J-isometric and J-unitary dilations of operator knot.

The method of dilations is used in the study of the non-unitary operators. Herewith, the theory of the unitary dilations of the contractions has been full enough developed in the works of B. Szokefalvi-Nagy and Ch. Foyash. Further, the $J$-unitary dilation of an arbitrary bounded operator was constructed by Ch. Davis and A. V. Kuzhel.
In the work “The point spectrum of unitary dilations in krein spaces” of Temme, D., and after that in the work “Analytic methods of spectral representations of non-self-adjoint and nonunitary operators” of V. A. Zolotarev the term of the operator knot has been introduced. Using this term, the $J$-unitary dilation of an arbitrary bounded operator has been constructed. Along with the open system construction has been used.
This paper provides a direct proof that the operator, built by the knot, is the $J$-unitary dilation of the main operator of the knot and its minimality.
Following the work “Harmonic Analysis of Operators on Hilbert Space” of B. Szokefalvi-Nagy and Ch. Foyash the $J$-isometric dilation is built preliminarily and some of its properties are proved.
A set of linear bounded operators acting from an entire Hilbert space $H_{1}$ into a Hilbert space $H_{2}$, will be denoted by $\left [ H_{1},H_{2} \right ]$.
Definition 1. The assembly of Hilbert spaces $H$, $ H_{-}$, $ H_{+}$ and operators $T\in \left [ H,H \right ], \Psi \in \left [ H,H_{+} \right ],\Phi \in \left [ H_{-},H \right ],K \in \left [ H_{-},H_{+} \right ],J_{-} \in \left [ H_{-},H_{-} \right ],J_{+} \in \left [ H_{+},H_{+} \right ], J_{-}=J_{-}^{*}=J_{-}^{-1}, J_{+}=J_{+}^{*}=J_{+}^{-1}$ is called the unitary metric knot [6] $\Delta $ (further, knot) $\Delta = \left ( J_{-} ,H\oplus H_{-},V=\begin{bmatrix}T \ \Phi \\ \Psi \ K \end{bmatrix} , H\oplus H_{+},J_{+}\right )$ if the following relations hold:
$T^{*}T+\Psi ^{*}J_{+}\Psi = I$ (1)
$T^{*}\Phi +\Psi ^{*}J_{+}K = 0 $ (2) $\left (\Phi ^{*}T +K ^{*}J_{+}\Psi = 0 \right ) $ (2*)
$\Phi ^{*}\Phi + K ^{*}J_{+}K = J_{-} $ (3)
$T T^{*} + \Phi J_{-}\Phi ^{*} = I $ (4)
$T \Psi + \Psi J_{-}K^{*} = 0 $ (5) $\left (\Psi T^{*} + K J_{-}\Phi ^{*} = 0 \right )$ (5*)
$\Psi \Psi ^{*} + K J_{-}K ^{*} = J_{+}$ (6)
These relations (1) - (6) can be written in a more compact form.
Let us introduce the spaces $ H\oplus H_{-}$, $H\oplus H_{+}$ and the operator $V=\begin{bmatrix}T \ \Phi \\ \Psi \ K \end{bmatrix}\in \left [ H\oplus H_{-} , H\oplus H_{+}\right ]$ : Then the conditions (1) - (3) can be written in the following form:
$V^{*}\begin{bmatrix}I_{H} \ \ 0 \\ 0 \ \ J_{+}\end{bmatrix}V=\begin{bmatrix}I_{H} \ \ 0 \\ 0 \ \ J_{-}\end{bmatrix}$, and conditions (4)-(6) can be written in the following form:
$V\begin{bmatrix}I_{H} \ \ 0 \\ 0 \ \ J_{-}\end{bmatrix}V^{*}=\begin{bmatrix}I_{H} \ \ 0 \\ 0 \ \ J_{+}\end{bmatrix}$.
Moreover, the operator $T$ is called the main operator of the knot, $\Phi$ ,$\Psi $ are the canal operators, $K$ are the deforming operator, and $J_{+}$, $J_{-}$ are the metric operators of the knot $\Delta $.
Let $T\in \left [ H,H \right ]$ be included in the knot $\Delta $.
