Let $T:D(T)\subset H \to H$ be selfadjoint unbounded linear operator on a complex Hilbertspace $H$. Show that $T^2$ is self-adjoint.
Since $T$ is selfadjoint it's spectrum must be real, so $T\pm i Id$ are both surjective. According to this answer I can conclude $T$ is densely defined. Furthermore it is $$ T^2+1Id=(T+iId)(T-iId), $$ so $T^2+1Id$ is surjective aswell. Now I need to show only two things
- $D(T^2)$ is dense (which can be proven identicaly as in answer) and
- $D((T^2)^*)\subset D(T^2)$ (other inclusion is trivial since $T^2$ is symmetric).
I am stuck here, how do I proceed?