)) as a sum of simpler operators and this is where perturbation theory enters the area of evolution equations. The general question is: given a generator (A,D(A)) and another linear operator (B,D(B)), under which conditions does the operator A+B generate a semigroup? When talking about one-parameter semigroups of linear operators on Banach spaces, mostly C0-semigroups come to mind. Nevertheless, there are operator semigroups which are not strongly continuous with respect to the norm on the Banach space but for some weaker additional locally convex topology. This is one of the reasons why people are interested in different continuity concepts of semigroups and more general solutions in order to overcome these limitations of strongly continuous semigroups. One of the auspicious approaches to this gives rise to so-called bi-continuous semigroups, which were introduced by Kühnemund.