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This thread is Q&A.

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$.

Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$

Denote their evolutions: $$U_\#(t)^*=U_\#(-t)=U_\#(t)^{-1}$$

For a bounded operator: $$J:\mathcal{H}_0\to\mathcal{H}:\quad\|J\|<\infty$$

Assume the limit: $$\Omega\varphi:=\lim_{t\to\infty}U(t)^*JU_0(t)\varphi\quad(\varphi\in\mathcal{H})$$

Polar decomposition: $$\Omega=J_\Omega|\Omega|:\quad\mathcal{N}J_\Omega=\left(\overline{\mathcal{R}|\Omega|}\right)^\perp$$

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

How can I prove this?

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Denote for shorthand: $$\Omega':=\Omega^*\Omega\in\mathcal{B}(\mathcal{H}):\quad |\Omega|=\lim_n\Omega'_n$$

Regard spectral measures: $$H=\int\lambda\mathrm{d}E(\lambda)\quad H_0=\int\lambda\mathrm{d}E_0(\lambda)$$

By a previous thread:* $$\Omega E_0(A)=E(A)\Omega\implies\Omega'E_0(A)=E_0(A)\Omega'\implies(\Omega')^n E_0=E_0(A)(\Omega')^n$$

So one arrives at: $$E_0(A)|\Omega|=E_0(A)\lim_n\Omega'_n=\lim_nE_0(A)\Omega'_n=\lim_n\Omega'_nE_0(A)=|\Omega|E_0(A)$$

So one obtains: $$J_\Omega E_0(A)|\Omega|=J_\Omega|\Omega|E_0(A)=\Omega E_0(A)=E_0(A)\Omega=E(A)J_\Omega|\Omega|$$

Regard an element: $$\psi=\lim_n|\Omega|\varphi_n\in\overline{\mathcal{R}|\Omega|}$$

By continuity one has: $$J_\Omega E_0(A)\psi=J_\Omega E_0(A)\lim_n|\Omega|\varphi_n=\lim_nJ_\Omega E_0(A)|\Omega|\varphi_n\\ =\lim_nE_0(A)J_\Omega|\Omega|\varphi_n=E(A)J_\Omega\lim_n|\Omega|\varphi_n=E(A)J_\Omega\psi$$

Regard an element: $$\psi\in\left(\overline{\mathcal{R}|\Omega}|\right)^\perp=\mathcal{N}|\Omega|=\mathcal{N}\Omega=\mathcal{N}J_\Omega$$

By reducibility one has:** $$J_\Omega E_0(A)\psi=J_\Omega1_{(\mathcal{N}\Omega)^\perp}E_0(A)\psi=J_\Omega E_0(A)1_{(\mathcal{N}\Omega)^\perp}\psi=0=E(A)J_\Omega\psi$$

By measurable calculus:*** $$J_\Omega E_0(A)=E(A)J_\Omega\quad(A\in\mathcal{B}(\mathbb{R}))\implies J_\Omega\eta(H_0)\subseteq\eta(H)J_\Omega\quad(\eta\in\mathcal{B}(\mathbb{R}))$$

Concluding the assertion.

*See the thread: Calculus (WO)

**See the thread: Reducibility (WO)

***See proof of: Reducibility (SM)

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