Editing Unitary operator

{{short description|Surjective bounded operator on a Hilbert space preserving the inner product}}
{{distinguish|Unitarity (physics)}}

In [[functional analysis]], a '''unitary operator''' is a [[surjective function|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 matrix|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 ==
'''Definition 1.''' A ''unitary operator'' is a [[bounded linear operator]] {{math|''U'' : ''H'' → ''H''}} on a Hilbert space {{mvar|H}} that satisfies {{math|1=''U''*''U'' = ''UU''* = ''I''}}, where {{math|''U''*}} is the [[Hermitian adjoint|adjoint]] of {{mvar|U}}, and {{math|''I'' : ''H'' → ''H''}} is the [[identity (mathematics)|identity]] operator.

The weaker condition {{math|1=''U''*''U'' = ''I''}} defines an ''[[isometry]]''. The other weaker condition, {{math|1=''UU''* = ''I''}}, defines a ''coisometry''. Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry,<ref>{{harvnb|Halmos|1982|loc=Sect. 127, page 69}}</ref> or, equivalently, a [[surjective function|surjective]] isometry.<ref>{{harvnb|Conway|1990|loc=Proposition I.5.2}}</ref>

An equivalent definition is the following:

'''Definition 2.''' A ''unitary operator'' is a bounded linear operator {{math|''U'' : ''H'' → ''H''}} on a Hilbert space {{mvar|H}} for which the following hold:
*{{mvar|U}} is [[surjective function|surjective]], and
*{{mvar|U}} preserves the [[inner product]] of the Hilbert space, {{mvar|H}}. In other words, for all [[vector space|vector]]s {{mvar|x}} and {{mvar|y}} in {{mvar|H}} we have: 
*:<math>\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.</math>

The notion of isomorphism in the [[Category theory|category]] of Hilbert spaces is captured if domain and range are allowed to differ in this definition. Isometries preserve [[Cauchy sequence]]s; hence the [[complete metric space|completeness]] property of Hilbert spaces is preserved<ref>{{harvnb|Conway|1990|loc=Definition I.5.1}}</ref>

The following, seemingly weaker, definition is also equivalent:

'''Definition 3.''' A ''unitary operator'' is a bounded linear operator {{math|''U'' : ''H'' → ''H''}} on a Hilbert space {{mvar|H}} for which the following hold:
*the range of {{mvar|U}} is [[dense set|dense]] in {{mvar|H}}, and
*{{mvar|U}} preserves the inner product of the Hilbert space, {{mvar|H}}. In other words, for all vectors {{mvar|x}} and {{mvar|y}} in {{mvar|H}} we have: 
*:<math>\langle Ux, Uy \rangle_H = \langle x, y \rangle_H.</math>

To see that definitions 1 and 3 are equivalent, notice that {{mvar|U}} preserving the inner product implies {{mvar|U}} is an [[isometry]] (thus, a [[bounded linear operator]]). The fact that {{mvar|U}} has dense range ensures it has a bounded inverse {{math|''U''<sup>−1</sup>}}. It is clear that {{math|1=''U''<sup>−1</sup> = ''U''*}}.

Thus, unitary operators are just [[automorphism]]s 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 (mathematics)|group]] of all unitary operators from a given Hilbert space {{mvar|H}} to itself is sometimes referred to as the ''Hilbert group'' of {{mvar|H}}, denoted {{math|Hilb(''H'')}} or {{math|''U''(''H'')}}.

