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Unitary Transformation




:T:H_1 o H_2\,

where H_1 and H_2 are Hilbert spaces, such that

:\langle Ux, Uy angle = \langle x, y angle

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

:T:H_1 o H_2\,

between two Complex Hilbert spaces such that

:\langle Ux, Uy angle = \overline{\langle x, y angle}=\langle y, x angle

for all x and y in H_1, where the horizontal bar represents the Complex Conjugate .


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