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

1. 1.

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

2. 2.

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

3. 3.

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

Remark: $A+\lambda$ represents the operator $A+\lambda I\colon D(A)\subset\mathscr{H}\to\mathscr{H}$, and $\operatorname{Ker}$ and $\operatorname{Ran}$ 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. 1.

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

2. 2.

$\operatorname{Ker}(A^{*}\pm i)=\{0\}$;

3. 3.

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

Title basic criterion for self-adjointness BasicCriterionForSelfadjointness 2013-03-22 14:53:02 2013-03-22 14:53:02 Koro (127) Koro (127) 5 Koro (127) Theorem msc 47B25