Is the following argument correct?
Suppose $A$ is a closed and densely defined operator on a Banach space $X$ such that $0$ is a simple pole of the resolvent $R(\cdot, A)$ with associated spectral projection denoted by $P.$ Since $0$ is an isolated spectral value, there exists $\mu <0$ such that $(\mu,0) \subseteq \rho(A).$ We know $$\lambda R(\lambda ,A) \stackrel{\text{strongly}}{\longrightarrow} P\qquad \text{ as } \lambda \to 0, \lambda \in (\mu,0).$$ Therefore there exists $K>0$ such that $$\sup_{\lambda \in (\mu,0)}\|\lambda R(\lambda,A)\|\leq K.$$
Edit: If yes, then does the conclusion still hold if one or more the assumptions are weakened?