On isomorphism of common type J-selfadjoint dilations for lin- ear operator with nonempty regular points set

The common approach to construction of \\textsf{J}-selfadjoint dilation for linear
operator with nonempty regular point set is considered in this article.

Let $A$~--- linear operator with nonempty regular point set $(-i\\in \\rho(A))$ and
$\\Clos{\\textmd{dom}(A)}=\\mathfrak{H}$, where $\\mathfrak{H}$~--- Hilbert space,
$$
B_{+}:=iR_{-i}-iR_{-i}^{*}-2R_{-i}^{*}R_{-i},
\\ \\ B_{-}:=iR_{-i}-iR_{-i}^{*}-2R_{-i}R_{-i}^{*},
$$
$Q_{\\pm}:=\\sqrt{|B_{\\pm}|}$, $B_{\\pm}=\\mathcal{J}_{\\pm}Q_{\\pm}$~--- polar decompositions of
$B_{\\pm}$, $\\mathfrak{Q}_{\\pm}=\\Clos(Q_{\\pm}\\mathfrak{H})$.

Let $\\mathfrak{D}_{\\pm}^{(r)},~r=1,2$~--- arbitrary Hilbert spaces and
$F_{\\pm}:\\textmd{dom}(F_{\\pm})\\longrightarrow \\mathfrak{D}_{\\pm}^{(1)} \\
(\\textmd{dom}(F_{\\pm})\\subset\\mathfrak{D}_{\\pm}^{(1)}),
G_{\\pm}:\\textmd{dom}(G_{\\pm})\\longrightarrow \\mathfrak{D}_{\\pm}^{(2)} \\
(\\textmd{dom}(G_{\\pm})\\subset\\mathfrak{D}_{\\pm}^{(2)}), $~--- simple maximal symmetric operators with defect
numbers $(\\mathfrak{q}_{-},0)$ and $(0,\\mathfrak{q}_{+})$ respectively, moreover
$\\dim\\mathfrak{Q}_{\\pm}=\\dim\\mathfrak{N_{\\pm}}^{(r)}=\\mathfrak{q}_{\\pm}, r=1.2$,
$\\Phi_{\\pm}:\\mathfrak{N}_{\\pm}^{(1)}\\rightarrow\\mathfrak{Q}_{\\pm},
Psi_{\\pm}:\\mathfrak{N}_{\\pm}^{(2)}\\rightarrow\\mathfrak{Q}_{\\pm}$ are isometries, $V_{\\pm}, W_{\\pm}$~--- Cayley
ansforms of $F_{\\pm}$ and $G_{\\pm}$ respectively.

Let $\\langle \\mathcal{H}_{\\pm}^{(r)},\\Gamma_{\\pm}^{(r)}\\rangle$ are the spaces of boundary values of operators
$F_{\\pm}^{*}$ and $G_{\\pm}^{*}$ i.e.:
$$
a_{F_{\\pm}})~\\forall f_{1},g_{1}\\in \\textmd{dom}(F_{\\pm}^{*}) \\ \\
(F_{\\pm}^{*}f_{1},g_{1})_{\\mathfrak{D}_{\\pm}^{1}}-(f_{1},F_{\\pm}^{*}g_{1})_{\\mathfrak{D}_{\\pm}^{1}}=\\mp
i(\\Gamma_{\\pm}^{(1)}f_{1},\\Gamma_{\\pm}^{(1)}g_{1})_{\\mathcal{H}_{\\pm}^{(1)}};
$$
$$
б_{F_{\\pm}})\\mbox{the transformations \\ } \\textmd{dom}(F_{\\pm}^{*})\
i f_{1}\\mapsto \\Gamma_{\\pm}^{(1)}f_{1}\\in
\\mathcal{H}_{\\pm}^{(1)} \\mbox{ \\ are surjective.}
$$
$$
a_{G_{\\pm}})~\\forall f_{2},g_{2}\\in \\textmd{dom}(G_{\\pm}^{*}) \\ \\
(G_{\\pm}^{*}f_{2},g_{2})_{\\mathfrak{D}_{\\pm}^{(2)}}-(f_{2},G_{\\pm}^{*}g_{2})_{\\mathfrak{D}_{\\pm}^{(2)}}=\\mp
i(\\Gamma_{\\pm}^{(2)}f_{2},\\Gamma_{\\pm}^{(2)}g_{2})_{\\mathcal{H}_{\\pm}^{(2)}};
$$
$$
б_{G_{\\pm}})\\mbox{the transformations \\ } \\textmd{dom}(G_{\\pm}^{*})\
i f_{2}\\mapsto \\Gamma_{\\pm}^{2}f_{2}\\in
\\mathcal{H}_{\\pm}^{(2)} \\mbox{ \\ are surjective.}
$$

Consider the Hilbert spaces $\\mathbb{H}^{(r)}=\\mathfrak{D}_{-}^{(r)}\\oplus\\mathfrak{H}\\oplus\\mathfrak{D}_{+}^{(r)}$.
Define in this spaces indefinite metrics $\\textsf{J}^{(r)}=J_{-}^{(r)}\\oplus I\\oplus J_{+}^{(r)}$ and selfadjoint dilations of operator $A$
$\\textsf{S}$:
$$
\\forall \\ h_{\\pm}^{(1)}=\\sum\\limits_{k=0}^{\\infty}V_{\\pm}^{k}n^{\\pm}_{k}\\in \\mathfrak{D}_{\\pm}^{(1)}, \\ \\
n^{\\pm}_{k}\\in\\mathfrak{N}_{\\pm}^{(1)}, \\ \\
J_{\\pm}^{(1)}\\left(\\sum\\limits_{k=0}^{\\infty}V_{\\pm}^{k}n^{\\pm}_{k}\\right):=
\\sum\\limits_{k=0}^{\\infty}V_{\\pm}^{k}\\Phi_{\\pm}^{-1}\\mathcal{J}_{\\pm}^{(1)}\\Phi_{\\pm}n^{\\pm}_{k}.
$$

Analogously defined operator $\\textsf{J}^{(2)}$.

The vector $\\textsf{h}_{1}=(h_{-}^{(1)},h_{0},h_{+}^{(1)})^{T} \\in dom(\\textsf{S}_{1})$ iff

• $h_{\\pm}^{(1)}\\in dom(F^{*}_{\\pm});$
• $\\varphi^{(1)}=h_{0}+Q_{-}\\Phi_{-}\\Gamma_{-}^{(1)}h_{-}^{(1)}\\in dom(A);$
• $\\Phi_{+}\\Gamma_{+}^{(1)}h_{+}^{(1)}=T^{*}\\Phi_{-}\\Gamma_{-}^{(1)}h_{-}^{(1)}+i\\mathcal{J}_{+}Q_{+}(A+i)\\varphi^{(1)}, \\mbox{ where }T^{*}=I+2iR_{-i}^{*}$.

If this conditions are fulfil, that for all $\\textsf{h}_{1}=(h_{-}^{(1)},h_{0},h_{+}^{(1)})^{T}\\in
dom(\\textsf{S}_{1})$
$$
\\textsf{S}_{1}\\textsf{h}_{1}=\\textsf{S}_{1}(h_{-}^{(1)},h_{0},h_{+}^{(1)})^{T}:=
(F^{*}_{-}h_{-}^{(1)},~~-ih_{0}+(A+i)\\varphi^{(1)},~~F^{*}_{+}h_{+}^{(1)})^{T}.
$$

Analogously defined operator $\\textsf{S}_{2}$.