# Unitarity and time evolution

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.