On common approach to the construction of $\textsf{J}$-selfadjoint dilation for a linear operator with a nonempty set of regular points

D. V. Tretyakov

Taurida Journal of Computer Science Theory and Mathematics, no. 4 (2019), pp. 92-106

Résumé

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 $\mathfrak{D}_{\pm}$ — arbitrary Hilbert spaces and $F_{\pm}:$dom$(F_{\pm})\longrightarrow \mathfrak{D}_{\pm} ($dom$(F_{\pm})\subset\mathfrak{D}_{\pm})$ — simple maximal symmetric operators with defect numbers $(\mathfrak{q}_{-},0)$ and $(0,\mathfrak{q}_{+})$ respectively, moreover $\dim\mathfrak{Q}_{\pm}=\dim\mathfrak{N_{\pm}}=\mathfrak{q}_{\pm}$, $\Phi_{\pm}:\mathfrak{N}_{\pm}\rightarrow\mathfrak{Q}_{\pm}$ are isometries, $V_{\pm}$ — Caley transformations of $F_{\pm}$. Let $\langle \mathcal{H}_{\pm},\Gamma_{\pm}\rangle$ are the spaces of boundary values of operators $F_{\pm}^{*}$, i.e.: 1) $\forall f,g\in $dom$(F_{\pm}^{*}) \ \ (F_{\pm}^{*}f,g)_{\mathfrak{D}_{\pm}}-(f,F_{\pm}^{*}g)_{\mathfrak{D}_{\pm}}=\mp i(\Gamma_{\pm}f,\Gamma_{\pm}g)_{\mathcal{H}_{\pm}}; $ 2) the transformations dom$(F_{\pm}^{*})\ni f\mapsto \Gamma_{\pm}f\in \mathcal{H}_{\pm}$ are surjective. Consider the Hilbert space $\mathbb{H}=\mathfrak{D}_{-}\oplus\mathfrak{H}\oplus\mathfrak{D}_{+}$. Define in this space indefinite metric $\textsf{J}=J_{-}\oplus I\oplus J_{+}$ and operator $\textsf{S}$: $$ \forall \ h_{\pm}=\sum\limits_{k=0}^{\infty}V_{\pm}^{k}n^{\pm}_{k}\in \mathfrak{D}_{\pm}, \ \ n^{\pm}_{k}\in\mathfrak{N}_{\pm}, \ \ J_{\pm}\left(\sum\limits_{k=0}^{\infty}V_{\pm}^{k}n^{\pm}_{k}\right):= \sum\limits_{k=0}^{\infty}V_{\pm}^{k}\Phi_{\pm}^{-1}\mathcal{J}_{\pm}\Phi_{\pm}n^{\pm}_{k}. $$ The vector $\textsf{h}=(h_{-},h_{0},h_{+})^{T} \in$dom$(\textsf{S})$ iff $h_{\pm}\in$dom$(F_{\pm}^{*});$ $\varphi=h_{0}+Q_{-}\Phi_{-}\Gamma_{-}h_{-}\in$dom$(A);$ $\Phi_{+}\Gamma_{+}h_{+}=T^{*}\Phi_{-}\Gamma_{-}h_{-}+i\mathcal{J}_{+}Q_{+}(A+i)\varphi, $ where $T^{*}=I+2iR_{-i}^{*}$. If this conditions are fulfil, that for all $\textsf{h}=(h_{-},h_{0},h_{+})^{T}\in $dom$(\textsf{S})$ $$ \textsf{S}\textsf{h}=\textsf{S}(h_{-},h_{0},h_{+})^{T}:=(F_{-}^{*}h_{-},~~-ih_{0}+(A+i)\varphi,~~F_{+}^{*}h_{+})^{T}. $$ Theorem. Operator $\textsf{S}$ is a $\textsf{J}$-sejfadjoint dilation of operator $A$. Different private cases of dilation $\textsf{S}$ are considered too.

Détail

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