Raychaudhuri equation - Revision history

77.69.221.233: Needless adjectives/embellishments are subjective and detract from the authority of the article and the subject scientists' referenced work.

2014-11-20T20:34:49Z

Needless adjectives/embellishments are subjective and detract from the authority of the article and the subject scientists' referenced work.

← Older revision		Revision as of 22:34, 20 November 2014
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	~~In [[mathematics]], a~~ '~~''nuclear operator'''~~ is ~~a [[compact operator]]~~ for ~~which~~ a ~~[[trace (linear algebra)\|trace]] may be defined, such that the trace is finite and independent~~ of ~~the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace).~~		Alyson is what my husband loves to call me but I don't like when individuals use my complete name. Office supervising is what she does for a residing. I am truly fond of handwriting but I can't make it my occupation truly. Her family members lives in Ohio but her spouse wants them to move.<br><br>Feel free to surf to my web page ... [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ online psychic]
	~~Nuclear operators are essentially the same as~~ '~~''[[trace class\|trace class operators]]''', though most authors reserve the term "trace class operator" for the special case of~~
	~~nuclear operators on [[Hilbert space]]s~~. The general definition for [[Banach space]]s was given by [[Grothendieck]]. This article concentrates on the general case of nuclear operators on Banach spaces; for the important special case of nuclear (=trace class) operators on Hilbert space see the article on [[trace class\|trace class operator]]s.

	~~==Compact operator==~~
	~~An operator <math>\mathcal{L}</math> on a [[Hilbert space]] <math>\mathcal{H}</math>~~

	~~:<math>\mathcal{L}:\mathcal{H} \to \mathcal{H}</math>~~

	~~is said to be a [[compact operator]] if it can be written~~ in ~~the form{{Citation needed\|date=September 2011}}~~

	~~:<math>\mathcal{L} = \sum_{n=1}^N \rho_n \langle f_n, \cdot \rangle g_n</math>~~

	where <math>1 \le N \le \infty</math> and <math>f_1,\ldots,f_N</math> and <math>g_1,\ldots,g_N</math> are (not necessarily complete) orthonormal sets. Here, <math>\rho_1,\ldots,\rho_N</math> are a set of real numbers, the [[singular value]]s of the operator, obeying <math>\rho_n \to ~~0</math> if <math>N = \infty</math>~~. ~~The bracket~~ <~~math~~>~~\langle\cdot,\cdot\rangle~~<~~/math~~> ~~is the scalar product on the Hilbert space; the sum on the right hand side must converge in norm.~~

	~~==Nuclear operator==~~
	~~An operator that is compact as defined above is said~~ to ~~be '''nuclear''' or '''trace-class''' if~~

	~~:<math>\sum_{n=1}^\infty \|\rho_n\| < \infty</math>~~

	~~==Properties==~~
	A nuclear operator on a Hilbert space has the important property that its [[trace class\|trace]] may be defined so that it is finite and is independent of the basis. Given any orthonormal basis <math>\{\psi_n\}</math> for the Hilbert space, one may define the trace as

	~~:<math>\mbox{Tr} \mathcal {L} = \sum_n \langle \psi_n , \mathcal{L} \psi_n \rangle</math>~~

	~~since the sum converges absolutely and is independent of the basis{{Citation needed\|date=September 2011}}. Furthermore, this trace is identical~~ to ~~the sum over the eigenvalues of <math>\mathcal{L}</math> (counted with multiplicity)~~.

	~~==On Banach spaces==~~
	~~:''See main article [[Fredholm kernel]]~~.''

	~~The definition of trace-class operator was extended to [[Banach space]]s by [[Alexander Grothendieck]] in 1955.~~

	Let ''A'' and ''B'' be Banach spaces, and ''A''' be the [[continuous dual space\|dual]] of ''A'', that is, the set of all [[continuous (topology)\|continuous]] or (equivalently) [[bounded linear functional]]s on ''A'' with the usual norm. ~~Then an operator~~

	~~:<math>\mathcal{L}:A \to B</math>~~

	is said to be '''nuclear of order ''q'' ''' if there exist sequences of vectors <math>\{g_n\} \in B</math> with <math>\Vert g_n \Vert \le 1</math>, functionals <math>\{f^_n\} \in A'</math> with <math>\Vert f^_n \Vert \le 1</math> and [~~[complex number]]s <math>\{\rho_n\}</math> with~~

	~~:<math>\inf \left\{ p\ge 1~~ : ~~\sum_n \|\rho_n\|^p < \infty \right\} = q,<~~/~~math>~~

	~~such that the operator may be written as~~

	~~:<math>\mathcal{L} = \sum_n \rho_n f^*_n(\cdot) g_n<~~/~~math>~~

	~~with the sum converging in the operator norm~~.

