Unitary transformation

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A unitary transformation is an isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function

[U:H_1\to H_2\,]

where [H_1] and [H_2] are Hilbert spaces, such that

[\langle Ux, Uy \rangle = \langle x, y \rangle]

for all [x] and [y] in [H_1]. A unitary transformation is an isometry, as one can see by setting [x=y] in this formula.

In the case when [H_1] and [H_2] are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.

A closely related notion is that of antiunitary transformation, which is a bijective function

[U:H_1\to H_2\,]

between two complex Hilbert spaces such that

[\langle Ux, Uy \rangle = \overline=\langle y, x \rangle]

for all [x] and [y] in [H_1], where the horizontal bar represents the complex conjugate.

Unitary transformation

See also