Let $H$ be a closed operator on a Hilbert space $\mathcal H$. The Lumer-Phillips theorem states that $H$ is a generator of a one-parameter contraction semigroup if and only if
$\Re\langle H\psi,\psi\rangle\leq 0$ for all $\psi\in\mathcal H$ and
there exists at most one $z$ in $\rho(H)$ with $\Re z>0$.
In particular, the numerical range of $H$ plays an important role in determining wether $H$ generates a contraction semigroup. My question: Does there exist a theorem relating the numerical range of an operator to it being a generator of an arbitrary bounded semigroup (not necessarily one of contractions)?