	~~With additional steps, a trace may be defined for such operators when ''A'' = ''B''.~~

	~~Operators that are nuclear of order 1 are called '''nuclear operators''': these are the ones for which the series ∑''ρ<sub>n<~~/~~sub>'' is absolutely convergent. Nuclear operators of order 2 are called [[Hilbert~~-~~Schmidt operator]]s.~~

	More generally, an operator from a [[locally convex topological vector space]] ''A'' to a Banach space ''B'' is called '''nuclear''' if it satisfies the condition above with all ''f<sub>n</sub><sup>*</sup>'' bounded by 1 on some fixed neighborhood of 0 and all ''g<sub>n</sub>'' bounded by 1 on some fixed neighborhood of 0.

	~~==References==~~
	* A. Grothendieck (1955), Produits tensoriels topologiques et espace nucléaires,''Mem. Am. Math.Soc.'' '''16'''. {{MR\|0075539}}
	* A. Grothendieck (1956), La theorie de Fredholm, ''Bull. Soc. Math. France'', '''84''':319-~~384. {{MR\|0088665}}~~
	* A. Hinrichs and A. Pietsch (2010), ''p''-~~nuclear operators in the sense of Grothendieck, ''Mathematische Nachrichen'' '''283''': 232~~-~~261. {{doi\|10.1002~~/~~mana.200910128}} {{MR\|2604120}}~~
	* {{springer\|id=Nuclear_operator\|author=G. L. Litvinov\|title=Nuclear operator }}

	~~{{Functional Analysis}}~~

	~~[[Category:Operator theory]~~]

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-29T03:51:30Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 05:51, 29 January 2014
Line 1:		Line 1:
	~~Greetings~~. The [~~http~~://~~Www~~.~~Justmommies~~.~~com~~/~~articles/sexually_transmitted_disease~~.~~shtml author~~'s] ~~name~~ is ~~Phebe~~ and ~~she feels comfortable when people use~~ the ~~full name~~. ~~Hiring has been my [http~~://~~www~~.~~medhelp~~.~~org/posts/STDs/Sore~~-~~Throat-STDs/show/245570 occupation~~] ~~for some time but I~~'~~ve currently applied~~ [~~http~~://~~Www.hotporn123.com~~/~~user~~/~~KGottscha std testing at home~~] ~~for an additional~~ 1~~. To collect cash is over~~ the ~~counter std test [http~~:/~~/www~~.~~gaysphere~~.~~net~~/~~blog/262619 home std test kit~~] ~~what his family members and him enjoy~~. ~~South Dakota~~ is ~~exactly std testing at home where I~~'~~ve always been residing.~~<br><br>~~Feel free to surf to my web site; [http:~~//~~tcciaarusha~~-~~bic~~.~~info/groups/simple~~-~~stuff~~-~~you-could-do-to-remove-candida~~/ ~~std testing at home~~]		In [[mathematics]], a '''nuclear operator''' is a [[compact operator]] for which a [[trace (linear algebra)\|trace]] may be defined, such that the trace is finite and independent of the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace).
			Nuclear operators are essentially the same as '''[[trace class\|trace class operators]]''', though most authors reserve the term "trace class operator" for the special case of
			nuclear operators on [[Hilbert space]]s. The general definition for [[Banach space]]s was given by [[Grothendieck]]. This article concentrates on the general case of nuclear operators on Banach spaces; for the important special case of nuclear (=trace class) operators on Hilbert space see the article on [[trace class\|trace class operator]]s.

