Editing Self-adjoint operator (section)

== Bounded self-adjoint operators ==
Let <math>H</math> be a [[Hilbert space]] and <math>A:\operatorname{Dom}(A) \to H </math> a symmetric operator. According to [[Hellinger–Toeplitz theorem]], if <math>\operatorname{Dom}(A)=H</math> then <math>A</math> is necessarily bounded.<ref>{{harvnb|Hall|2013}} Corollary 9.9</ref>
A [[bounded operator]] <math>A : H \to H</math> is self-adjoint if 
: <math>\langle Ax, y\rangle = \langle x, Ay\rangle, \quad \forall x,y\in H.</math>
Every bounded operator <math>T:H\to H</math> can be written in the [[complex number|complex]] form <math>T = A + i B</math> where <math>A:H\to H</math> and <math>B:H\to H</math> are bounded self-adjoint operators.{{sfn | Griffel | 2002 | p=238 |ps=}}

Alternatively, every [[Positive operator (Hilbert space)|positive]] [[bounded linear operator]] <math>A:H \to H </math> is self-adjoint if the Hilbert space <math>H</math> is ''complex''.{{sfn | Reed | Simon | 1980 | p=195 |ps=}}

=== Properties ===
A bounded self-adjoint operator <math>A : H \to H</math> defined on <math>\operatorname{Dom}\left( A \right) = H</math> has the following properties:{{sfn | Rudin | 1991 | pp=326-327 |ps=}}{{sfn | Griffel | 2002 | pp=224-230|ps=}} 
* <math>A : H \to \operatorname{Im} A \subseteq H</math> is invertible if the [[Image (mathematics)|image]] of <math>A</math> is dense in <math>H.</math>
* The [[operator norm]] is given by <math>\left\| A \right\| = \sup \left\{ |\langle x, A x \rangle| : \| x \| = 1 \right\}</math>
* If <math>\lambda</math> is an [[eigenvalue]] of <math>A</math> then <math>| \lambda | \leq \sup \left\{ |\langle x, A x \rangle| : \| x \| \leq 1 \right\}</math>; the eigenvalues are real and the corresponding [[eigenvector]]s are orthogonal. 
Bounded self-adjoint operators do not necessarily have an eigenvalue. If, however, <math>A</math> is a [[Compact operator on Hilbert space#Compact self-adjoint operator|compact self-adjoint operator]] then it always has an eigenvalue <math>| \lambda | = \| A \|</math> and corresponding normalized eigenvector.{{sfn | Griffel | 2002 | p=241|ps=}}