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,
⟨Ux, Uy⟩ = ⟨x, y⟩.
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.