What is self-adjoint anyway? The adjoint $A^*$ of an operator $A$ (always densely defined on a Hilbert space) is given by all pairs $(y,z)$ (the graph of $A^*$) that obey \[ \langle y,Ax \rangle = \langle z,x \rangle \quad\forall x \in D(A) \] and thus it holds $z = A^*y$ and $y \in D(A^*)$. If $A=A^*$ on $D(A)$ the operator is called symmetric and one easily gets $D(A) \subseteq D(A^*)$. If further the domains coincide $D(A) = D(A^*)$ the operator is self-adjoint.