Wave Operators: Calculus Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$.
Consider Hamiltonians:
$$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$
Denote their evolutions:
$$U_\#(t)^*=U_\#(-t)=U_\#(t)^{-1}$$
Regard a bounded operator:
$$J:\mathcal{H}_0\to\mathcal{H}:\quad\|J\|<\infty$$
Assume the limit:
$$\Omega\varphi:=\lim_{t\to\infty}U(t)^*JU_0(t)\varphi\quad(\varphi\in\mathcal{H})$$

Then one has:
  $$\eta\in\mathcal{B}(\mathbb{C}):\quad\Omega\eta(H_0)\subseteq\eta(H)\Omega$$

How can I prove this?
 A: Thanks really alot to T.A.E.!!!
Inclusion
They are bounded:
$$\|\Omega\varphi\|=\lim_{t\to\infty}\|U(t)^*JU_0(t)\varphi\|\leq\|J\|\cdot\|\varphi\|$$
By intertwining relations:
$$\int_{-\infty}^{+\infty}e^{-it\lambda}\mathrm{d}\langle E(\lambda)\Omega\varphi,\chi\rangle=\langle U(t)\Omega\varphi,\chi\rangle=\langle\Omega U_0(t)\varphi,\chi\rangle=\langle U_0(t)\varphi,\Omega^*\chi\rangle\\=\int_{-\infty}^{+\infty}e^{-it\lambda}\mathrm{d}\langle E_0(\lambda)\varphi,\Omega^*\chi\rangle=\int_{-\infty}^{+\infty}e^{-it\lambda}\mathrm{d}\langle\Omega E_0(\lambda)\varphi,\chi\rangle$$
By Fourier uniqueness:*
$$\Omega E_0(A)=E(A)\Omega\quad(A\in\mathcal{B}(\mathbb{R}))$$
By measurable calculus:**
$$\Omega\eta(H_0)\subseteq\eta(H)\Omega\quad(\eta\in\mathcal{B}(\mathbb{R}))$$
Concluding inclusion.
Strictness
Consider a Hamiltonian:
$$H_0:\mathcal{D}(H_0)\to\mathcal{H}_0:\quad\mathcal{D}(H_0)\subsetneq\mathcal{H}_0$$
Then for trivial operator:
$$J=0:\quad\mathcal{D}(\Omega H_0)=\mathcal{D}(H_0)\subsetneq\mathcal{H}_0=\mathcal{D}(0)=\mathcal{D}(H\Omega)$$
Concluding strictness.
*See the thread: Fourier Uniqueness
**See proof of: Reducibility
