In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces.

Definition

Definition 1. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H that satisfies UU = UU = I , where U* is the adjoint of U, and I : H → H is the identity operator. The weaker condition UU = I defines an isometry. The other weaker condition, UU = I , defines a coisometry. Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry, or, equivalently, a surjective isometry. An equivalent definition is the following: Definition 2. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: The notion of isomorphism in the category of Hilbert spaces is captured if domain and range are allowed to differ in this definition. Isometries preserve Cauchy sequences; hence the completeness property of Hilbert spaces is preserved The following, seemingly weaker, definition is also equivalent: Definition 3. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: To see that definitions 1 and 3 are equivalent, notice that U preserving the inner product implies U is an isometry (thus, a bounded linear operator). The fact that U has dense range ensures it has a bounded inverse U−1 . It is clear that U−1 = U* . Thus, unitary operators are just automorphisms of Hilbert spaces, i.e., they preserve the structure (the vector space structure, the inner product, and hence the topology) of the space on which they act. The group of all unitary operators from a given Hilbert space H to itself is sometimes referred to as the Hilbert group of H, denoted Hilb(H) or U(H) .

Examples

R2 are the simplest nontrivial example of unitary operators. Rotations do not change the length of a vector or the angle between two vectors. This example can be expanded to R3 . C of complex numbers, multiplication by a number of absolute value 1 , that is, a number of the form eiθ for θ ∈ R , is a unitary operator. θ is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of θ modulo 2π does not affect the result of the multiplication, and so the independent unitary operators on C are parametrized by a circle. The corresponding group, which, as a set, is the circle, is called U(1) . Rn . ℓ2 indexed by the integers is unitary. In general, any operator in a Hilbert space that acts by permuting an orthonormal basis is unitary. In the finite dimensional case, such operators are the permutation matrices. UU = UU = I , where I is the multiplicative identity element.

Linearity

The linearity requirement in the definition of a unitary operator can be dropped without changing the meaning because it can be derived from linearity and positive-definiteness of the scalar product: Analogously we obtain

Properties

. This can be seen as a consequence of the spectral theorem for normal operators. By the theorem, U is unitarily equivalent to multiplication by a Borel-measurable f on L2(μ) , for some finite measure space (X, μ) . Now UU* = I implies , μ-a.e. This shows that the essential range of f, therefore the spectrum of U, lies on the unit circle.

Footnotes