Unitary operator
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In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying
- U*U=UU*=I
where I is the identity operator. This property is equivalent to any of the following:
- U is a surjective isometry
- U is surjective and preserves the inner product on the Hilbert space, so that for all vectors x and y in the Hilbert space,
- <math>\langle Ux, Uy \rangle = \langle x, y \rangle.<math>
Unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalisation of the notion of a unitary matrix.
Unitary operators implement isomorphisms between operator algebras.