Essentially self adjoint operators definition equivalences

Question

Quick question about this post here. If I recall correctly, $T$ being essentially self-adjoint means that its closure is self-adjoint. In other words, we have $\text{cl}(T)^* = \text{cl}(T)$. However, in this post they are claiming that $T$ being essentially self-adjoint is the same as $T^* = \text{cl}(T)$. How is this true? I can see why $T^*$ is an extension of $\text{cl}(T)$, but not necessarily the other way around.

Answer 1

When $T$ is densely defined and closable (aka has a closure), we have $T^{**} = \text{cl}(T)$ and $T^*$ is closed, thus: $$T^* = \text{cl}(T^*) = (T^*)^{**} = (T^{**})^* = (\text{cl}(T))^*$$ This can be seen in theorem VIII.1 of Functional Analysis by Reed and Simon for example.