# Difference between revisions of "Hermitian operator"

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''symmetric operator'' | ''symmetric operator'' | ||

− | A linear operator | + | A linear operator $A$ on a Hilbert space $H$ with a dense domain of definition $D(A)$ and such that $\langle Ax,y\rangle = \langle x,Ay\rangle$ for any $x,y\in D(A)$. This condition is equivalent to: 1) $D(A)\subset D(A^*)$; and 2) $Ax = A^*x$ for all $x\in D(A)$, where $A^*$ is the adjoint of $A$, that is, to $A\subset A^*$. A bounded Hermitian operator is either defined on the whole of $H$ or can be so extended by continuity, and then $A=A^*$, that is, $A$ 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==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman (1981) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman (1981) (Translated from Russian)</TD></TR> | ||

+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)</TD></TR> | ||

+ | </table> |

## Revision as of 18:31, 26 April 2016

*symmetric operator*

A linear operator $A$ on a Hilbert space $H$ with a dense domain of definition $D(A)$ and such that $\langle Ax,y\rangle = \langle x,Ay\rangle$ for any $x,y\in D(A)$. This condition is equivalent to: 1) $D(A)\subset D(A^*)$; and 2) $Ax = A^*x$ for all $x\in D(A)$, where $A^*$ is the adjoint of $A$, that is, to $A\subset A^*$. A bounded Hermitian operator is either defined on the whole of $H$ or can be so extended by continuity, and then $A=A^*$, that is, $A$ 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=29566