Dissipative operators and additive perturbations in locally convex spaces

Let (A, D(A)) be a densely defined operator on a Banach space X. Characterizations of when (A, D(A)) generates a C-0-semigroup on X are known. The famous result of Lumer and Phillips states that it is so if and only if (A, D(A)) is dissipative and rg(lambda I - A) subset of X is dense in X for some lambda > 0. There exists also a rich amount of Banach space results concerning perturbations of dissipative operators. In a recent paper Tyran-Kaminska provides perturbation criteria of dissipative operators in terms of ergodic properties. These results, and others, are shown to remain valid in the setting of general non-normable locally convex spaces. Applications of the results to concrete examples of operators on function spaces are also presented. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim