I am trying to formulated a lemma, because my main result demainds a change of norm.
I have an operator $A\colon D(A)\subset X\to X$, $X$ Banach, and $(0,\infty)\subset \rho(A)$. This operator has the propriety that the resolvent is bounded, i.e. there exists $C>0$ such that, $$\|(\lambda I-A)^{-n}\|\leq \frac{C}{\lambda}, \,\forall n, \ \ \ \forall \lambda>0$$
Can I constructe an equivalent norm of $|\cdot|$ on $X$, $|\cdot|$, such that $ |(\lambda -A)^{-1}x|\leq \frac{1}{\lambda}|x|$ for all $x\in D(A)$? Thanks for any help.