nLab
unitary operator

Definition

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

Explicitly, this means the following.

Unitary operators

A unitary operator $U$ on a Hilbert space $\mathscr{H}$ is a bounded linear operator $U:\mathscr{H}\to \mathscr{H}$ 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 $⟨\cdot ,\cdot ⟩$ 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 $U(\mathscr{H})$ 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 ${ℂ}^n$ with respect to the respective inner product. $U$ is also a normal matrix? whose eigenvalue?s 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, $U^{†}$ is exclusively used for the adjoint while $U^{*}$ is interpreted as the same thing as $\overline{U}$, i.e. the complex conjugate.