Let $\Omega \subset \mathbb{R}^n$ with smooth boundary. Let $A=\Delta$ laplace operator on $\mathcal{D}(A)=H^{2}(\Omega) \cap H_{0}^{1}(\Omega)$.
Then in [Li, Xungjing, and Jiongmin Yong. Optimal control theory for infinite dimensional systems. 1995] says $(I-A)^{-1}: L^{2}(\Omega) \rightarrow \mathcal{D}(A)$ exists as an operator on $L^2$.
It' not clear to me why the range of $(I-A)^{-1}$ is $D(A)$, can someone explain or give some references?