On isomorphism of common type $\textsf{J}$-selfadjoint dilations for linear operator with nonempty regular points set

D. V. Tretyakov

Taurida Journal of Computer Science Theory and Mathematics, no. 2 (2020), pp. 76-87

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 $\mathfrak{D}_{\pm}^{(r)},~r=1,2$ — 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.

Détail

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