basic criterion for self-adjointness

Let $A:D(A)\subset \mathscr{H}\to \mathscr{H}$ be a symmetric operator on a Hilbert space. The following are equivalent:

1. $A=A^{*}$ (i.e $A$ is self-adjoint);

2. $\mathrm{Ker}(A^{*}\pm i)=\{0\}$ and $A$ is closed;

3. $\mathrm{Ran}(A\pm i)=\mathscr{H}$.

Remark: $A+\lambda$ represents the operator $A+\lambda I:D(A)\subset \mathscr{H}\to \mathscr{H}$, and $\mathrm{Ker}$ and $\mathrm{Ran}$ stand for kernel and range, respectively.

A similar version for essential self-adjointness is an easy corollary of the above. The following are equivalent:

1. $\overline{A}=A^{*}$ (i.e. $A$ is essentially self-adjoint);

2. $\mathrm{Ker}(A^{*}\pm i)=\{0\}$;

3. $\mathrm{Ran}(A\pm i)$ is dense in $\mathscr{H}$.