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88.108.232.88: 10/3.46 is not 2.71 but 2.89 so I changed it.

2015-01-02T19:54:30Z

10/3.46 is not 2.71 but 2.89 so I changed it.

← Older revision		Revision as of 21:54, 2 January 2015
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	~~In mathematics~~, ~~and specifically in [[operator theory]], a '''positive-definite function~~ on a group''' relates the notions of positivity, in the context of [[Hilbert space]]s, and algebraic [[group (mathematics)\|group]]s. It can be viewed as a particular type of [[positive-definite kernel]] where the underlying set has the additional group structure.		Hi there, I am Alyson Boon even though it is not the title on my beginning certificate. My spouse and I reside in Kentucky. I am really fond of to go to karaoke but I've been taking on new issues recently. Office supervising is exactly where her primary income arrives from but she's currently utilized for another one.<br><br>Here is my site ... [https://www-ocl.gist.ac.kr/work/xe/?document_srl=605236 psychic love readings]

	~~== Definition ==~~

	~~Let ''G'' be a group, ''H'' be a complex Hilbert space,~~ and ~~''L''(''H'') be the bounded operators on ''H''.~~
	~~A '''positive-definite function''' on ''G'' is a function {{nowrap\|''F'': ''G'' → ''L''(''H'')}} that satisfies~~

	~~:<math>\sum_{s,t \~~in ~~G}\langle F(s^{-1}t) h(t), h(s) \rangle \geq 0 ,</math>~~

	~~for every function ''h'': ''G'' → ''H'' with finite support (''h'' takes non-zero values for only finitely many ''s'')~~.

	~~In other words, a function ''F'': ''G'' → ''L''(''H'') is said~~ to ~~be a positive function if the kernel~~ '~~'K'': ''G'' × ''G'' → ''L''(''H'') defined by ''K''(''s'', ''t'') = ''F''(''s''<sup>−1</sup>''t'') is a positive-definite kernel~~.

	~~== Unitary representations ==~~

	~~A '''[[unitary representation]]'''~~ is ~~a unital homomorphism Φ: ''G'' → ''L''(''H'')~~ where ~~Φ('~~'s~~'') is a unitary operator~~ for ~~all ''s''. For such Φ, Φ(''s''<sup>−1</sup>) = Φ(''s'')*.~~

	Positive-definite functions on ''G'' are intimately related to unitary representations of ''G''. Every unitary representation of ''G'' gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one ~~can define a unitary representation of ''G'' in a natural way.~~

	Let Φ: ''G'' → ''L''(''H'') be a unitary representation of ''G''. If ''P'' ∈ ''L''(''H'') is the projection onto a closed subspace ''H`'' of ''H''. Then ''F''(''s'') = ''P'' Φ(''s'') is a positive-definite function on ''G'' with values in ''L''(''H`''). ~~This can be shown readily:~~

	:<~~math~~>~~\begin{align}~~
	~~\sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle~~
	~~& =\sum_{s,t \in G}\langle P \Phi (s^{-1}t) h(t), h(s) \rangle \\~~
	~~{} & =\sum_{s,t \in G}\langle \Phi (t) h(t), \Phi(s)h(s) \rangle \\~~
	~~{} & = \left\langle \sum_{t \in G} \Phi (t) h(t), \sum_{s \in G} \Phi(s)h(s) \right\rangle \\~~
	~~{} & \geq 0~~
	~~\end{align}~~
	<~~/math~~>

	~~for every ''h'': ''G'' → ''H`'' with finite support. If ''G'' has a topology and Φ~~ is ~~weakly(resp~~. ~~strongly) continuous, then clearly so is ''F''~~.

	~~On the other hand, consider now a positive-definite function ''F'' on ''G''~~. ~~A unitary representation of ''G'' can be obtained as follows. Let ''C''<sub>00</sub>(''G'', ''H'') be the family of functions ''h''~~: ~~''G'' → ''H'' with finite support. The corresponding positive kernel ''K''(''s'', ''t'') = ''F''(''s''<sup>−1<~~/~~sup>''t'') defines a (possibly degenerate) inner product on ''C''<sub>00<~~/~~sub>(''G'', ''H'')~~. ~~Let the resulting Hilbert space be denoted by ''V''~~.