			==Compact operator==
			An operator <math>\mathcal{L}</math> on a [[Hilbert space]] <math>\mathcal{H}</math>

			:<math>\mathcal{L}:\mathcal{H} \to \mathcal{H}</math>

			is said to be a [[compact operator]] if it can be written in the form{{Citation needed\|date=September 2011}}

			:<math>\mathcal{L} = \sum_{n=1}^N \rho_n \langle f_n, \cdot \rangle g_n</math>

			where <math>1 \le N \le \infty</math> and <math>f_1,\ldots,f_N</math> and <math>g_1,\ldots,g_N</math> are (not necessarily complete) orthonormal sets. Here, <math>\rho_1,\ldots,\rho_N</math> are a set of real numbers, the [[singular value]]s of the operator, obeying <math>\rho_n \to 0</math> if <math>N = \infty</math>. The bracket <math>\langle\cdot,\cdot\rangle</math> is the scalar product on the Hilbert space; the sum on the right hand side must converge in norm.

			==Nuclear operator==
			An operator that is compact as defined above is said to be '''nuclear''' or '''trace-class''' if

			:<math>\sum_{n=1}^\infty \|\rho_n\| < \infty</math>

			==Properties==
			A nuclear operator on a Hilbert space has the important property that its [[trace class\|trace]] may be defined so that it is finite and is independent of the basis. Given any orthonormal basis <math>\{\psi_n\}</math> for the Hilbert space, one may define the trace as

			:<math>\mbox{Tr} \mathcal {L} = \sum_n \langle \psi_n , \mathcal{L} \psi_n \rangle</math>

			since the sum converges absolutely and is independent of the basis{{Citation needed\|date=September 2011}}. Furthermore, this trace is identical to the sum over the eigenvalues of <math>\mathcal{L}</math> (counted with multiplicity).

			==On Banach spaces==
			:''See main article [[Fredholm kernel]].''

			The definition of trace-class operator was extended to [[Banach space]]s by [[Alexander Grothendieck]] in 1955.

			Let ''A'' and ''B'' be Banach spaces, and ''A''' be the [[continuous dual space\|dual]] of ''A'', that is, the set of all [[continuous (topology)\|continuous]] or (equivalently) [[bounded linear functional]]s on ''A'' with the usual norm. Then an operator

			:<math>\mathcal{L}:A \to B</math>

			is said to be '''nuclear of order ''q'' ''' if there exist sequences of vectors <math>\{g_n\} \in B</math> with <math>\Vert g_n \Vert \le 1</math>, functionals <math>\{f^_n\} \in A'</math> with <math>\Vert f^_n \Vert \le 1</math> and [[complex number]]s <math>\{\rho_n\}</math> with

			:<math>\inf \left\{ p\ge 1 : \sum_n \|\rho_n\|^p < \infty \right\} = q,</math>

			such that the operator may be written as

			:<math>\mathcal{L} = \sum_n \rho_n f^*_n(\cdot) g_n</math>

			with the sum converging in the operator norm.

			With additional steps, a trace may be defined for such operators when ''A'' = ''B''.

			Operators that are nuclear of order 1 are called '''nuclear operators''': these are the ones for which the series ∑''ρ<sub>n</sub>'' is absolutely convergent. Nuclear operators of order 2 are called [[Hilbert-Schmidt operator]]s.

			More generally, an operator from a [[locally convex topological vector space]] ''A'' to a Banach space ''B'' is called '''nuclear''' if it satisfies the condition above with all ''f<sub>n</sub><sup>*</sup>'' bounded by 1 on some fixed neighborhood of 0 and all ''g<sub>n</sub>'' bounded by 1 on some fixed neighborhood of 0.

			==References==
			* A. Grothendieck (1955), Produits tensoriels topologiques et espace nucléaires,''Mem. Am. Math.Soc.'' '''16'''. {{MR\|0075539}}
			* A. Grothendieck (1956), La theorie de Fredholm, ''Bull. Soc. Math. France'', '''84''':319-384. {{MR\|0088665}}
			* A. Hinrichs and A. Pietsch (2010), ''p''-nuclear operators in the sense of Grothendieck, ''Mathematische Nachrichen'' '''283''': 232-261. {{doi\|10.1002/mana.200910128}} {{MR\|2604120}}
			* {{springer\|id=Nuclear_operator\|author=G. L. Litvinov\|title=Nuclear operator }}

			{{Functional Analysis}}

			[[Category:Operator theory]]

en>Bibcode Bot: Adding 1 arxiv eprint(s), 0 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2012-08-23T05:24:35Z

Adding 1 arxiv eprint(s), 0 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

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