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:

• 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:

  \(\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.\)

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:

• 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:

  \(\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.\)

Condensed: the full section is in Wikipedia.

• The identity function is trivially a unitary operator.
• Rotations in R are the simplest nontrivial example of unitary operators. Rotations do not change the length of a vector or the angle between two vectors. This example can be expanded to R. In even higher dimensions, this can be extended to the Givens rotation.
• Reflections, like the Householder transformation.
• \(\frac{1}{\sqrt{n}}\) times a Hadamard matrix.
• In general, any operator in a Hilbert space that acts by permuting an orthonormal basis is unitary. In the finite dimensional case, such operators are the permutation matrices.
• On the vector space C of complex numbers, multiplication by a number of absolute value 1, that is, a number of the form e for θ ∈ R, is a unitary operator. θ is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of θ modulo 2π does not affect the result of the multiplication, and so the independent unitary operators on C are parametrized by a circle. The corresponding group, which, as a set, is the circle, is called U(1).
• The Fourier operator is a unitary operator, i.e. the operator that performs the Fourier transform (with proper normalization). This follows from Parseval's theorem.
• Quantum logic gates are unitary operators. Not all gates are Hermitian.
• More generally, unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalization of the notion of a unitary matrix. Orthogonal matrices are the special case of unitary matrices in which all entries are real. They are the unitary operators on R.
• The bilateral shift on the sequence space ℓ indexed by the integers is unitary.
• The unilateral shift (right shift) is an isometry; its conjugate (left shift) is a coisometry.
• Unitary operators are used in unitary representations.
• 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 multiplicative identity element.
• Any composition of the above.

The linearity requirement in the definition of a unitary operator can be dropped without changing the meaning because it can be derived from linearity and positive-definiteness of the scalar product:

\(\begin{aligned} \| \lambda U(x) -U(\lambda x) \|^2 &= \langle \lambda U(x) -U(\lambda x), \lambda U(x)-U(\lambda x) \rangle \\[5pt] &= \| \lambda U(x) \|^2 + \| U(\lambda x) \|^2 - \langle U(\lambda x), \lambda U(x) \rangle - \langle \lambda U(x), U(\lambda x) \rangle \\[5pt] &= |\lambda|^2 \| U(x)\|^2 + \| U(\lambda x) \|^2 - \overline{\lambda} \langle U(\lambda x), U(x) \rangle - \lambda \langle U(x), U(\lambda x) \rangle \\[5pt] &= |\lambda|^2 \| x \|^2 + \| \lambda x \|^2 - \overline{\lambda} \langle \lambda x, x \rangle - \lambda \langle x, \lambda x \rangle \\[5pt] &= 0 \end{aligned}\)

Analogously we obtain

\(\| U(x+y)-(Ux+Uy)\| = 0.\)

• The spectrum of a unitary operator U lies on the unit circle. That is, for any complex number λ in the spectrum, one has |λ| = 1. This can be seen as a consequence of the spectral theorem for normal operators. By the theorem, U is unitarily equivalent to multiplication by a Borel-measurable f on L(μ), for some finite measure space (X, μ). Now UU* = I implies |f(x)| = 1, μ-a.e. This shows that the essential range of f, therefore the spectrum of U, lies on the unit circle.
• A linear map is unitary if it is surjective and isometric. (Use Polarization identity to show the only if part.)