An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$
$$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$
On a Hilbert space this is equivalent to saying that $\Re\langle Ax,x\rangle \le 0.$
In particular, if the spectrum of $A$ fulfills $\sigma(A)\subset (-\infty,0]$, then $A$ is dissipative.
I ask: Is the same true on Banach spaces, i.e. is any operator $A$ on a Banach space with$\sigma(A)\subset (-\infty,0]$ dissipative?