# 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.