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 formula_1 and formula_2 are bounded operators on a Hilbert space formula_3, the following are equivalent:
Moreover, if these equivalent conditions hold, then there is a unique operator formula_7 such that
A generalization of Douglas' lemma for unbounded operators on a Banach space was proved by Forough (2014).
References.
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