# Unitary operator

From Encyclopedia of Mathematics

A linear operator mapping a normed linear space onto a normed linear space such that . The most important unitary operators are those mapping a Hilbert space onto itself. Such an operator is unitary if and only if for all . Other characterizations of a unitary operator are: 1) , i.e. ; and 2) the spectrum of lies on the unit circle and there is the spectral decomposition . The set of unitary operators acting on forms a group.

Examples of unitary operators and their inverses on the space are the Fourier transform and its inverse.

#### References

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

[2] | N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1 , Pitman (1980) (Translated from Russian) |

[3] | A.I. Plessner, "Spectral theory of linear operators" , F. Ungar (1965) (Translated from Russian) |

**How to Cite This Entry:**

Unitary operator. V.I. Sobolev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Unitary_operator&oldid=14352

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098