self-adjoint operator

A densely defined linear operator $A:𝒟(A)\subset \mathscr{H}\to \mathscr{H}$ on a Hilbert space $\mathscr{H}$ is a Hermitian or symmetric operator if $(Ax,y)=(x,Ay)$ for all $x,y\in 𝒟(A)$. This means that the adjoint $A^{*}$ of $A$ is defined at least on $𝒟(A)$ 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.

Title	self-adjoint operator
Canonical name	SelfadjointOperator
Date of creation	2013-03-22 13:48:23
Last modified on	2013-03-22 13:48:23
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	8
Author	Koro (127)
Entry type	Definition
Classification	msc 47B15
Classification	msc 47B25
Related topic	HermitianMatrix
Defines	Hermitian operator
Defines	symmetric operator
Defines	essentially self-adjoint
Defines	self-adjoint