Could someone verify whether the solution of the following problem is correct? Thanks in advance for any help.
Problem: Let $T$ be a closed linear operator in the Hilbert space $\mathcal{H}$, and $\mathcal{D}$ a vector subspace of $\mathcal{D}_T$ (domain of $T$). Then $\overline{T|_{\mathcal{D}}}=T$ if and only if $\overline{|T||_{\mathcal{D}}}=|T|$.
Solution: $\begin{align} &\overline{\{(\xi,T\xi):\xi\in\mathcal{D}\}}=\{(\xi,T\xi):\xi\in\mathcal{D}_T\}\\ \iff & \overline{\{(\xi,v|T|\xi):\xi\in\mathcal{D}\}}=\big\{(\xi,v|T|\xi):\xi\in D_{|T|}\big\}\,(\text{We use the polar decomposition } T=v|T|,\,v^*v=s(|T|) \text{ and the fact } \mathcal{D}=\mathcal{D}_T)\\ \iff & \overline{(Id\,\times v)\{(\xi,|T|\xi):\xi\in\mathcal{D}\}}=(Id\,\times v)\big\{(\xi,|T|\xi):\xi\in\mathcal{D}_{|T|}\big\}\\ \iff & (Id\,\times v)\overline{\{(\xi,|T|\xi):\xi\in\mathcal{D}\}}=(Id\,\times v)\big\{(\xi,|T|\xi):\xi\in\mathcal{D}_{|T|}\big\}\,(\text{We use the fact that $\overline{v(A)}=v(\overline{A})$ for each $A\subset\mathcal{H}$})\\ \iff & (Id\,\times v)^*(Id\,\times v)\overline{\{(\xi,|T|\xi):\xi\in\mathcal{D}\}}=(Id\,\times v)^*(Id\,\times v)\big\{(\xi,|T|\xi):\xi\in\mathcal{D}_{|T|}\big\}\\ \iff & \overline{\{(\xi,|T|\xi):\xi\in\mathcal{D}\}}=\big\{(\xi,|T|\xi):\xi\in\mathcal{D}_{|T|}\big\}\,(\text{Using the fact that $v^*v=1-Ker(|T|$}). \end{align}$