# self-adjoint operator

A densely defined linear operator^{} $A:\mathcal{D}(A)\subset \mathscr{H}\to \mathscr{H}$ on a Hilbert space $\mathscr{H}$ is a *Hermitian* or *symmetric ^{}* operator

^{}if $(Ax,y)=(x,Ay)$ for all $x,y\in \mathcal{D}(A)$. This means that the adjoint

^{}${A}^{*}$ of $A$ is defined at least on $\mathcal{D}(A)$ 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 |