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 H1 and H2 are Hilbert spaces, such that

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

for all x and y in H1. A unitary transformation is an isometry, as one can see by setting x = y in this formula.

In the case when H1 and H2 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 x, y \rangle}=\langle y, x \rangle

for all x and y in H1, where the horizontal bar represents the complex conjugate.

[edit] See also

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