Isomorphism of spectral and translational presentations of selfadjoint dilation of dissipative operator

Let $A$ be a linear dissipative operator with dense domain $\mathfrak{D}(A)$ in Hilbert space $\mathfrak{H}$ and $−i \in \rho(A)$. We consider the self-adjoint operators $B = iR−iR^\ast−2R^\ast R, \widetilde{B} = iR−iR^\ast−2RR^\ast,$ where $R = (A + iI)^{−1}.$ Let $Q = \sqrt{B}, \widetilde{Q} = \sqrt{\widetilde{B}}, \mathfrak{H}_1 = \overline{Q\mathfrak{H}}, \mathfrak{H}_1 = \overline{\widetilde{Q}\mathfrak{H}}.$
1. Spectral presentation. We consider the Hilbert spaces $H_+ = L_2(0, \infty; \mathfrak{H}_1), H_− = L_2(−\infty, 0; \mathfrak{H}_2), H = H_−\oplus \mathfrak{H}\oplus H_+$ and operator $S, h = (h_−, h_0, h_+) \in \mathfrak{D}(S)$ if and only if
a) $\left\{ h_\pm, \frac{dh_\pm}{dt} \right\} \subset H\pm,$
b) $\phi = h_0 + Qh_−(0) \in \mathfrak{D}(A),$
c) $h_+(0) = T^\ast h_−(0) + iD\phi,$
where $T^\ast = I + 2iR^\ast, D = Q(A + iI). S(h_−, h_0, h_+) = \left( i\frac{dh_−}{dt}, −ih_0 + (A + iI)\phi, \frac{dh_+}{dt} \right). S$ is dilatation of $A.$
2. Translational presentation. We consider the Hilbert spaces $\mathfrak{H}_− = \oplus_{-\infty}^{-1}\mathfrak{H}_2, \mathfrak{H}_+ = \oplus_{\infty}^{1}\mathfrak{H}_1$ and $\textbf{H} = \mathfrak{H}_−\oplus\mathfrak{H}\oplus\mathfrak{H}_+, f = (\dots, f_{-1}, f_0, f_1, \dots) \in \textbf{H}$ if and only if $\sum_{-\infty}^\infty \|f_n\|^2 < \infty, f_0 \in \mathfrak{H}, f_n \in \mathfrak{H}_1, f_{−n} \in \mathfrak{H}_2, n \in \mathbb{N}.$ We consider the operators $S_+f = \sum_{k=1}^\infty = f_k, S_−f=f_{−k}. f \in \mathfrak{D}(S_T)$ if and only if
a) $f \in \mathfrak{D}(S_+) \cap \mathfrak{D}(S_−), \sum_{n=1}^\infty \|S_nf\|^2 < \infty, \sum_{n=1}^\infty \|S_{−n}f\|^2 < \infty,$ where $S_nf = −\frac{1}{2}f_n−\sum_{k=n+1}^\infty f_k, S_{−n}f = −\frac{1}{2}f_n−\sum_{k=n+1}^\infty f_{−k}.$
b) $\phi^\prime = f_0 + \widetilde{Q}S_−f \in \mathfrak{D}(A).$
c) $S_+f = T^\ast S_−f + iD\phi^\prime$ and $S_Tf = (\dots, g_{−1}, g_0, g_1, \dots),$ where $g_0=−if_0+(A+iI)\phi^\prime, g_n = iS_nf, n \in \mathbb{Z}\setminus{0}. S_T$ is dilatation of $A$.
$\textbf{Theorem 1.}$ If the spaces $\mathfrak{H}_1$ and $\mathfrak{H}_2$ are separable, then the dilations $S$ and $S_T$ are isomorphic.

Keywords: dissipative operator, self-adjoint dilation, isomorphism of dilations.