Let $E$ be a $\mathbb R$-Banach space, $(A,\mathcal D(A))$ be a linear operator and $\lambda\in\mathbb R$ such that $$A_\lambda:=\lambda\operatorname{id}_{\mathcal D(A)}-A$$ is injective, $A_\lambda\mathcal D(A)$ is dense and $R_\lambda:=A_\lambda^{-1}$ is bounded. We know that $R_\lambda$ has a unique bounded linear extension to $E$.
However, $R_\lambda$ is only on $A_\lambda\mathcal D(A)$ the proper inverse of $A_\lambda$. So, if $y\in E$ and $x:=R_\lambda y$, what is $A_\lambda y$?
I guess $R_\lambda A_\lambda$ is kome kind of projection onto $\mathcal D(A)$.