Let $[a,b] \subset \rho(T)$ and $T$ be a self-adjoint operator
then I want to show that $0=\frac{1}{\pi} \lim_{\varepsilon \downarrow 0} \lim_{\delta \downarrow 0} \int_{a+\delta}^{b+\delta} Im(\langle (T-s-i\varepsilon)^{-1}x,x \rangle ds$
My idea is that for small $\delta$ we have $[a+\delta , b+\delta] \subset \rho(T)$ and therefore $(T-s)^{-1}$ exists and $\langle (T-s)^{-1}x,x \rangle $ is purely real, as this operator is then also self-adjoint, but I don't quite see how to make this rigorous.
If $\lambda\in\mathbb{R}$, then $$ (x,(T-\lambda I)x) \in\mathbb{R},\;\;\; x \in \mathcal{D}(T). $$ Therefore, if $\lambda\in\mathbb{R}\cap\rho(T)$, you can set $x = (T-\lambda I)^{-1}y$ for any $y \in X$ in order to obtain $$ ((T-\lambda I)^{-1}y,y) \in\mathbb{R}. $$