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Unitary operator
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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. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitary matrices. 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.
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 weaker condition, UU* = I, defines a coisometry. Thus a unitary operator is a bounded linear operator that 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:
The notion of isomorphism in the category 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:
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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. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitary matrices. 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.
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 weaker condition, UU* = I, defines a coisometry. Thus a unitary operator is a bounded linear operator that 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:
The notion of isomorphism in the category 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: