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| {{Hatnote|For unitarity in physics, see [[Unitarity (physics)]].}}
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| In [[functional analysis]], a branch of [[mathematics]], a '''unitary operator''' (not to be confused with a unity operator) is a [[bounded linear operator]] ''U'' : ''H'' → ''H'' on a [[Hilbert space]] ''H'' satisfying
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| :<math>U^*U=UU^*=I \!</math>
| | <br><br>Today the need of IT or computer support has end up being the first need of any business organization. In corporate field the competition has increased a property. Companies are trying to produce maximum amount of items and options. The most necessary issue is the task should not be stopped any kind of condition. Much more many issues of having the work blocked. The biggest issue is technology issue.<br><br>Time Saving - No waiting online for the fax workout machine. You won't to be able to watch the fax machine to make sure it did things. Send and receive multiple faxes at minute.<br><br>The following are important points which have learned from multiple startup operations that utilize offshore IT resources. Follow these tips for success discover ways to dramatically chances for success.<br><br>When they sign up for your newsletter, they'll give you their email address contact information. Now you will automatically send them your IT newsletter once 30 days. If the newsletter contains useful info, the bank appreciative. And if a interest in an it support Provider arises, you can be sure that they're going to be considering of you and might give that you' call.<br><br>'Rob, if last brought your S2000 in to find a service, we forgot to name something that could help you a. Surely as a dealer you exactly what make of car I drive? Obviously you do - so use everything.<br><br>Change Passwords - Employees come and go and, even worse, write their passwords down and pass them around to other staff. You ought to have a policy that forces all the employees to change their passwords every ninety days. Again, you can set this up centrally to cause it to be fully currency trading. Staff won't like it, but it is not their business, they will not suffer the loss.<br><br>It is essential never to underestimate the flexibility of a booming enterprise contact. Any where a an entrepreneur spends money, is in order to look their way fondly. Would like them to come back. Each and every they can pass work their way, they will be able to. Many businesses learn that the places they order from actually refer work for. This can be a business relationship that does work and it is a real win-win situation for everyone that is involved.<br><br>If you adored this information and you would certainly such as to get more information regarding [http://www.amj-uk.com/-IT-Support-.html London IT Support] kindly check out our website. |
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| where ''U''<sup>∗</sup> is the [[Hermitian adjoint|adjoint]] of ''U'', and ''I'' : ''H'' → ''H'' is the [[identity (mathematics)|identity]] operator. This property is equivalent to the following:
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| #''U'' preserves the [[inner product]] 〈 , 〉 of the Hilbert space, i.e., for all [[vector space|vector]]s ''x'' and ''y'' in the Hilbert space, <math>\langle Ux, Uy \rangle = \langle x, y \rangle</math>, and
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| #''U'' is [[surjective function|surjective]].
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| It is also equivalent to the seemingly weaker condition:
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| #''U'' preserves the [[inner product]], and
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| #the range of ''U'' is [[dense set|dense]].
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| To see this, notice that ''U'' preserves the inner product implies ''U'' is an [[isometry]] (thus, a [[bounded linear operator]]). The fact that ''U'' has dense range ensures it has a bounded inverse ''U''<sup>−1</sup>. It is clear that ''U''<sup>−1</sup> = ''U''<sup>∗</sup>.
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| Thus, unitary operators are just [[automorphism]]s of [[Hilbert space]]s, i.e., they preserve the structure (in this case, the linear 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 ''H'' to itself is sometimes referred to as the '''Hilbert group''' of ''H'', denoted Hilb(''H'').
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| The weaker condition ''U''<sup>∗</sup>''U'' = ''I'' defines an ''isometry''. Another condition, ''U'' ''U''<sup>∗</sup> = ''I'', defines a ''coisometry''.<ref>{{harv|Halmos|1982|loc=Sect. 127, page 69}}</ref>
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| A '''unitary element''' is a generalization of a unitary operator. In a [[unital algebra|unital]] [[*-algebra]], an element ''U'' of the algebra is called a unitary element if
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| :<math>U^*U=UU^*=I</math>
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| where ''I'' is the identity element.<ref>
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| {{cite book | last = Doran | first = Robert S. |coauthors = Victor A. Belfi | title = Characterizations of C*-Algebras: The Gelfand-Naimark Theorems | publisher = Marcel Dekker | location = New York | year = 1986 | isbn = 0-8247-7569-4 }}
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| </ref>{{Rp|55}}
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| ==Examples==
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| * The [[identity function]] is trivially a unitary operator.
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| * Rotations in '''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 2 vectors. This example can be expanded to '''R'''<sup>''3''</sup>.
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| * On the [[vector space]] '''C''' of [[complex number]]s, multiplication by a number of [[absolute value]] 1, that is, a number of the form ''e''<sup>''i θ''</sup> 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 2''π'' 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).
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| * 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. They are the unitary operators on '''R'''<sup>''n''</sup>.
