Let $T_0 :H^1_0(]0,1[) → L^2(]0,1[)$ be the Dirac operator defined by $T_0(f)=if’$ and $T:D → L^2(]0,1[)$ be an extension of $T_0$ where $D={{f \in{H^1(]0,1[)} ;f(0)=f(1)}}$
I have proved that $T_0$ is not self adjoint and $T$ is self adjoins, now I want to show that $T_0$ is essentialy self adjoint,so how can I prove that $T$ is the unique extension of $T_0$?
$T_0$ is not essentially selfadjoint because $T_0$ is closed and is not selfadjoint. There are many selfadjoint extensions of $T_0$. For example, if $0 \le \theta \lt \pi$, then $T_{\theta}$ is selfadjoint, where $T_{\theta}f=if'$ and where $\mathcal{D}(T_{\theta})$ consists of all elements $f$ of $H^1(0,1)$ for which $\cos\theta f(0)+\sin\theta f(1)=0$.