On Common Approach to the Construction of J-selfadjoint Dilation for a Linear Operator with a Nonempty Set of Regular Points

The common approach to construction of 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 $(−i ∈ ρ(A))$ and Clos dom(A) =$ \mathfrak{H}$,
where $\mathfrak{H}$ — Hilbert space,

$B_+ := iR_{−i} − iR^*_{−i} − 2R^∗_{−i}R_{−i}, B_− := iR_{−i} − iR^∗_{−i} − 2R_{−i}R^∗_{−i}$,

$Q_\pm := \sqrt{(|B_\pm|)}, B_\pm = \mathfrak{J}_\pm Q_\pm$ - polar decompositions of $B_\pm, \mathfrak{Q}_\pm = Clos(Q_\pm\mathfrak{H})$.

Let $\mathfrak{D}_\pm$ — arbitrary Hilbert spaces and $F_\pm : dom(F_\pm) → \mathfrak{D}_\pm (dom(F_\pm) ⊂ \mathfrak{ D}_\pm)$ — simple
maximal symmetric operators with defect numbers $(q_−, 0)$ and $(0, q_+)$ respectively, moreover
dim $\mathfrak{Q}_\pm= $ dim $ \mathfrak{N}_\pm = q_\pm, \Phi_\pm : \mathfrak{N}_\pm$ → $\mathfrak{Q}_\pm$ are isometries, $V_\pm$ — Caley transformations of $F_\pm$.
Let $\langle\mathcal{H}_\pm, Γ_\pm\rangle$ are the spaces of boundary values of operators $F^∗_\pm$, i.e.:

1. ∀ $f, g \in$ dom$(F^∗_\pm)$ $(F^∗_\pm f, g)_{\mathfrak{D}_\pm} − (f, F^∗_\pm g)_{\mathfrak{D}_\pm} = \mp i(Γ_\pm f, Γ_\pm g)_{\mathcal{H}_\pm}$;
2. the transformations dom$(F^∗_\pm) \in f \mapsto Γ_\pm f \in \mathcal{H}_\pm$ are surjective.

Consider the Hilbert space $\mathbb{H} = \mathfrak{D}_− ⊕ \mathfrak{H} ⊕ \mathfrak{D}_+$. Define in this space indefinite metric
$J = J_− ⊕ I ⊕ J_+ $and operator $\mathcal{S}:$

∀ $h_\pm = \sum\limits_{k=0}^∞ V^k_\pm n^\pm_k \in \mathfrak{D}_\pm, n^\pm_k \in \mathfrak{N}_\pm, J_\pm (\sum\limits_{k=0}^∞ V^k_\pm n^\pm_k \in \mathfrak{D}_\pm) := \sum\limits_{k=0}^∞ V^k_\pm \Phi^{−1}_\pm \mathfrak{J} _\pm \Phi_\pm n^\pm_k$.

The vector $h = (h_−, h_0, h_+)^T \in$ dom($\mathcal{S]) if

1. $h_\pm \in$ dom$(F^∗_\pm)$;
2. $\phi = h_0+Q_−\Phi_−Γ_−h_− \in$ dom$(A)$
3. $\Phi_+ Γ_+h_+ = T^*\Phi_−Γ_−h_− + i \mathfrak{J}_+Q_+(A + i)_\phi, where T^∗ = I + 2iR^∗_{−i}$.
   .

If this conditions are fulfil, that for all $h = (h_−, h_0, h_+)^ T \in$ dom($\mathcal{S}$)

$\mathcal{Sh} = \mathcal{S}(h_−, h_0, h_+)^T := (F^∗_−h_−, −ih_0 + (A + i)\phi, F^∗_+h_+)^T.$

Theorem. Operator $\mathcal{S}$ is a $\mathcal{J}$-sejfadjoint dilation of operator $A$.
Different private cases of dilation $\mathcal{S}$ are considered too.

Keywords:$ \mathcal{J}$-selfadjoint dilation, maximal closed symmetric operator, defect operators.