Domain
Denote common domain:
$$B^\eta_R:=B_R\cap\{|\eta|\leq R\}:\quad\mathcal{D}_0^\eta:=\bigcup_{R>0}\mathcal{R}E(B^\eta_R)$$
It is included in both since:
$$\|\eta(N)\varphi\|^2=\int|\eta|^2\mathrm{d}\nu_\varphi\leq R_\varphi^2\|\varphi\|^2<\infty$$
$$\|\Lambda_s(N)\varphi\|^2=\int\Lambda_s^2\mathrm{d}\nu_\varphi\leq(1+ R_\varphi^2)^{|s|}\|\varphi\|^2<\infty$$
It is dense in scale space:
$$\|\varphi-E(B^\eta_R)
\varphi\|_s^2=\int(1-1^\eta_R)\Lambda_s^2\mathrm{d}\nu_\varphi\stackrel{R\to\infty}{\to}0$$
They remain invariant:
$$E(B^{\eta}_R)\eta(N)\varphi=\eta(N)E(B^{\eta}_R)\varphi=\eta(N)\varphi$$
Concluding preparation.
Operator
Define maximal operator:
$$\eta(N):\mathcal{D}\eta(N)\cap\mathcal{D}\Lambda_s(N)\to\mathcal{D}\Lambda_{s+a}(N)$$
It is well defined and bounded:
$$\|\eta(N)\varphi\|_{s+a}^2=\int\Lambda_{s+a}^2|\eta|^2\mathrm{d}\nu_\varphi\leq\|\Lambda_a\eta\|_\infty^2\int\Lambda_s^2|\eta|^2\mathrm{d}\nu_\varphi=\|\Lambda_a\eta\|_\infty^2\|\varphi\|_s^2$$
By density it extends to all:
$$\overline{\mathcal{D}\eta(N)\cap\mathcal{D}\Lambda_s(N)}^s=\overline{\mathcal{D}^\eta_0}^s=\mathcal{H}^s$$
Concluding operator.
Product
As they remain invariant:
$$\omega:=|\eta|+|\vartheta|:\quad\eta(N)\mathcal{D}_0^\omega\subseteq\mathcal{D}^\omega_0\subseteq\mathcal{D}\vartheta(N)$$
By measurable calculus:
$$\varphi\in\mathcal{D}_0^\omega:\quad\eta(N)\vartheta(N)\varphi=\vartheta(N)\eta(N)\varphi$$
By density it extends to all:
$$\overline{\mathcal{D}\vartheta(N)\eta(N)\cap\mathcal{D}\Lambda_s(N)}^s=\overline{\mathcal{D}_0^\omega}^s=\mathcal{H}^s$$
And it splits into product:
$$\overline{\vartheta(N)\eta(N)}\subseteq\overline{\vartheta(N)}\cdot\overline{\eta(N)}\subseteq\overline{\vartheta(N)\eta(N)}$$
Concluding product.
Unitaries
They are isometric since:
$$\|\Lambda_a(N)\varphi\|_{s-a}=\|\Lambda_{s-a}(N)\Lambda_a(N)\varphi\|=\|\Lambda_s(N)\varphi\|=\|\varphi\|_s$$
As well as surjective since:
$$\overline{\Lambda_a(N)}\cdot\overline{\Lambda_{-a}(N)}=\overline{\Lambda_a(N)\Lambda_{-a}(N)}=\overline{1}=1_{s-a}$$
Concluding unitarity.
Resolvent
It is bounded for both:*
$$\delta_-(|z|)\stackrel{z\notin\sigma(N)}{\leq}\frac{|\lambda-z|^2}{1+|\lambda|^2}\leq\delta_+(|z|)$$
So one obtains inverses:
$$\overline{R(z)}\cdot\overline{(N-z)}=\overline{R(z)(H-z)}=\overline{1_{\mathcal{D}N}}=1_s$$
$$\overline{(N-z)}\cdot\overline{R(z)}=\overline{(N-z)R(z)}=\overline{1_\mathcal{H}}=1_{s-1}$$
Concluding resolvent.
Adjoint
Regard dense elements:
$$\varphi,\psi\in\mathcal{D}_0^\eta\subseteq\mathcal{D}\eta(N),\mathcal{D}\eta(N)^*,\mathcal{D}\Lambda_a(N),\mathcal{D}\Lambda_s(N)$$
By measurable calculus:
$$\langle\Lambda_a(N)\eta(N)\varphi,\psi\rangle_s=\langle\Lambda_s(N)\Lambda_a(N)\eta(N)\varphi,\Lambda_s(N)\psi\rangle\\
=\langle\Lambda_s(N)\varphi,\Lambda_s(N)\Lambda_a(N)\eta^*(N)\psi\rangle=\langle\varphi,\Lambda_a(N)\eta^*(N)\psi\rangle_s$$
So the adjoint writes as:
$$\overline{\Lambda_a(N)}\cdot\overline{\eta^*(N)}=\bigg(\overline{\Lambda_a(N)}\cdot\overline{\eta(N)}\bigg)^*=\overline{\eta(N)}^*\cdot\overline{\Lambda_a(N)}^*=\overline{\eta(N)}^*\cdot\overline{\Lambda_{-a}(N)}$$
Concluding adjoint.
*See the thread: Decay Behavior
