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.

Definition

Definition 1. A unitary operator is a bounded linear operator Template:Math on a Hilbert space Template:Mvar that satisfies Template:Math, where Template:Math is the adjoint of Template:Mvar, and Template:Math is the identity operator.

The weaker condition Template:Math defines an isometry. The other weaker condition, Template:Math, 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 Template:Math on a Hilbert space Template:Mvar for which the following hold:

• Template:Mvar is surjective, and
• Template:Mvar preserves the inner product of the Hilbert space, Template:Mvar. In other words, for all vectors Template:Mvar and Template:Mvar in Template:Mvar we have:

  $⟨Ux,Uy{⟩}_H=⟨x,y{⟩}_H.$

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 Template:Math on a Hilbert space Template:Mvar for which the following hold:

• the range of Template:Mvar is dense in Template:Mvar, and
• Template:Mvar preserves the inner product of the Hilbert space, Template:Mvar. In other words, for all vectors Template:Mvar and Template:Mvar in Template:Mvar we have:

  $⟨Ux,Uy{⟩}_H=⟨x,y{⟩}_H.$

To see that definitions 1 and 3 are equivalent, notice that Template:Mvar preserving the inner product implies Template:Mvar is an isometry (thus, a bounded linear operator). The fact that Template:Mvar has dense range ensures it has a bounded inverse Template:Math. It is clear that Template:Math.

Thus, unitary operators are just automorphisms of Hilbert spaces, i.e., they preserve the structure (the vector space structure, the inner product, and hence the topology) of the space on which they act. The group of all unitary operators from a given Hilbert space Template:Mvar to itself is sometimes referred to as the Hilbert group of Template:Mvar, denoted Template:Math or Template:Math.

Examples

• The identity function is trivially a unitary operator.
• Rotations in Template:Math 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 Template:Math. 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 Template:Math of complex numbers, multiplication by a number of absolute value Template:Math, that is, a number of the form Template:Math for Template:Math, is a unitary operator. Template:Mvar is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of Template:Mvar modulo Template:Math does not affect the result of the multiplication, and so the independent unitary operators on Template:Math are parametrized by a circle. The corresponding group, which, as a set, is the circle, is called Template:Math.
• 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 Template:Math.
• The bilateral shift on the sequence space Template:Math 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 Template:Mvar of the algebra is called a unitary element if Template:Math, where Template:Mvar is the multiplicative identity element.
• Any composition of the above.

Linearity

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:

$\begin{array}{cc}\Vert \lambda U(x)-U(\lambda x){\Vert }^2& =⟨\lambda U(x)-U(\lambda x),\lambda U(x)-U(\lambda x)⟩\\ & =\Vert \lambda U(x){\Vert }^2+\Vert U(\lambda x){\Vert }^2-⟨U(\lambda x),\lambda U(x)⟩-⟨\lambda U(x),U(\lambda x)⟩\\ & =|\lambda {|}^2\Vert U(x){\Vert }^2+\Vert U(\lambda x){\Vert }^2-\bar{\lambda }⟨U(\lambda x),U(x)⟩-\lambda ⟨U(x),U(\lambda x)⟩\\ & =|\lambda {|}^2\Vert x{\Vert }^2+\Vert \lambda x{\Vert }^2-\bar{\lambda }⟨\lambda x,x⟩-\lambda ⟨x,\lambda x⟩\\ & =0\end{array}$

Analogously we obtain

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

Properties

• The spectrum of a unitary operator Template:Mvar lies on the unit circle. That is, for any complex number Template:Mvar in the spectrum, one has Template:Math. This can be seen as a consequence of the spectral theorem for normal operators. By the theorem, Template:Mvar is unitarily equivalent to multiplication by a Borel-measurable Template:Mvar on Template:Math, for some finite measure space Template:Math. Now Template:Math implies Template:Math, Template:Mvar-a.e. This shows that the essential range of Template:Mvar, therefore the spectrum of Template:Mvar, 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.)

See also

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References

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