	~~We notice that the "matrix elements" ''K''(''s'', ''t'') = ''K''(''a''<sup>−1</sup>''s'', ''a''<sup>−1</sup>''t'') for all ''a'', ''s'', ''t'' in ''G''~~. ~~So ''U<sub>a<~~/~~sub>h''(''s'') = ''h''(''a''<sup>−1<~~/~~sup>''s'') preserves the inner product on ''V'', i.e. it is unitary in ''L''(''V''). It is clear that the map Φ(''a'') = ''U''<sub>a<~~/~~sub> is a representation of ''G'' on ''V''.~~

	~~The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:~~

	~~:<math>V~~ = ~~\bigvee_{s \in G} \Phi(s)H \, </math>~~

	~~where <math>\bigvee</math> denotes the closure of the linear span.~~

	Identify ''H'' as elements (possibly equivalence classes) in ''V'', whose support consists of the identity element ''e'' ∈ ''G'', and let ''P'' be the projection onto this subspace. Then we have ''PU<sub>a</sub>P'' = ''F''(''a'') for all ''a'' ∈ ''G''.

	~~== Toeplitz kernels ==~~

	Let ''G'' be the additive group of integers '''Z'''. The kernel ''K''(''n'', ''m'') = ''F''(''m'' − ''n'') is called a kernel of ''Toeplitz'' type, by analogy with [[Toeplitz matrix\|Toeplitz matrices]]. If ''F'' is of the form ''F''(''n'') = ''T<sup>n</sup>'' where ''T'' is a bounded operator acting on some Hilbert space. One can show that the kernel ''K''(''n'', ''m'') is positive if and only if ''T'' is a [[Contraction (operator theory)\|contraction]]. By the discussion from the previous section, we have a unitary representation of '''Z''', Φ(''n'') = ''U''<sup>''n''</sup> for a unitary operator ''U''. Moreover, the property ''PU<sub>a</sub>P'' = ''F''(''a'') now translates to ''PU<sup>n</sup>P'' = ''T<sup>n</sup>''. This is precisely [[Sz.-Nagy's dilation theorem]] and hints at an important dilation-theoretic characterization of positivity that leads to a parametrization of arbitrary positive-definite kernels.

	~~== References ==~~
	*Christian Berg, Christensen, Paul Ressel''Harmonic Analysis on Semigroups'', GTM, Springer Verlag.
	*T. Constantinescu, ''Schur Parameters, Dilation and Factorization Problems'', Birkhauser Verlag, 1996.
	*B. Sz.-Nagy and C. Foias, ''Harmonic Analysis of Operators on Hilbert Space,'' North-Holland, 1970.
	*Z. Sasvári, ''Positive Definite and Definitizable Functions'', Akademie Verlag, 1994
	*Wells, J. H.; Williams, L. R. ''Embeddings and extensions in analysis''. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.

	~~[[Category:Operator theory]]~~
	~~[[Category:Representation theory of groups]~~]

en>RomanSpa at 14:01, 10 December 2013

2013-12-10T14:01:40Z

← Older revision		Revision as of 16:01, 10 December 2013
Line 1:		Line 1:
	~~Claude~~ is ~~her name and she completely digs~~ that ~~title~~. ~~Bookkeeping~~ is ~~what she does~~. ~~The preferred hobby~~ for ~~my kids~~ and me is ~~playing crochet and~~ now I'~~m trying to make cash~~ with it. ~~Her spouse and her selected~~ to ~~reside~~ in ~~Alabama~~.<br><br>~~Take~~ a ~~look at my website~~ ... [~~http:~~//14.63.~~168~~.~~193/index.php?document_srl~~=~~249897&mid~~=~~freeboard extended car warranty~~]		In mathematics, and specifically in [[operator theory]], a '''positive-definite function on a group''' relates the notions of positivity, in the context of [[Hilbert space]]s, and algebraic [[group (mathematics)\|group]]s. It can be viewed as a particular type of [[positive-definite kernel]] where the underlying set has the additional group structure.

			== Definition ==

			Let ''G'' be a group, ''H'' be a complex Hilbert space, and ''L''(''H'') be the bounded operators on ''H''.
			A '''positive-definite function''' on ''G'' is a function {{nowrap\|''F'': ''G'' → ''L''(''H'')}} that satisfies

			:<math>\sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle \geq 0 ,</math>

			for every function ''h'': ''G'' → ''H'' with finite support (''h'' takes non-zero values for only finitely many ''s'').

			In other words, a function ''F'': ''G'' → ''L''(''H'') is said to be a positive function if the kernel ''K'': ''G'' × ''G'' → ''L''(''H'') defined by ''K''(''s'', ''t'') = ''F''(''s''<sup>−1</sup>''t'') is a positive-definite kernel.

			== Unitary representations ==

			A '''[[unitary representation]]''' is a unital homomorphism Φ: ''G'' → ''L''(''H'') where Φ(''s'') is a unitary operator for all ''s''. For such Φ, Φ(''s''<sup>−1</sup>) = Φ(''s'')*.

			Positive-definite functions on ''G'' are intimately related to unitary representations of ''G''. Every unitary representation of ''G'' gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of ''G'' in a natural way.

