Consider the operator $H_0=-\Delta$ acting on $L^2(\mathbb R^n)$. The spectrum of $H_0$ is $\sigma(H_0)=\sigma_{ac}(H_0)=[0,+\infty)$.
I've been said that it is straightforward to show that if $I$ is an interval not containing $0$ then exist a constant $c>0$ such that $$E_I(H_0)H_0 E_I(H_0)\geq cE_I(H_0).$$ where $E(H_0)$ are the spectral projections (SP) of $H_0$. Why is that?
I know this is a basic consequence of the relation between $H_0$ and its SP but I'm not totally confortable about how to handle SP under AC spectrum.