Spectral mapping theorems of differentiable C0 semigroups

Abstract

Let $(T(t){)}_{t\ge 0}$ be a $C_0$ semigroup on a Banach space $X$ with infinitesimal generator $A$. In this work, we give conditions for which the spectral mapping theorem ${\sigma }_{\ast }(T(t))\backslash \{0\}=\{e^{\lambda s},\lambda \in {\sigma }_{\ast }(A)\}$ holds, where ${\sigma }_{\ast }$ can be equal to the essential, Browder and Kato spectrum. Also, we will be interested in the relations between the spectrum of $A$ and the spectrum of the nth derivative $T(t{)}^{(n)}$ of a differentiable $C_0$ semigroup $(T(t){)}_{t\ge 0}$.