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Concept
Unitary operator
Summary
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 U of the algebra is called a unitary element if UU = UU = I,
where I is the identity element.
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 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 → H on a Hilbert space H for which the following hold:
U is surjective, and
U preserves the inner product of the Hilbert space, H. In other words, for all vectors x and y in H we have:
The notion of isomorphism in the 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:
the range of U is dense in H, and
U preserves the inner product of the Hilbert space, H. In other words, for all vectors x and y in H we have:
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.
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