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In [[mathematics]], '''Stone's theorem''' on [[one-parameter group|one-parameter]] [[unitary group]]s is a basic theorem of [[functional analysis]] that establishes a one-to-one correspondence between [[self-adjoint operator]]s on a [[Hilbert space]] <math> \mathcal{H} </math> and one-parameter families
He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been [http://En.Search.Wordpress.com/?q=worried worried] by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is.<br><br>I keep conjuring up images of him in 1983, trying to reignite the passion. <br>I told him, before he started wriggling, that I think that memorable evening, when after our game of squash he had asked me to take his racquet home for him because he had a date, he had already started seeing the woman he would marry.<br><br>He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.<br><br>The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen.<br><br>99                  &#9733;&#9733;&#9733;&#9733;&#9733;Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�s famous examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329.<br><br>And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. �<br>This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you.<br><br>Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx� <br>The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message�<br>�The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road!<br><br>She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week.<br><br>The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum<br>But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7.<br><br>In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth.<br><br>Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3D printer.<br><br>Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too.<br><br>WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES<br>The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals <br>With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. &#9733;&#9733;&#9733;&#9733;&#9733;<br><br>For more info about [http://nouveauclashofclanstriche.blogspot.com/ clash of clans triche gemmes] look into our own web-site.
 
:<math> (U_{t})_{t \in \mathbb{R}} </math>
 
of [[unitary operators]] that are [[strong operator topology|strongly continuous]], i.e.,
 
:<math> \forall t_{0} \in \mathbb{R}, ~ \xi \in \mathcal{H}: \quad \lim_{t \rightarrow t_{0}} U_{t} \xi = U_{t_{0}} \xi </math>
 
and are homomorphisms, i.e.,
 
:<math> \forall s,t \in \mathbb{R}: \quad U_{t + s} = U_{t} U_{s}. </math>
 
Such one-parameter families are ordinarily referred to as '''strongly continuous one-parameter unitary groups'''.
 
The theorem was proved by {{harvs|txt|authorlink=Marshall Stone|first=Marshall |last=Stone|year1=1930|year2=1932}}, and {{harvtxt|Von Neumann|1932}} showed that the requirement that <math> (U_{t})_{t \in \mathbb{R}} </math> be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.
 
This is a very stunning theorem, as it allows to define the derivative of the mapping <math> t \mapsto U_{t} </math>, which is only supposed to be continuous. It is also related to the theory of [[Lie groups]] and [[Lie algebras]].
 
== Formal statement ==
Let <math> (U_{t})_{t \in \mathbb{R}} </math> be a [[compact operator|strongly continuous]] one-parameter unitary group. Then there exists a unique (not necessarily bounded) self-adjoint operator <math> A </math> such that
 
:<math> \forall t \in \mathbb{R}: \quad U_{t} = e^{i t A}. </math>
 
Conversely, let <math> A </math> be a (not necessarily bounded) self-adjoint operator on a Hilbert space <math> \mathcal{H} </math>.  Then the one-parameter family <math> (U_{t})_{t \in \mathbb{R}} </math> of unitary operators defined by (using the Spectral Theorem for Self-Adjoint Operators)
 
:<math> \forall t \in \mathbb{R}: \quad U_{t} := e^{i t A} </math>
 
is a strongly continuous one-parameter group.
 
The [[Lie group#The Lie algebra associated to a Lie group|infinitesimal generator]] of <math> (U_{t})_{t \in \mathbb{R}} </math> is defined to be the operator <math> i A </math>. This mapping is a bijective correspondence. Furthermore, <math> A </math> will be a bounded operator if and only if the operator-valued mapping <math> t \mapsto U_{t} </math> is [[Norm (mathematics)|norm]]-continuous.
 
Stone's Theorem can be recast using the language of the [[Fourier transform]]. The real line <math> \mathbb{R} </math> is a locally compact abelian group. Non-degenerate <math> * </math>-representations of the [[group algebra|group C*-algebra]] <math> {C^{*}}(\mathbb{R}) </math> are in one-to-one correspondence with strongly continuous unitary representations of <math> \mathbb{R} </math>, i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a <math> * </math>-isomorphism from <math> {C^{*}}(\mathbb{R}) </math> to <math> {C_{0}}(\mathbb{R}) </math>, the C*-algebra of complex-valued continuous functions on the real line that vanish at infinity. Hence, there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and <math> * </math>-representations of <math> {C_{0}}(\mathbb{R}) </math>. As every <math> * </math>-representation of <math> {C_{0}}(\mathbb{R}) </math> corresponds uniquely to a self-adjoint operator, Stone's Theorem holds.
 
