Adapting the Lanczos algorithm to matrices with almost continuous spectra

We consider approximating $B^T(A+sI)^{-1}B$, where $A\in\mathbb{R}^{n\times n}$ is large, symmetric positive definite, and $B\in\mathbb{R}^{n\times p}$ with $p\ll n$. We focus on the case where $A$ has an almost continuous (dense) spectrum: its eigenvalues fill one or more intervals so densely that