$u\rightarrow u’$ is self-adjoint on $H^{1}_{0}(0,1)$?

I am reading the note, p.21.

I think the claim there is not true.

The claim: Let \begin{align*} A_{0}: H^{1}_{0}(0,1) & \rightarrow L^{2}(0,1)\ u & \rightarrow u’ \end{align*} be an unbounded operator on $ L^{2}(0,1)$ . Then $ A_{0}$ is a self-adjoint operator on $ L^{2}(0,1)$ .

I think that $ A_{0}$ is not self-adjoint since the domain of $ A_{0}^{\star}$ is not $ H^{1}_{0}(0,1)$ .

Could anyone help me with this concern? Thank so much.