Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving 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.

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 $U * U = UU * = I$ , where $I$ is the identity element.

Definition 1. A unitary operator is a bounded linear operator $U : H \to H$ on a Hilbert space $H$ that satisfies $U * U = UU * = I$ , where $U *$ is the adjoint of $U$ , and $I : H \to H$ is the identity operator.

The weaker condition $U * U = I$ defines an isometry. The other condition, $UU * = I$ , defines a coisometry. Thus a unitary operator is a bounded linear operator which 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 $U : H \to H$ on a Hilbert space $H$ for which the following hold:

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