Perturbations of bi-continuous semigroups

Abstract

initial value problems (Cauchy problems) are usually studied via operator semigroups. In many cases, the well-developed theory of C0-semigroups, i.e., one-parameter operator semigroups which are strongly continuous for the norm on a Banach space X , suffices and provides a powerful machinery to study such problems. The applications range from partial differential equations, Volterra integro-differential equations and dynamic boundary problems to delay equations. It seems that a linear (autonomous) evolution equation like { u′(t) = Au(t) u(0) = x, (EE) that is an equation that describes a system evolving from an initial state ”in time”, can be handled via the theory of C0-semigroups. However, as the most trivial example shows this is not the case. Consider the left shift semigroup S on the space of bounded, continuous functions Cb(R) S(t)f(s) := f(t+ s). It is clear that the orbit t 7→ S(t)f is continuous for the supremum norm, if and only if f is uniformly continuous. This shows that the shift semigroup does not fit into the framework of C0-semigroups, nevertheless S clearly describes an evolution system on R. The situation is not so bad either. If we replace the norm topology by the topology τc of uniform convergence on compact sets, then the orbits become τc-continuous. This leads us to weakening the notion of C0-semigroups, thus allowing to consider such ”pathological” cases, these therefore turn out to be less unpleasant than thought previously. There are numerous generalisations of the theory of C0-semigroups. Among these we find the approach of introducing new continuity notions, this method is we want to follow. Investigations on semigroups on locally convex spaces were started fairly long ago, see e.g., [14], [51], [52] and [64]. A nice exposition on the historical aspects can be found in [56]. To exploit the Banach space structure and the same time to introduce coarser topologies, the notion of bi-continuous semigroups was introduced recently by Kuhnemund in [56]. The theory developed therein covers a large part of previously known results, and puts these concrete examples in an abstract framework. The general theory is then applicable in concrete cases.