Let $B$ be a densely defined operator on a Banach space $X$ and let $B$ generate an analytic semigroup on $X$. I know that the right half-plane satisfies $\{Re\ z>0\}\subset \rho(B)$ ($\rho(B)$ denotes the resolvent set of $B$) and $||R(z,B)||\leq \frac{C}{1+|z|}$ for all $Re\ z>0$. Why is then $0\in\rho(B)$?
I know that the resolvent function $z\mapsto R(z,B)$ is holomorphic but how does that help? Thanks.