Form the Hilbert space $\Gamma $ consists of the vectors $h=\left ( h_{0},h_{1},h_{2},... \right )$ with components $h_{0}\in H,h_{n}\in H_{+} \left ( n\geq 1 \right )$. Every such $h\in \Gamma $ if (and only if) $\sum_{n=1}^{\infty }\left \| h_{n} \right \|^{2}< \infty$ and $\left ( h,{h}' \right )=\sum_{k=1}^{\infty }\left ( h_{k},{h_{k} }'\right )$, where ${h}'=\left ( {h_{0}}', {h_{1}}',{h_{2}}',...\right )$.Let identify the vector $\left ( h_{0},0,0,... \right )\equiv h_{0}$, then $H\subset \Gamma $ and the operator $Ph=\left ( h_{0},0,0,... \right )=h_{0}$ is orthogonal projection from $\Gamma $ onto $H$.
Let us introduce $J$-metric by use of the operator $J$ in the space $\Gamma $: $J\left ( h_{0},h_{1},h_{2},... \right )=\left ( h_{0},J_{+}h_{1},J_{+}h_{2},... \right )$, where $J=J^{*}=J^{-1}$ and $\left [ h,{h}' \right ]=\left ( Jh,{h}' \right )$.
Let consider the operator $V$ : $Vh=V\left ( h_{0},h_{1},h_{2},... \right )=\left ( Th_{0},\Psi h_{1},h_{2},... \right )$ in the space $\Gamma $, where the operator $\Psi $ from the knot $\Delta$.
Theorem 1.The operator $V$ is the $J$-isometric dilation of the operator $T$ (and hence, of the knot $\Delta$ ).
Definition 2.The $J$-isometric dilation $V\in \left [ \Gamma ,\Gamma \right ]$ of the operator $T\in \left [ H,H \right ]$ s called minimal if $\Gamma =span\left \{ V^{n}H\mid n\in \mathbb{N}\bigcup \left \{ 0 \right \} \right \}$.
Theorem 2.The $J$-isometric dilation $V$ is minimal if $H_{+}=\overline{\Psi H}$.
Theorem 3.The minimal $J$-isometric dilation of the operator $T$ is determined up to $J$-unitary isomorphism.
Let T\in \left [ H,H \right ] and $T$ be inclueded in the knot $\Delta$.
Form the Hilbert space $\Gamma $, whose elements are the vectors $h=\left ( ...,h_{-2},h_{-1},\underline{ h_{0}},h_{1},h_{2},... \right )$ (an underline indicates, that the element, which placed in it, is situated on zero position), where $h_{0}\in H, h_{n}\in H_{+},h_{-n}\in H_{-} \left ( n\in \mathbb{N} \right )$.
$h\in \Gamma $ if (and only if) $\sum_{k=-\infty }^{\infty }\left \| h_{k} \right \|^{2}< \infty $ and $\left ( h,{h}' \right )=\sum_{k=-\infty }^{\infty }\left ( h_{k},{h_{k}}' \right )$, where ${h}'=\left ( {h_{-2}}',{h_{-1}}',\underline{{h_{0}}'},{h_{1}}',{h_{2}}',... \right )$. Let identify the vector $\left ( ...,0,0,\underline{h_{0}},0,0,.. \right )\equiv h_{0}$, thus we obtain $H\subset \Gamma $ and the operator $Ph=\left ( ...,0,0,\underline{h_{0}},0,0,... \right )=h_{0}$ is the orthogonal projection from $\Gamma $ onto $H$.
Let consider the operator $U$:
$Uh=\left ( ...,h_{-3},h_{-2},\underline{Th_{0}+\Phi h_{-1}},\Psi h_{0}+K h_{-1},h_{1},... \right )$.
Theorem 4.The operator $U$ is the $J$-unitary dilation of the operator knot $\Delta$.
Definition 3.The $J$-unitary dilation $U\in \left [ \Gamma ,\Gamma \right ]$ of operator the $T\in \left [ H,H \right ]$ is called minimal if $\Gamma =\overline{span\left \{ U^{n}H \mid n\in \mathbb{Z} \right \}}$, where $U^{-1}=JU^{*}J$ .
Theorem 5.If $H_{+}=\overline{\Psi H}$, $H_{-}=\overline{J_{-}\Phi^{*} H}$, then the $J$-unitary dilation $U$ is minimal.
Theorem 6.The minimal $J$-unitary dilation of the operator $T\in \left [ H,H \right ]$ is determined up to $J$-unitary isomorphism.

Keywords: minimal $J$-isometric dilatation, minimal $J$-unitary dilatation, $J$-unitary isomorphism of dilatations.