Suppose $T$ is a (unbounded) symmetric operator on Hilbert space $\mathscr{H}$, defined on $D$. Let $D_1 \subseteq D$ is a linear subset dense in $D$, $T|_{D_1}$ be the restriction of $T$ on $D_1$. Show that if $T|_{D_1}$ is essentially self-adjoint, then $T$ is essentially self-adjoint, and $\overline{T} = \overline{T|_{D_1}}$. I have the answer following.
It suffices to show that $\overline{\Gamma (T)} \subseteq \overline{\Gamma (T|_{D_1})}$, then I choose a sequence in $D_1$ converge to x $\in D$, using diagnal technique.Simply let $x_n(\in D) \rightarrow x$, while $T(x_n) \rightarrow y$ and let $x_k^{(n)}(\in D_1)\rightarrow x_n$ then $x^{(n)}_n \rightarrow x$, but how can I show that $T(x^{(n)}_n) \rightarrow y$? Thanks a lot in advance!