Let $\lambda>0$ and $n\geq 1$. Prove that the operator
$-\Delta+\lambda I:H^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$
is invertible and find the norm
$\left|\left|\left(-\Delta+\lambda I\right)^{-1}\right|\right|_{L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)}.$
Futhermore show that
$\left(-\Delta+\lambda I\right)^{-1}:H^s(\mathbb{R}^n)\to H^{s+2}(\mathbb{R}^n)\qquad s\in\mathbb{R}$
is bounded and find the norm.
I don't know how to find norm of inverse when $\lambda$ is not $0$ and I'm not sure if invertibility can be proved in same way.