Spectral mapping theorems of differentiable C0 semigroups

Abstract: 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 $(T(t))_{t\geq0}$.