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.