# Minimality of selfadjoint dilation of operator knot of dissipative operator.

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Let A is dissipative densely defined operator in the space $\mathfrak{H}$ and $-i\in \rho (A)$. Let denote $R= (A+iI)^{-1}$ and consider the defect operators
\begin{align}
B = iR - iR^{*} - 2R^{*}R,\\
\widetilde{B} = iR - iR^{*} - 2RR^{*},\\
T = I - 2iR.\end{align}
A set of linear bounded operators acting from an entire Hilbert space $H_{1}$ into a Hilbert space $H_{2}$ will be denoted by $L(H_{1},H_{2}).$
Definition 1. The assembly of Hilbert spaces $\mathfrak{H}$, $E_{-}$ и $E_{+}$ and operators $A : \mathfrak{H} \rightarrow \mathfrak{H}$, $\Phi \in L(E_{-},\mathfrak{H})$, $\Psi \in L(E_{-},E_{+})$, $K \in L(E_{-},E_{+})$ is called the operator knot, which has been introduced in work of U. L. Kudryashov «Selfadjoint dilation of operator knot of dissipative operator» in «Dynamic systems», 3(31), №1-2, 2013, p. $45-48^{[1] }$.
$Q=(A,\Phi ,K,\Psi ,\mathfrak{H},E_{-},E_{+})$, if the following relations hold:
\begin{align}B=\Psi^{*}\Psi ;\\
\widetilde{B}=\Phi \Phi^{*};\\
T^{*}\Phi + \Psi ^{*}K=0;\\
T\Psi ^{*}+\Phi K^{*}=0;\\
2\Phi ^{*}\Phi + K^{*}K=I;\\
2\Psi \Psi ^{*}+KK^{*}=I.\end{align}
Selfadjoint dilation S of dissipative operator A is constructed using the knot $\Theta$ in [1] in the following manner.
The spaces $H_{-}=L_{2}((-\infty ,0],E_{-}),H_{+}=L_{2}([0,+\infty ))$ and $H=H_{-}\oplus \mathfrak{H}\oplus H_{+}$ are considered.
The vector $h=(h_{-},h_{0},h_{+})\in \mathfrak{D}(S)$ if and only if
1. $\begin{Bmatrix} h_{\pm },\frac{dh_{\pm }(t)}{dt} \end{Bmatrix}\subset H_{\pm }$;
2. $\widetilde{h}=h_{0}+\Phi h_{-}(0)\in \mathfrak{D}(A)$;
3. $h_{+}(0)=-Kh_{-}(0)+i\Psi (A+iI)\widetilde{h}$.
Theorem 1. The dilation S is minimal, i.e.
\begin{align}H=\overline{span\left \{ R_{\pm i}(S)h|h\in \mathfrak{H},n \in \left \{ 0 \right \}\cup \mathbb{N} \right \}}\end{align}
if the spaces $E_{+}=\overline{\Psi \mathfrak{H}}, E_{-}= \overline{\Phi^{*} \mathfrak{H}}$ are separable.
The following expressions was used for the proof:
\begin{align}R_{-i}^{n}(S)\begin{pmatrix}
0\\
h_{0}\\
0
\end{pmatrix}=\begin{pmatrix}
0\\
a_{n}\\
b_{n}
\end{pmatrix} \end{align}
where $n\in \mathbb{N},a_{n}=R^{n}h_{0}$,
\begin{align}b_{n}=e^{-t}\sum_{k=1}^{n}\frac{t^{n-k}}{(n-k)!i^{n-k-1}}\Psi R^{k-1}h_{0}.\\
R_{i}^{n}(S)\begin{pmatrix}
0
\\ h_{0}
\\ 0
\end{pmatrix}=\begin{pmatrix}
c_{n}
\\ d_{n}
\\ 0
\end{pmatrix},\end{align}
where $d_{n}=R^{*n}h_{0}$,
\begin{align}c_{n}=e^{t}\sum_{k=1}^{n}\frac{t^{n-k}}{(n-k)!(-i)^{n-k-1}}\Phi ^{*}R^{*k-1}h_{0},\end{align}
where $h_{0}\in \mathfrak{H}$.

Keywords:dilation, self-adjoint operator, unbounded dissipative operator, minimality, operator knot

UDC:
517.432