The resolvent set of an operator is always contained in the angular sector of angle $\pi$?

Question

I have a query about the resolving set that has arisen studying the Laplacian operator.

In abstract, if $A:D(A)\subset X\to X$ with $X$ Banach space. The resolvent set of $A$ is in always contained in the set $\left\{\lambda\in\mathbb{C}\setminus 0: |\text{arg}(z)|<\pi\right\}$?