Let $T(t)$ be a $C_{0}$-semigroup with the generator $A$. Now, does the so called integral representation of the resolvent $$ (\lambda - A)^{-1} = \int_{0}^{\infty} e^{-t\lambda}T(t) dt $$ hold for all $\lambda$ whose real part is strictly greater than the spectral bound $S(A)$ of the generator? (It is clear this representation holds if $S(A)$ is replaced by the growth bound, $\omega_{0}$, of the semigroup.)

This result holds for positive semigroups but how about in general?