Pixel connectivity - Revision history

103.27.8.45: /* 2-dimensional */

2015-01-07T10:25:21Z

2-dimensional

← Older revision		Revision as of 12:25, 7 January 2015
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	In [[operator theory]], an area of mathematics, '''Douglas' lemma''' relates [[Matrix decomposition\|factorization]], range inclusion, and majorization <!-- [[Majorization]] discusses a different topic, so is not currently ~~appropriate to link to--> of [[Hilbert space]] operators~~. ~~It is generally attributed to [[Ronald G~~. ~~Douglas]], although Douglas acknowledges that aspects of the result may already~~ have been ~~known. The statement of the result is~~ as ~~follows:~~		Brunilda is how I am known as even though I do not truly like currently being known as like that. Pennsylvania has usually been my residence. For many years I have been operating as a generation and [http://Www.Reddit.com/r/howto/search?q=distribution+officer distribution officer]. I am truly fond of marbles but I have not produced a dime with it. You can come across my web-site in this article: http://www.verso.me/datatool/assets/fonts/thumbs/salomon-snowboardschoenen.cfm<br><br>My blog post :: [http://www.verso.me/datatool/assets/fonts/thumbs/salomon-snowboardschoenen.cfm Salomon Snowboardschoenen]

	~~'''Theorem''': If ''A''~~ and ~~''B'' are~~ [~~[bounded operator]]s on a Hilbert space ''H'', the following are equivalent~~:
	~~# <math>\text{im}(A)\subseteq\text{im}(B)~~.~~\,<~~/~~math>~~
	~~# <math>AA^\leq\lambda^2 BB^<~~/~~math> for some <math>\lambda\geq 0.\,<~~/~~math>~~
	~~# There exists a bounded operator ''C'' on ''H'' such that ''A'' ~~=~~ ''BC''~~.
	~~Moreover, if these equivalent conditions hold, then there is~~ a ~~unique operator ''C'' such that~~
	* <math>\Vert C \Vert^2=\inf\{\mu :~~\,AA^\leq\mu BB^\}~~.</~~math>~~
	* ker(''A'') = ker(''C'')
	* <~~math~~>~~\text{im}(C)\subseteq\overline{\text{im}(B^*)}~~<~~/math~~>

	~~==See also==~~
	[~~[Positive operator]]~~

	~~==References==~~
	*Douglas, R.G.~~: "On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space"~~. ~~''Proceedings of the American Mathematical Society'' '''17''', 413–415 (1966)~~
	~~{{mathanalysis-stub}}~~

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

14.139.69.85: /* 4-connected */

2014-01-24T16:34:23Z

4-connected

← Older revision		Revision as of 18:34, 24 January 2014
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	Hi, ~~еverybody~~! <br>~~I'm Portuguese male ;=~~). <br>~~Ι really love American Dad!~~<br>~~http:~~//~~www~~.~~southfieldwatch~~.~~org.za~~/~~UserFiles/2011hogan/sitemap-1.xml~~<br><br>~~My web site -~~ [~~http~~:~~//www~~.~~immobiliaremazzola.com/UserFiles/2011hogan/sitemap~~-~~1.xml louis vuitton seattle~~]		In [[operator theory]], an area of mathematics, '''Douglas' lemma''' relates [[Matrix decomposition\|factorization]], range inclusion, and majorization <!-- [[Majorization]] discusses a different topic, so is not currently appropriate to link to--> of [[Hilbert space]] operators. It is generally attributed to [[Ronald G. Douglas]], although Douglas acknowledges that aspects of the result may already have been known. The statement of the result is as follows:

			'''Theorem''': If ''A'' and ''B'' are [[bounded operator]]s on a Hilbert space ''H'', the following are equivalent:
			# <math>\text{im}(A)\subseteq\text{im}(B).\,</math>
			# <math>AA^\leq\lambda^2 BB^</math> for some <math>\lambda\geq 0.\,</math>
			# There exists a bounded operator ''C'' on ''H'' such that ''A'' = ''BC''.
			Moreover, if these equivalent conditions hold, then there is a unique operator ''C'' such that
			* <math>\Vert C \Vert^2=\inf\{\mu :\,AA^\leq\mu BB^\}.</math>
			* ker(''A'') = ker(''C'')
			* <math>\text{im}(C)\subseteq\overline{\text{im}(B^*)}</math>

			==See also==
			[[Positive operator]]

			==References==
			*Douglas, R.G.: "On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space". ''Proceedings of the American Mathematical Society'' '''17''', 413–415 (1966)
			{{mathanalysis-stub}}

			[[Category:Operator theory]]

en>Helpful Pixie Bot: ISBNs (Build KC)

2012-05-05T23:57:18Z

ISBNs (Build KC)

New page

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