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.
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Unitary operator. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Unitary_operator&oldid=14352