basic criterion for self-adjointness

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

1.

$A=A^{*}$ (i.e $A$ is self-adjoint);
2.

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

$\overline{A}=A^{*}$ (i.e. $A$ is essentially self-adjoint);
2.

$\operatorname{Ker}(A^{*}\pm i)=\{0\}$ ;
3.

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

Title	basic criterion for self-adjointness
Canonical name	BasicCriterionForSelfadjointness
Date of creation	2013-03-22 14:53:02
Last modified on	2013-03-22 14:53:02
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	5
Author	Koro (127)
Entry type	Theorem
Classification	msc 47B25