nLab
unitary operator

Contents

Definition

A unitary operator is a unitary morphism in the †-category Hilb.

Explicitly, this means the following.

Unitary operators

A unitary operator UU on a Hilbert space \mathcal{H} is a bounded linear operator U:U: \mathcal{H} \to \mathcal{H} that satisfies

U *U=UU *=IU^{*}U=U U^{*}=I

where U *U^{*} is the Hilbert space adjoint? of UU and II is the identity operator. This property is equivalent to saying that the range of UU is dense and that UU preserves the inner product ,\langle \cdot,\cdot\rangle on the Hilbert space. An operator is unitary if and only if U 1=U *U^{-1}=U^{*}.

Unitary operators are the isomorphisms of Hilbert spaces since they preserve the basic structure of the space, e.g. the topology. The group of all unitary operators from a given Hilbert space to itself is sometimes called the unitary group U()U(\mathcal{H}) or Hilbert group of HH and is denoted Hilb(HH).

Sometimes operators may only obey the isometry U *U=IU^{*}U=I or the coisometry? UU *=IU U^{*}=I.

The generalization of a unitary operator is called a unitary element of a unital *-algebra.

Unitary matrices

If a basis for a finite dimensional Hilbert space is chosen, the defnition of unitary operator reduces to that of unitary matrix.

A unitary matrix is an n×nn \times n matrix with complex entries that satisfies the condition

U *U=UU *=I nU^{*}U=U U^{*}=I_{n}.

This is equivalent to saying that both the rows and the columns of UU form an orthonormal basis in n\mathbb{C}^{n} with respect to the respective inner product. UU is also a normal matrix? whose eigenvalues lie on the unit circle.

Notation

The notation used here for the adjoint, U *U^{*}, is commonly used in linear algebraic circles (as is U HU^{H}). In quantum mechanics, U U^{\dagger} is exclusively used for the adjoint while U *U^{*} is interpreted as the same thing as U¯\bar{U}, i.e. the complex conjugate.

Revised on April 26, 2010 11:21:21 by Urs Schreiber (131.211.232.147)