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?
No, this cannot hold in general. An argument is the following.
If a Laplace transform representation of the resolvent holds for all $\Re\lambda>\omega$, then $\omega\geq \omega_1(A)$, see
Arendt, Batty, Hieber, Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Proposition 5.1.5. (and the definitions in Section 5.1).
Example 5.1.10 in the same book provides an example of a semigroup (in a Hilbert space) such that $$s(A)<\omega_1(A)$$ showing that the Laplace transform representation formula cannot be true for all $\Re\lambda>s(A)$.
