# Hermitian operator

From Encyclopedia of Mathematics

*symmetric operator*

A linear operator on a Hilbert space with a dense domain of definition and such that for any . This condition is equivalent to: 1) ; and 2) for all , where is the adjoint of , that is, to . A bounded Hermitian operator is either defined on the whole of or can be so extended by continuity, and then , that is, is a self-adjoint operator. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a Hermitian matrix.

#### References

[1] | N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian) |

[2] | F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French) |

**How to Cite This Entry:**

Hermitian operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hermitian_operator&oldid=13375

This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article