Semigroups and infinitesimal generators

Question

I am studying semigroups, however I don't know how to prove the authors following claim (he uses this fact as part of a bigger proof, but he doesn't show it):

Let $A : D(A) \subset X \rightarrow X$ be an operator, the infinitesimal generator of $\{T(t)\}$, which satisfies the following properties:

1. $A$ is closed;
2. $\exists\omega\in\Bbb{R}:Z:=\{ \lambda\in\Bbb{C} : \Re\lambda > \omega\}\subset \rho(A)$;
3. $\exists M>0:\|R(\lambda,A)\|_{L(X)} \le M|\lambda-\omega|^{-1}$, $\forall\lambda\in Z$.

Then $L_1=A-\beta I$ and $L_2=\beta A$ also satisfy (1),(2),(3).

Why is this true? Any hints are appreciated.