In functional analysis, a **unitary operator** is a bounded linear operator *U* on a Hilbert space satisfying

*U* * *U* = *U**U* * = *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,

⟨*U**x*, *U**y*⟩ = ⟨*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.

Category:Evolutions and Operations Category:Handbook of Quantum Information

## Last modified:

Monday, October 26, 2015 - 17:56