  Let $(T(t))_{t\geq 0}$ 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 nth derivative
$T(t)^{(n)}$ of a differentiable $C_0$ semigroup $(T(t))_{t\geq0}$.
