Isomorphism of spectral and translational presentations of self-adjoint 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 $Q=\sqrt{B},$ $\widetilde{Q}=\sqrt{\widetilde{B}},$ $\mathfrak{H}_{1}=\overline{Q \mathfrak{H}},$ $\mathfrak{H}_{2}=\overline{\widetilde{Q} \mathfrak{H}}.$

1. Spectral presentation. We consider the Hilbert spaces $H_{+}=L_{2}(0, \infty; \mathfrak{H_{1}}),$ $H_{-}=L_{2}(-\infty, 0; \mathfrak{H_{2}}),$ $H=H_{-}\bigoplus \mathfrak{H}\bigoplus H_{+}$ and operator $S$, $h=(h_{-}, h_{0}, h_{+}) \in \mathfrak{D}(S)$ if and only if

$a) \ \left\lbrace h_{\pm}, \dfrac{dh_{\pm}}{dt} \right\rbrace \subset H{\pm},$

$b) \ \varphi=h_{0}+\widetilde{Q}h_{-}(0) \in \mathfrak{D}(A),$

$c) \ h_{+}(0)=T^{\ast}h_{-}(0)+iD\varphi,$
where $T^{\ast}=I+2iR^{\ast},$ $D=Q(A+iI).$ $S(h_{-}, h_{0}, h_{+})=\left( i\dfrac{dh_{-}}{dt}, -ih_{0}+(A+iI)\varphi, \dfrac{dh_{+}}{dt} \right).$ $S$ is dilatation of $A.$

2. Translational presentation. We consider the Hilbert spaces $\mathfrak{H}_{-}=\underset{-\infty }{\overset{-1}{\mathop \oplus }}\mathfrak{H}_{2},$ $\mathfrak{H}_{+}=\underset{\infty }{\overset{1}{\mathop \oplus }}\mathfrak{H}_{1}$ and $\mathbf{H}=\mathfrak{H}_{-}\oplus\mathfrak{H}\oplus\mathfrak{H}_{+},$ $f=(\ldots, f_{-1}, f_{0}, f_{1}, \ldots) \in \mathbf{H}$ 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 $S_{+}f=\sum_{k=1}^{\infty}f_{k},$ $S_{-}f=\sum_{k=1}^{\infty}f_{-k}.$ \[f\in \mathfrak{D}(S_{\mathbf{T}})\] if and only if

$a) \ f\in \mathfrak{D}(S_{+})\bigcap \mathfrak{D}(S_{-}),$ $\sum_{n=1}^{\infty}\|S_{n}f\|^{2}<\infty,$ $\sum_{n=1}^{\infty}\|S_{-n}f\|^{2}<\infty,$
where $S_{n}f=-\dfrac{1}{2}f_{n}-\sum_{k=n+1}^{\infty}f_{k},$ $S_{-n}=\dfrac{1}{2}f_{n}+\sum_{k=n+1}^{\infty}f_{-k}.$

$b) \ \varphi'=f_{0}+\widetilde{Q}S_{-}f\in\mathfrak{D}(A).$

$c) \ S_{+}f=T^{\ast}S_{-}f+iD\varphi'$ and $S_{\mathrm{T}}f=(... g_{-1}, g_{0}, g_{1}, ...),$
where $g_{0}=-if_{0}+(A+iI)\varphi',$ $g_{n}=iS_{n}f,$ $n\in \mathbb{Z}\backslash\{0\}.$ $S_{\mathrm{T}}$ is dilatation of $A.$

Theorem. If the spaces $\mathfrak{H}_{1}$ and $\mathfrak{H}_{2}$ are separable, then the dilations $S$ and $S_{\mathrm{T}}$ are isomorphic.