If $Af(x)=xf(x)$ is defined on $D(A)=\{f\in L^2(\mathbb{R}):xf\in L^2(\mathbb{R})\}$, then $A$ is known to be self-adjoint and so $D(A^*)=D(A)$. The proof is available is many textbooks. One proof may be found in the classical textbook by Akhiezer-Glazman (theory of linear oeprators in Hilbert space). The "big" problem is on Page 104: Indeed, they say that "if we restrict $A$ to a smaller domain, i.e. to $D'\subset D(A)$ and call this restriction $A'$, then similar arguments show that $A'^*=A$ and so $A=\overline{A'}$!!!".
I am not sure at all about this last statement. Indeed, if $A'$ is a (proper) closed restriction of $A$, then according to the last statement: $A=A'$ which is impossible!
Any explanation is most welcome!
Thanks in advance.
Math