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| * The [[bilateral shift]] on the [[Lp space|sequence space]] <math>\ell^2</math> indexed by the [[integer]]s is unitary. In general, any operator in a Hilbert space which acts by shuffling around an [[orthonormal basis]] is unitary. In the finite dimensional case, such operators are the [[permutation matrix|permutation matrices]]. The [[unilateral shift]] is an isometry; its conjugate is a coisometry.
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| * The [[Fourier operator]] is a unitary operator, i.e. the operator which performs the [[Fourier transform]] (with proper normalization). This follows from [[Parseval's theorem]].
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| * Unitary operators are used in [[unitary representation]]s.
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| ==Linearity==
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| 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]]:
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| :<math> \langle \lambda\cdot Ux-U(\lambda\cdot x), \lambda\cdot Ux-U(\lambda\cdot x) \rangle </math>
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| :<math> = \| \lambda \cdot Ux \|^2 + \| U(\lambda \cdot x) \|^2 - \langle U(\lambda\cdot x), \lambda\cdot Ux \rangle - \langle \lambda\cdot Ux, U(\lambda\cdot x) \rangle </math> | |
| :<math> = |\lambda|^2 \cdot \| Ux \|^2 + \| U(\lambda \cdot x) \|^2 - \overline{\lambda}\cdot \langle U(\lambda\cdot x), Ux \rangle - \lambda\cdot \langle Ux, U(\lambda\cdot x) \rangle </math>
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| :<math> = |\lambda|^2 \cdot \| x \|^2 + \| \lambda \cdot x \|^2 - \overline{\lambda}\cdot \langle \lambda\cdot x, x \rangle - \lambda\cdot \langle x, \lambda\cdot x \rangle </math>
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| :<math> = 0</math>
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| :Analogously you obtain <math>\langle U(x+y)-(Ux+Uy), U(x+y)-(Ux+Uy) \rangle = 0 </math>.
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| ==Properties==
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| * The [[spectrum (functional analysis)|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 operator]]s. By the theorem, ''U'' is unitarily equivalent to multiplication by a Borel-measurable ''f'' on ''L''²(''μ''), for some finite measure space (''X'', ''μ''). Now ''U U*'' = ''I'' implies |''f''(''x'')|² = 1 ''μ''-a.e. This shows that the essential range of ''f'', therefore the spectrum of ''U'', lies on the unit circle.
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| ==See also==
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| *[[Unitary matrix]]
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| *[[Unitary transformation]]
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| *[[Antiunitary]]
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| ==Footnotes==
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| {{Reflist}}
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| ==References==
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| * {{cite book
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| | authorlink=Serge Lang
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| | last = Lang
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| | first = Serge
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| | title = Differential manifolds
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| | publisher = Addison-Wesley Publishing Co., Inc.
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| | location = Reading, Mass.–London–Don Mills, Ont.
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| | year = 1972
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| }}
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| * {{cite book
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| | authorlink=Paul Halmos
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| | last = Halmos
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| | first = Paul
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| | title = A Hilbert space problem book
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| | publisher = Springer
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| | year = 1982
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| }}
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| {{Functional Analysis}}
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| [[Category:Operator theory]]
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| [[Category:Unitary operators]]
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| [[de:Unitäre Abbildung]]
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Today the need of IT or computer support has end up being the first need of any business organization. In corporate field the competition has increased a property. Companies are trying to produce maximum amount of items and options. The most necessary issue is the task should not be stopped any kind of condition. Much more many issues of having the work blocked. The biggest issue is technology issue.
Time Saving - No waiting online for the fax workout machine. You won't to be able to watch the fax machine to make sure it did things. Send and receive multiple faxes at minute.
The following are important points which have learned from multiple startup operations that utilize offshore IT resources. Follow these tips for success discover ways to dramatically chances for success.
When they sign up for your newsletter, they'll give you their email address contact information. Now you will automatically send them your IT newsletter once 30 days. If the newsletter contains useful info, the bank appreciative. And if a interest in an it support Provider arises, you can be sure that they're going to be considering of you and might give that you' call.
'Rob, if last brought your S2000 in to find a service, we forgot to name something that could help you a. Surely as a dealer you exactly what make of car I drive? Obviously you do - so use everything.
Change Passwords - Employees come and go and, even worse, write their passwords down and pass them around to other staff. You ought to have a policy that forces all the employees to change their passwords every ninety days. Again, you can set this up centrally to cause it to be fully currency trading. Staff won't like it, but it is not their business, they will not suffer the loss.
It is essential never to underestimate the flexibility of a booming enterprise contact. Any where a an entrepreneur spends money, is in order to look their way fondly. Would like them to come back. Each and every they can pass work their way, they will be able to. Many businesses learn that the places they order from actually refer work for. This can be a business relationship that does work and it is a real win-win situation for everyone that is involved.
If you adored this information and you would certainly such as to get more information regarding London IT Support kindly check out our website.