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 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.