Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the generator of a semigroup $T(t)$ with growth bound $\omega$, i.e., $\|T(t)\|\le Ce^{\omega t}$, we have $s(A) \le \omega$. Let $\alpha > s(A)$ be a real numer. Let $R(\lambda, A) = (\lambda - A)^{-1}$ be the resolvent. Do we have the following spectral representation of the semigroup $e^{tA}$? $$e^{tA} = \int_{\Gamma}e^{tz}R(z,A)dz,$$ where $\Gamma \subset \mathbb{C}$ is the vertical line $\{z\in\mathbb{C}, Re(z) = \alpha\}$.

Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the generator of a semigroup $T(t)$ with growth bound $\omega$, i.e., $\|T(t)\|\le Ce^{\omega t}$, we have $s(A) \le \omega$. Let $\alpha > s(A)$ be a real numer. Let $R(\lambda, A) = (\lambda - A)^{-1}$ be the resolvent. Do we have the following spectral representation of the semigroup $e^{tA}$? $$e^{tA} = \frac{1}{2\pi i}\int_{\Gamma}e^{tz}R(z,A)dz,$$ where $\Gamma \subset \mathbb{C}$ is the vertical line $\{z\in\mathbb{C}, Re(z) = \alpha\}$.