In the context of the following fact:
Fact: For an operator $A$ on $X$ (Banach space), are equivalet; A generates a bounded holomorphic semigroup on $X$ if an only if $\left\{z\in\mathbb{C}:\text{Re}(z)>0\right\}\subset \rho(A)$ and $\sup_{\text{Re}(\lambda)}\left\|\lambda R(\lambda,A)\right\|<\infty$.
Question. If $A$ is an arbitrary operator. When operator $A$ has a resolvent set $\rho(A)$ is only made up of real numbers?
I made this question in order of relaxing the condition $\left\{z\in\mathbb{C}:\text{Re}(z)>0\right\}$ by $\left\{z\in\mathbb{R}:z>0\right\}\subset\rho(A)$.
Thanks.