Pixel connectivity: Difference between revisions

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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:
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'''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''&nbsp;=&nbsp;''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'')&nbsp;=&nbsp;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&ndash;415 (1966)
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[[Category:Operator theory]]

Latest revision as of 11:25, 7 January 2015

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