In several papers I've seen the following argument and I don't understand it.
Let $H$ be a self-adjoint defined on the sobolev space $H^2(\mathbb R^d)$ with absolutely continuos spectrum. Moreover, it does not have positive eigenvalues.
My questions are:
1) If I got that $v(x)=O((1+|x|)^{-\alpha})$ for some $\alpha>0$, how can I prove that $v(x)(H+i)^{-1}$ is compact?
2) Knowing that, how can I claim that $E_I v(x) E_I$ is compact, where $I\subset (0,\infty)$ and $E_I$ is the spectral projection of $H$.
I guess is something related to the compact embedding of $H^2$ onto $L^2$, and the second part would be consequence of the ideal structure of the space of compact operators, but I can figure it out really.