Abstract:
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
$Closdom(A)=\mathfrak{H}$, 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}:dom(F_{\pm})\longrightarrow \mathfrak{D}_{\pm}^{(1)}(dom(F_{\pm})\subset\mathfrak{D}_{\pm}^{(1)}), G_{\pm}:dom(G_{\pm})\longrightarrow \mathfrak{D}_{\pm}^{(2)}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 transforms 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.:
$a_{F_{\pm}})~\forall f_{1},g_{1}\in dom(F_{\pm}^{*}) \ (F_{\pm}^{*}f_{1},g_{1})_{\mathfrak{D}_{\pm}^{1}}-(f_{1},F_{\pm}^{*}g_{1})_{\mathfrak{D}_{\pm}^{1}}=\mp i(\Gamma_{\pm}^{(1)}f_{1},\Gamma_{\pm}^{(1)}g_{1})_{\mathcal{H}_{\pm}^{(1)}};$
$b_{F_{\pm}})dom(F_{\pm}^{*})\ni f_{1}\mapsto \Gamma_{\pm}^{(1)}f_{1}\in\mathcal{H}_{\pm}^{(1)}$ are surjective.
$a_{G_{\pm}})~\forall f_{2},g_{2}\in dom(G_{\pm}^{*}) \ (G_{\pm}^{*}f_{2},g_{2})_{\mathfrak{D}_{\pm}^{(2)}}-(f_{2},G_{\pm}^{*}g_{2})_{\mathfrak{D}_{\pm}^{(2)}}=\mp i(\Gamma_{\pm}^{(2)}f_{2},\Gamma_{\pm}^{(2)}g_{2})_{\mathcal{H}_{\pm}^{(2)}};$
$b_{G_{\pm}})$ the transformations dom$(G_{\pm}^{*})\ni f_{2}\mapsto \Gamma_{\pm}^{2}f_{2}\in\mathcal{H}_{\pm}^{(2)}$ 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
- $h_{\pm}^{(1)}\in dom(F^{*}_{\pm});$
- $\varphi^{(1)}=h_{0}+Q_{-}\Phi_{-}\Gamma_{-}^{(1)}h_{-}^{(1)}\in dom(A);$
- $\Phi_{+}\Gamma_{+}^{(1)}h_{+}^{(1)}=T^{*}\Phi_{-}\Gamma_{-}^{(1)}h_{-}^{(1)} +i\mathcal{J}_{+}Q_{+}(A+i)\varphi^{(1)},$ 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
$\textsf{S}_{2}$.
Definition. Let
$L_{1}$ and
$L_{2}$ are
$J_{1}$-selfadjoint and
$J_{2}$-selfadjoint dilations of operator
$A$.
$L_{1}$ and
$L_{2}$ acting in Hilbert spaces
$\mathscr{H}_{1}$ and
$\mathscr{H}_{2}$ respectively and operator
$A$ is density defined in Hilbert space
$\mathfrak{H}\subset\mathscr{H}_{r},~r=1,2$. Operators
$L_{1}$ and
$L_{2}$ are called isomorphic if exist unitary operator
$U:\mathscr{H}_{1}\rightarrow \mathscr{H}_{2}$ that:
- $Uh=h~~\forall h\in \mathfrak{H}$;
- $UL_{1}\subseteq L_{2}U$;
- $\forall~ \textsf{h}_{1}\in \mathscr{H}_{1}:~~UJ_{1}\textsf{h}_{1}=J_{2}U\textsf{h}_{1}$.
Theorem. Operators
$\textsf{S}_{1}$ and
$\textsf{S}_{2}$ are isomorphic. Some theorem's corollaries are proved.