==Examples==
* The [[identity function]] is trivially a unitary operator.
* [[Rotation|Rotations]] in {{math|'''R'''<sup>2</sup>}} 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 {{math|'''R'''<sup>3</sup>}}.  In even higher dimensions, this can be extended to the [[Givens rotation]].
* Reflections, like the [[Householder transformation]].
* <math>\frac{1}{\sqrt{n}}</math> 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 matrix|permutation matrices]].
* On the [[vector space]] {{math|'''C'''}} of [[complex number]]s, multiplication by a number of [[absolute value]] {{math|1}}, that is, a number of the form {{math|''e<sup>iθ</sup>''}} for {{math|''θ'' ∈ '''R'''}}, is a unitary operator. {{mvar|θ}} is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of {{mvar|θ}} modulo {{math|2''π''}} does not affect the result of the multiplication, and so the independent unitary operators on {{math|'''C'''}} are parametrized by a circle.  The corresponding group, which, as a set, is the circle, is called {{math|[[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 gate]]s are unitary operators. Not all gates are [[Self-adjoint operator|Hermitian]].
* More generally, [[unitary matrix|unitary matrices]] are precisely the unitary operators on finite-dimensional [[Hilbert space]]s, so the notion of a unitary operator is a generalization of the notion of a unitary matrix.  [[Orthogonal matrix|Orthogonal matrices]] are the special case of unitary matrices in which all entries are real.<ref>{{harvnb|Roman|2008|loc=p. 238 §10}}</ref> They are the unitary operators on {{math|'''R'''<sup>''n''</sup>}}.
* The [[bilateral shift]] on the [[Lp space|sequence space]] {{math|''ℓ''<sup>2</sup>}} indexed by the [[integer]]s is unitary.
* The [[unilateral shift]] (right shift) is an isometry; its conjugate (left shift) is a coisometry.
* Unitary operators are used in [[unitary representation]]s.
* A '''unitary element''' is a generalization of a unitary operator. In a [[unital algebra]], an element {{mvar|U}} of the algebra is called a unitary element if {{math|''U''*''U'' {{=}} ''UU''* {{=}} ''I''}}, where {{mvar|I}} is the multiplicative [[identity element]].<ref>{{harvnb|Doran|Belfi|1986|p=55}}</ref>
* 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|positive-definiteness]] of the [[scalar product]]:

:<math>\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}</math>

Analogously we obtain

:<math>\| U(x+y)-(Ux+Uy)\| = 0.</math>

==Properties==
* The [[spectrum (functional analysis)|spectrum]] of a unitary operator {{mvar|U}} lies on the [[unit circle]]. That is, for any complex number {{mvar|λ}} in the spectrum, one has {{math|{{!}}''λ''{{!}} {{=}} 1}}. This can be seen as a consequence of the [[spectral theorem]] for [[normal operator]]s. By the theorem, {{mvar|U}} is unitarily equivalent to multiplication by a [[Borel measurable|Borel-measurable]] {{mvar|f}} on {{math|''L''<sup>2</sup>(''μ'')}}, for some finite [[measure space]] {{math|(''X'', ''μ'')}}. Now {{math|''UU''* {{=}} ''I''}} implies {{math|{{!}}''f''(''x''){{!}}<sup>2</sup> {{=}} 1}}, {{mvar|μ}}-a.e. This shows that the essential range of {{mvar|f}}, therefore the spectrum of {{mvar|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.)

==See also==

* {{annotated link|Antiunitary}}
* {{annotated link|Crinkled arc}}
* {{annotated link|Quantum logic gate}}
* {{annotated link|Unitary matrix}}
* {{annotated link|Unitary transformation}}

==Footnotes==
{{Reflist}}

==References==
* {{cite book|last=Conway|first=J. B.|author-link=John B. Conway|title=A Course in Functional Analysis|year=1990|series=Graduate Texts in Mathematics|volume=96|publisher=[[Springer Verlag]]|isbn=0-387-97245-5}}

* {{cite book | last1 = Doran | first1 = Robert S. |first2=Victor A.|author1link = Robert S. Doran|last2= Belfi  | title = Characterizations of C*-Algebras: The Gelfand-Naimark Theorems | publisher = Marcel Dekker | location = New York | year = 1986 | isbn = 0-8247-7569-4 }}

* {{cite book|author-link=Paul Halmos|last = Halmos|first = Paul|title = A Hilbert space problem book|publisher = Springer Verlag|series=Graduate Texts in Mathematics|volume=19|year = 1982|edition=2nd|isbn=978-0387906850}}

* {{cite book |author-link=Serge Lang|last = Lang|first = Serge|title = Differential manifolds|publisher = Addison-Wesley Publishing Co., Inc.|location = Reading, Mass.&ndash;London&ndash;Don Mills, Ont.|year = 1972|isbn=978-0387961132}}

*{{citation | last=Roman | first=Stephen
 | title=Advanced Linear Algebra | edition=Third | series=[[Graduate Texts in Mathematics]] | publisher  = Springer | date=2008| pages= | isbn=978-0-387-72828-5 |author-link=Steven Roman}}

{{Functional analysis}}
{{Hilbert space}}

[[Category:Operator theory]]
[[Category:Unitary operators| ]]
[[Category:Linear operators]]