Unitary operator

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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.


[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:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article