Spectral mapping theorems for differentiable $$C_0$$ semigroups

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 $$(T(t))_{t\ge 0}$$ .