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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:
 
'''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]]

Revision as of 17:34, 24 January 2014

In operator theory, an area of mathematics, Douglas' lemma relates factorization, range inclusion, and majorization 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 operators on a Hilbert space H, the following are equivalent:

  1. im(A)im(B).
  2. AA*λ2BB* for some λ0.
  3. 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

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)

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