A densely defined linear operator $A\colon\mathscr{D}(A)\subset \mathscr{H}\to\mathscr{H}$ on a Hilbert space $\mathscr{H}$ is a Hermitian or symmetric operator if $(Ax,y) = (x,Ay)$ for all $x,y\in \mathscr{D}(A)$ This means that the adjoint $A^*$ of $A$ is defined at least on $\mathscr{D}(A)$ and that its restriction to that set coincides with $A$ This fact is often denoted by $A\subset A^*$

The operator $A$ is self-adjoint if it coincides with its adjoint, i.e. if $A=A^*$ If $A$ is closable and its closure coincides with its adjoint (i.e. $\overline{A}=A^*$ , then $A$ is said to be essentially self-adjoint.