**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical)

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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

Explicitly, this means the following.

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 *linear isometry* condition $U^{*}U=I$ or the *coisometry?* condition $U U^{*}=I$.

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

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.

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.

- In harmonic analysis, Parseval's theorem says that the Fourier transform is a unitary operator.

See also:

- Wikipedia,
*Unitary operator*

Last revised on November 9, 2023 at 08:52:21. See the history of this page for a list of all contributions to it.