Douglas' lemma
Encyclopedia
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 operator
s on a Hilbert space H, the following are equivalent:
Moreover, if these equivalent conditions hold, then there is a unique operator C such that
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, an area of mathematics, Douglas' lemma relates factorization
Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...
, range inclusion, and majorization of Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
operators. It is generally attributed to Ronald G. Douglas
Ronald G. Douglas
Ronald George Douglas is an American mathematician, best known for his work on operator algebras.Douglas was born in Osgood, Indiana. He was an undergraduate at the Illinois Institute of Technology, and received his Ph.D. in 1962 from Louisiana State University as a student of Pasquale Porcelli...
, 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
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s on a Hilbert space H, the following are equivalent:
-
-
for some 
- 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
-
- ker(A) = ker(C)
-
