Let $\sigma\in S^m$ with $m>0$. Then $T_{\sigma}:\mathcal{S}\subset L^p\to L^p$ is the pseudo-differential operators asociated with the symbol $\sigma$. By a Proposition, $T_{\sigma}$ is closable.
Let $T_{\sigma,0}$ the smallest closed extension of $T_{\sigma}$ and $T_{\sigma,1}$ the largest closed extension of $T_{\sigma}$.
By a Proposition, if $m>0$ and $\sigma\in S^m$ elliptic, then $T_{\sigma,0}=T_{\sigma,1}$.
My question is: If $T_{\sigma}$ is closed in $\mathcal{S}$ and $\sigma\in S^m,\, m>0$ elliptic. Then $T_{\sigma}=T_{\sigma,0}=T_{\sigma,1}$?