basic criterion for selfadjointness
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 selfadjoint);

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 selfadjointness is an easy corollary of the above. The following are equivalent:

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

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

3.
$\mathrm{Ran}(A\pm i)$ is dense in $\mathscr{H}$.
Title  basic criterion for selfadjointness 

Canonical name  BasicCriterionForSelfadjointness 
Date of creation  20130322 14:53:02 
Last modified on  20130322 14:53:02 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  5 
Author  Koro (127) 
Entry type  Theorem 
Classification  msc 47B25 