Spectral mapping theorems for differentiable $C_0$ 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 $\sigma _{*}(T(t))\backslash \{0\}=\{e^{\lambda s}, \lambda \in \sigma _{*}(A)\}$ holds, where $\sigma _*$ 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 $n^{th}$ derivative $T(t)^{(n)}$ of a differentiable $C_0$ semigroup $(T(t))_{t\ge 0}$.