Let $T : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ be a possibly unbounded self adjoint operator. Let $R_\varepsilon$ denote the resolvent $(T - i \varepsilon)^{-1}$, $\varepsilon > 0$. Suppose that, for all $\varepsilon \in (0, 1]$, there exists some $M > 0$ so that the weighted operator norm
$$\| (1 + |x|)^{-1} R_{\varepsilon}(1 + |x|)^{-1} \|_{L^2 \to L^2} \le M .$$
Then, is it true that $(1 + |x|)^{-1} R_{\varepsilon}(1 + |x|)^{-1}$ has a limit in the uniform topology as $ \varepsilon \to 0?$ More generally, if we replace the weight $(1 + |x|)^{-1}$ by an arbitrary bounded operator $B$, do we have that $\lim_{\varepsilon \to 0} BR_\varepsilon B $ exists in the uniform topology?
This statement is some type of "weighted" limiting absorption principle. I have been trying to prove this using the resolvent formula:
$$R_{\varepsilon_2} - R_{\varepsilon_1} = R_{\varepsilon_1}(i\varepsilon_1 - i\varepsilon_2)R_{\varepsilon_2},$$
To show that sequences $(1 + |x|)^{-1} R_{\varepsilon_k}(1 + |x|)^{-1}$ are Cauchy when $\varepsilon_k \to 0$. The idea is to use the resolvent formula to get to a place where we can take advantage of the bound $M$:
$$\| (1 + |x|)^{-1} R_{\varepsilon_2}(1 + |x|)^{-1} - (1 + |x|)^{-1} R_{\varepsilon_1}(1 + |x|)^{-1} \| = \| (1 + |x|)^{-1} R_{\varepsilon_1}(i\varepsilon_1 - i\varepsilon_2)R_{\varepsilon_2}(1 + |x|)^{-1}\| = \\ \| (1 + |x|)^{-1} R_{\varepsilon_1}(1 + |x|)^{-1}(1 + |x|)^{1}(i\varepsilon_1 - i\varepsilon_2)(1 + |x|)^{1}(1 + |x|)^{-1}R_{\varepsilon_2}(1 + |x|)^{-1}\|. $$
But I am stuck at this point because I have had to introduce the unbounded operator $(1 + |x|)^{1}$, and even though $(i\varepsilon_1 - i\varepsilon_2)$ can be small, I have little control over what $(1 + |x|)^{1}(i\varepsilon_1 - i\varepsilon_2)(1 + |x|)^{1}$ does to $L^2$ functions.
Solutions are greatly appreciated!