If have the following problem:

Let $A : \mathcal{H} \to \mathcal{H}$ be a bounded, self-adjoint operator on some Hilbert space $\mathcal{H}$. Let $B: \mathcal{H} \to \mathcal{H}$ be a bounded, strictly positive operator $B \geq c > 0$. Is the following statement true or false: $$ 0 \in \sigma_{ac}(A) \Leftrightarrow 0 \in \sigma_{ac}(BAB) \quad ? $$ Or can one say something about the relation of $\lim_{y \searrow 0} \Im \langle f, (A- i y)^{-1} f \rangle$ and $\lim_{y \searrow 0} \Im \langle f, (A- iB y)^{-1} f \rangle$. $A$ and $B$ do not commute.

I'm aware of a similar question: Spectrum of the operator PAP, with A self-adjoint and P strictly positive. I hope that the one I state here is a little more specific as to come up with a counterexample or a proof.