I am new to unbounded operators and their adjoints. So far I have understood that unlike bounded operators, when $T$ is a densely defined unbounded operator $T^{**} \neq T$ in general.
In Reed & Simon's book on functional analysis that state that if $T$ is essentially self-adjoint, then $(T^{**})^* = T^{**}$?, but I cannot see why this is true. I am guessing part of my confusion is due to not properly understanding the relationship between the adjoint and the closure of $T$, if there is one.
More generally, when does $T = T^{**}$ for unbounded operators? Does this only hold when $T$ is self-adjoint?
If $T$ is essentially selfadjoint then by definition $T^*=\overline{T}.$ Then $\overline{T}$ is selfadjoint. Indeed $T\subset\overline{T}$ hence $$\overline{T}^*\subset T^*=\overline{T}$$ The reverse containment follows from the symmetry of $\overline{T}.$ Next $$T^{**}=\overline{T}^*=\overline{T}$$ and consequently $$(T^{**})^*=\overline{T}^*=\overline{T}$$