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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 U*U = 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 U*U = UU* = I, where U* is the adjoint of U, and I : H → 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 → 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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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 U of the algebra is called a unitary element if U*U = UU* = I, where I is the identity element. Definition 1.

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Normal operator

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