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Unitary operator
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In functional analysis, a unitary operator is a surjective on a that preserves the . Non-trivial examples include rotations, reflections, and the . Unitary operators generalize . Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of between Hilbert spaces.


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

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

  • is surjective, and
  • preserves the of the Hilbert space, . In other words, for all and in we have:
  • :\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.

The notion of isomorphism in the of Hilbert spaces is captured if domain and range are allowed to differ in this definition. Isometries preserve ; 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 on a Hilbert space for which the following hold:

  • the range of is in , and
  • preserves the inner product of the Hilbert space, . In other words, for all vectors and in we have:
  • :\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.

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

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


Examples
  • The identity function is trivially a unitary operator.
  • in 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 . In even higher dimensions, this can be extended to the .
  • Reflections, like the Householder transformation.
  • \frac{1}{\sqrt{n}} times a .
  • 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 of , multiplication by a number of , that is, a number of the form for , is a unitary operator. is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of modulo does not affect the result of the multiplication, and so the independent unitary operators on are parametrized by a circle. The corresponding group, which, as a set, is the circle, is called .
  • The 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, are precisely the unitary operators on finite-dimensional , 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 .
  • The on the indexed by the is unitary.
  • The (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 , an element of the algebra is called a unitary element if , where is the multiplicative .
  • Any composition of the above.


Linearity
The 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 :

\begin{align}
\| \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{align}

Analogously we obtain

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


Properties
  • The spectrum of a unitary operator lies on the . That is, for any complex number in the spectrum, one has . This can be seen as a consequence of the for . By the theorem, is unitarily equivalent to multiplication by a on , for some finite . Now implies , -a.e. This shows that the essential range of , therefore the spectrum of , 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

Footnotes

  • (1972). 9780387961132, Addison-Wesley Publishing Co., Inc..

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