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Let $(T(t))_{t\geq 0}$ be a $C_0$ semigroup on a Banach space $X$ withinfinitesimal generator $A$. In this work, we give conditions for which thespectral mapping theorem $\sigma_{*}(T(t))\backslash \{0\}=\{e^{\lambda s},\lambda\in\sigma_{*}(A)\}$ holds, where $\sigma_*$ can be equal to theessential, Browder and Kato spectrum. Also, we will be interested in therelations between the spectrum of $A$ and the spectrum of the nth derivative$T(t)^{(n)}$ of a differentiable $C_0$ semigroup $(T(t))_{t\geq0}$.

Spectral mapping theorems of differentiable C0 semigroups