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)