Editing Self-adjoint operator (section)

== Definitions ==
Let <math>H</math> be a [[Hilbert space]] and <math>A</math> an [[unbounded operator|unbounded]] (i.e. not necessarily bounded) [[linear operator]] with a [[dense set|dense]] [[Domain of a function|domain]] <math>\operatorname{Dom}A \subseteq H.</math> This condition holds automatically when <math>H</math> is [[Euclidean space|finite-dimensional]] since <math>\operatorname{Dom}A = H</math> for every linear operator on a finite-dimensional space.

The '''[[Graph of a function|graph]]''' of an (arbitrary) operator <math>A</math> is the set <math>G(A) = \{(x,Ax) \mid x \in \operatorname{Dom}A\}.</math> An operator <math>B</math> is said to '''extend''' <math>A</math> if <math>G(A) \subseteq G(B).</math>{{sfn | Reed | Simon | 1980 | p=250}} This is written as <math>A \subseteq B.</math>

Let the inner product <math>\langle \cdot, \cdot\rangle</math> be [[conjugate linear]] on the ''second'' argument. The '''[[adjoint operator]]''' <math>A^*</math> acts on the subspace <math>\operatorname{Dom} A^* \subseteq H</math> consisting of the elements <math>y</math> such that
: <math> \langle Ax,y \rangle = \langle x,A^*y \rangle, \quad \forall x \in \operatorname{Dom} A.</math> 

The [[densely defined]] operator <math>A</math> is called [[Extensions of symmetric operators#Symmetric operators|'''symmetric''']] (or '''Hermitian''') if <math>A \subseteq A^*</math>, i.e., if <math>\operatorname{Dom} A \subseteq \operatorname{Dom} A^*</math> and <math>Ax =A^*x</math> for all <math>x \in \operatorname{Dom} A</math>. Equivalently, <math>A</math> is symmetric if and only if
: <math> \langle Ax , y \rangle =  \lang x , Ay \rangle, \quad \forall x,y\in \operatorname{Dom}A.</math>
Since <math>\operatorname{Dom} A^* \supseteq \operatorname{Dom} A</math> is dense in <math>H</math>, symmetric operators are always [[closable operator|closable]] (i.e. the closure of <math>G(A)</math> is the graph of an operator).{{sfn|Pedersen|1989|loc=5.1.4}} If <math>A^*</math> is a closed extension of <math>A</math>, the smallest closed extension <math>A^{**}</math> of <math>A</math> must be contained in <math>A^*</math>. Hence,
: <math>A \subseteq A^{**} \subseteq A^*</math>
for symmetric operators and
: <math>A = A^{**} \subseteq A^*</math>
for closed symmetric operators.{{sfn | Reed | Simon | 1980 | pp=255–256}}

The densely defined operator <math>A</math> is called '''self-adjoint''' if <math>A = A^*</math>, that is, if and only if <math>A</math> is symmetric and <math>\operatorname{Dom}A = \operatorname{Dom}A^*</math>. Equivalently, a closed symmetric operator <math>A</math> is self-adjoint if and only if <math>A^*</math> is symmetric. If <math>A</math> is self-adjoint, then <math>\left\langle x, A x \right\rangle</math> is real for all <math>x \in \operatorname{Dom}A</math>, i.e.,{{sfn | Griffel | 2002 | pp=224 |ps=}} 
: <math>\langle x, Ax\rangle = \overline{\langle A x, x\rangle}=\overline{\langle x,Ax\rangle} \in \mathbb{R}, \quad \forall x \in \operatorname{Dom}A.</math>

A symmetric operator <math>A</math> is said to be '''essentially self-adjoint''' if the closure of <math>A</math> is self-adjoint. Equivalently, <math>A</math> is essentially self-adjoint if it has a ''unique'' self-adjoint extension. In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator.

In physics, the term '''Hermitian''' refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.