Therefore, the procedure for obtaining the infinitesimal generator of a strongly continuous one-parameter unitary group is as follows.
* Let <math> (U_{t})_{t \in \mathbb{R}} </math> be a strongly continuous unitary representation of <math> \mathbb{R} </math> on a [[Hilbert space]] <math> \mathcal{H} </math>.
* Integrate this unitary representation to yield a non-degenerate <math> * </math>-representation <math> \rho </math> of <math> {C^{*}}(\mathbb{R}) </math> on <math> \mathcal{H} </math> by first defining <math> \displaystyle \rho(f) := \int_{\mathbb{R}} f(t) U_{t} \, \mathrm{d}{t} </math> for all <math> f \in {C_{c}}(\mathbb{R}) </math> and then extending <math> \rho </math> to all of <math> {C^{*}}(\mathbb{R}) </math> by continuity.
* Use the Fourier transform to obtain a non-degenerate <math> * </math>-representation <math> \tau </math> of <math> {C_{0}}(\mathbb{R}) </math> on <math> \mathcal{H} </math>.
* By the [[Riesz–Markov–Kakutani representation theorem|Riesz-Markov Theorem]], <math> \tau </math> gives rise to a [[projection-valued measure]] on <math> \mathbb{R} </math> that is the resolution of the identity of a unique [[self-adjoint operator]] <math> A </math>, which may be unbounded.
* Then <math> i A </math> is the infinitesimal generator of <math> (U_{t})_{t \in \mathbb{R}} </math>.
 
The precise definition of <math> {C^{*}}(\mathbb{R}) </math> is as follows. Consider the <math> * </math>-algebra <math> {C_{c}}(\mathbb{R}) </math>, the complex-valued continuous functions on <math> \mathbb{R} </math> with compact support, where the multiplication is given by [[convolution]]. The completion of this <math> * </math> algebra with respect to the <math> L^{1} </math>-norm is a Banach <math> * </math>-algebra, denoted by <math> ({L^{1}}(\mathbb{R}),\star) </math>. Then <math> {C^{*}}(\mathbb{R}) </math> is defined to be the '''enveloping C*-algebra''' of <math> ({L^{1}}(\mathbb{R}),\star) </math>, i.e., its completion with respect to the largest possible C*-norm. It is a non-trivial fact that, via the Fourier transform, <math> {C^{*}}(\mathbb{R}) </math> is isomorphic to <math> {C_{0}}(\mathbb{R}) </math>. A result in this direction is the [[Riemann–Lebesgue lemma|Riemann-Lebesgue Lemma]], which says that the Fourier transform maps <math> {L^{1}}(\mathbb{R}) </math> to <math> {C_{0}}(\mathbb{R}) </math>.
 
== Example ==
 
The family of translation operators
 
:<math> [T_t \psi](x) = \psi(x + t) \quad </math>
 
is a  one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an [[extensions of symmetric operators|extension]] of the differential operator
 
:<math> \frac{d}{dx} = i \frac{1}{i} \frac{d}{dx} </math>
 
defined on the space of complex-valued continuously differentiable functions of [[compact support]] on '''R'''.  Thus
 
:<math> T_t = e^{t \, {d}/{dx}}. \quad </math>
 
In other words, motion on the line is generated by the [[momentum operator]].
 
== Applications ==
{{Expand section|date=April 2009}}
 
Stone's theorem has numerous applications in [[quantum mechanics]]. For instance, given an isolated quantum mechanical system, with Hilbert space of states ''H'', [[time evolution]] is a strongly continuous one-parameter unitary group on ''H''.  The infinitesimal generator of this group is the system [[Hamiltonian (quantum theory)|Hamiltonian]].
 
== Generalizations ==
The [[Stone–von Neumann theorem]] generalizes Stone's theorem to a ''pair'' of self-adjoint operators, ''Q, P'' satisfying the [[canonical commutation relation]], and shows that these are all unitarily equivalent to the [[position operator]] and [[momentum operator]] on L<sup>2</sup>('''R''').
 
The [[Hille–Yosida theorem]] generalizes Stone's theorem to strongly continuous one-parameter semigroups of [[contraction (operator theory)|contraction]]s on [[Banach space]]s.
 
== References ==
*{{Citation | last1=Neumann | first1=J. von | title=Über einen Satz von Herrn M. H. Stone | publisher=Annals of Mathematics | language=German | series=Second Series | year=1932 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=33 | issue=3 | pages=567–573 | jstor=1968535 | doi=10.2307/1968535}}
*{{Citation | last1=Stone | first1=M. H. | title=Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory | publisher=National Academy of Sciences | year=1930 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=16 | issue=2 | pages=172–175 | jstor=85485 | doi=10.1073/pnas.16.2.172}}
* {{citation|first=M. H. |last=Stone|title= On one-parameter unitary groups in Hilbert Space|journal= Annals of Mathematics |volume=33|issue= 3|pages= 643–648|year=1932|jstor=1968538|doi=10.2307/1968538}}
 
* K. Yosida, ''Functional Analysis'', Springer-Verlag, (1968)
 
{{DEFAULTSORT:Stone's Theorem On One-Parameter Unitary Groups}}
[[Category:Functional analysis]]
[[Category:Theorems in functional analysis]]

Latest revision as of 12:04, 24 August 2014

He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been worried by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is.

I keep conjuring up images of him in 1983, trying to reignite the passion.
I told him, before he started wriggling, that I think that memorable evening, when after our game of squash he had asked me to take his racquet home for him because he had a date, he had already started seeing the woman he would marry.

He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.

The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen.

99 ★★★★★Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�s famous examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329.

And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. �
This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you.

Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx�
The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message�
�The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road!

She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week.

The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum
But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7.

In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth.

Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3D printer.

Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too.

WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES
The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals
With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. ★★★★★

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