I need to work with this operator, with $X=L^2(\Omega)$, $\Omega\subset \mathbb{R}^n$, $\lambda>0$ and $D(A)=\{u\in H^2(\Omega): \frac{\partial u}{\partial n}=0\}$:
\begin{align} A:D(A)\subset X\to X\\ Au=-\Delta u +\lambda u \end{align} I proved that $A$ is sectorial. Now, to considerating all possible power space, I need to enlarge the space to $H^{-1}$. How is the argue that should I to use? Integration by parts? I also need that $A$ continues sectorial.
Recall that an operator is sectorial if there exists $a\in \mathbb{R}, \phi\in (0,\pi/2)$ and $M>0$ such that:
- the resolvent set contains the sector $S=\{\lambda\in \mathbb{C}; \phi\leq |\arg(\lambda-a)|\leq \pi, \lambda\neq a\}$;
- $\|(\lambda I-A)^{-1}\|\leq \frac{M}{|\lambda-a|}$, $\forall \lambda \in S$.