unitary operator

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. A unitary element is a generalization of a unitary operator. In a unital algebra, an element of the algebra is called a unitary element if , where is the identity element. Definition Definition 1. A unitary operator is a bounded linear operator on a Hilbert space that satisfies , where is the adjoint of , and is the identity operator. The weaker condition defines an isometry. The other condition, , 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.