# Unitary operator

A linear operator $ U $
mapping a normed linear space $ X $
onto a normed linear space $ Y $
such that $ \| Ux \| _ {Y} = \| x \| _ {X} $.
The most important unitary operators are those mapping a Hilbert space onto itself. Such an operator is unitary if and only if $ ( x, y) = ( Ux, Uy) $
for all $ x, y \in X $.
Other characterizations of a unitary operator $ U: H \rightarrow ^ {\textrm{ onto } } H $
are: 1) $ U ^ {*} U = UU ^ {*} = I $,
i.e. $ U ^ {-} 1 = U ^ {*} $;
and 2) the spectrum of $ U $
lies on the unit circle and there is the spectral decomposition $ U = \int _ {0} ^ {2 \pi } e ^ {i \phi } dE _ \phi $.
The set of unitary operators acting on $ H $
forms a group.

Examples of unitary operators and their inverses on the space $ L _ {2} (- \infty , \infty ) $ 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Unitary_operator&oldid=49084