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

$U^{*}U=U U^{*}=I$

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

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$U(\mathcal{H})$ or Hilbert group of $H$ and is denoted Hilb($H$).

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

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 $n \times n$ matrix with complex entries that satisfies the condition

$U^{*}U=U U^{*}=I_{n}$.

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

Notation

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

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