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 $rg(\lambda I − A) \subseteq X$ 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.

### Dissipative operators and additive perturbations in locally convex spaces

#### Abstract

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 $rg(\lambda I − A) \subseteq X$ 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11587/408290
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