On stability of $C_0$-semigroups

We prove that if $T(t)$ is a $C_0$-semigroup on a Hilbert space $E$, then (a) $1\in\rho(T(\omega))$ if and only if $\sup\{\Vert\int^t_0\exp\{(2\pi ik)/\omega\}T(s)x\,ds\Vert\colon t\geq 0, k\in\mathbf{Z}\}<\infty$, for all $x\in E$, and (b) $T(t)$ is exponentially stable if and only if $\sup\{\Vert\int^t_0\exp\{i\lambda t\}T(s)x\,ds\Vert\colon t\geq 0, \lambda\in\mathbf{R}\}<\infty$, for all $x\in E$. Analogous, but weaker, statements also hold for semigroups on Banach spaces.