Editing Self-adjoint operator (section)

=== 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=}}