Setting: Let $(\mathcal{H}, \langle\cdot,\cdot\rangle)$ be a complex Hilbert space and consider two (possibly unbounded) self-adjoint operators $A = A^\ast\colon \mathcal{D}_A \mapsto \mathcal{H}$ and $B = B^\ast\colon \mathcal{D}_B \mapsto \mathcal{H}$. Suppose $B$ is invertible (consequently with a continuous inverse $B^{-1} \in \mathcal{L}(\mathcal{H}, \mathcal{D}_B)$) and $A$ is bounded relative to $B$, that is $\|A B^{-1}\| < \infty$.
Question: Under which circumstances can I conclude boundedness of the operators $B^{-1/2}AB^{-1/2}$ and/or $A^{1/2}BA^{1/2}$ (assuming non-negativity of $B$ and $A$, respectively)?