Unitary Operator

In functional analysis, a branch of mathematics, a unitary operator (not to be confused with a unity operator) is a bounded linear operator U : H → H on a Hilbert space H satisfying

where U∗ is the adjoint of U, and I : H → H is the identity operator. This property is equivalent to the following:

1. U preserves the inner product 〈, 〉 of the Hilbert space, i.e., for all vectors x and y in the Hilbert space, and
2. U is surjective (a.k.a. onto).

It is also equivalent to the seemingly weaker condition:

1. U preserves the inner product, and
2. the range of U is dense.

To see this, notice that U preserves 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 (in this case, the linear 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).

The weaker condition U∗U = I defines an isometry. Another condition, U U∗ = I, defines a coisometry.

A unitary element is a generalization of a unitary operator. In a unital *-algebra, an element U of the algebra is called a unitary element if

where I is the identity element.