			Let Φ: ''G'' → ''L''(''H'') be a unitary representation of ''G''. If ''P'' ∈ ''L''(''H'') is the projection onto a closed subspace ''H`'' of ''H''. Then ''F''(''s'') = ''P'' Φ(''s'') is a positive-definite function on ''G'' with values in ''L''(''H`''). This can be shown readily:

			:<math>\begin{align}
			\sum_{s,t \in G}\langle F(s^{-1}t) h(t), h(s) \rangle
			& =\sum_{s,t \in G}\langle P \Phi (s^{-1}t) h(t), h(s) \rangle \\
			{} & =\sum_{s,t \in G}\langle \Phi (t) h(t), \Phi(s)h(s) \rangle \\
			{} & = \left\langle \sum_{t \in G} \Phi (t) h(t), \sum_{s \in G} \Phi(s)h(s) \right\rangle \\
			{} & \geq 0
			\end{align}
			</math>

			for every ''h'': ''G'' → ''H`'' with finite support. If ''G'' has a topology and Φ is weakly(resp. strongly) continuous, then clearly so is ''F''.

			On the other hand, consider now a positive-definite function ''F'' on ''G''. A unitary representation of ''G'' can be obtained as follows. Let ''C''<sub>00</sub>(''G'', ''H'') be the family of functions ''h'': ''G'' → ''H'' with finite support. The corresponding positive kernel ''K''(''s'', ''t'') = ''F''(''s''<sup>−1</sup>''t'') defines a (possibly degenerate) inner product on ''C''<sub>00</sub>(''G'', ''H''). Let the resulting Hilbert space be denoted by ''V''.

			We notice that the "matrix elements" ''K''(''s'', ''t'') = ''K''(''a''<sup>−1</sup>''s'', ''a''<sup>−1</sup>''t'') for all ''a'', ''s'', ''t'' in ''G''. So ''U<sub>a</sub>h''(''s'') = ''h''(''a''<sup>−1</sup>''s'') preserves the inner product on ''V'', i.e. it is unitary in ''L''(''V''). It is clear that the map Φ(''a'') = ''U''<sub>a</sub> is a representation of ''G'' on ''V''.

			The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:

			:<math>V = \bigvee_{s \in G} \Phi(s)H \, </math>

			where <math>\bigvee</math> denotes the closure of the linear span.

			Identify ''H'' as elements (possibly equivalence classes) in ''V'', whose support consists of the identity element ''e'' ∈ ''G'', and let ''P'' be the projection onto this subspace. Then we have ''PU<sub>a</sub>P'' = ''F''(''a'') for all ''a'' ∈ ''G''.

			== Toeplitz kernels ==

			Let ''G'' be the additive group of integers '''Z'''. The kernel ''K''(''n'', ''m'') = ''F''(''m'' − ''n'') is called a kernel of ''Toeplitz'' type, by analogy with [[Toeplitz matrix\|Toeplitz matrices]]. If ''F'' is of the form ''F''(''n'') = ''T<sup>n</sup>'' where ''T'' is a bounded operator acting on some Hilbert space. One can show that the kernel ''K''(''n'', ''m'') is positive if and only if ''T'' is a [[Contraction (operator theory)\|contraction]]. By the discussion from the previous section, we have a unitary representation of '''Z''', Φ(''n'') = ''U''<sup>''n''</sup> for a unitary operator ''U''. Moreover, the property ''PU<sub>a</sub>P'' = ''F''(''a'') now translates to ''PU<sup>n</sup>P'' = ''T<sup>n</sup>''. This is precisely [[Sz.-Nagy's dilation theorem]] and hints at an important dilation-theoretic characterization of positivity that leads to a parametrization of arbitrary positive-definite kernels.

			== References ==
			*Christian Berg, Christensen, Paul Ressel''Harmonic Analysis on Semigroups'', GTM, Springer Verlag.
			*T. Constantinescu, ''Schur Parameters, Dilation and Factorization Problems'', Birkhauser Verlag, 1996.
			*B. Sz.-Nagy and C. Foias, ''Harmonic Analysis of Operators on Hilbert Space,'' North-Holland, 1970.
			*Z. Sasvári, ''Positive Definite and Definitizable Functions'', Akademie Verlag, 1994
			*Wells, J. H.; Williams, L. R. ''Embeddings and extensions in analysis''. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.

			[[Category:Operator theory]]
			[[Category:Representation theory of groups]]

en>Phil Boswell: convert dodgy URL to ID using AWB

2012-08-30T21:29:45Z

convert dodgy URL to ID using AWB

New page

Claude is her name and she completely digs that title. Bookkeeping is what she does. The preferred hobby for my kids and me is playing crochet and now I'm trying to make cash with it. Her spouse and her selected to reside in Alabama.<br><br>Take a look at my website ... [http://14.63.168.193/index.php?document_srl=249897&mid=freeboard extended car warranty]