Petr–Douglas–Neumann theorem

The Petr–Douglas–Neumann theorem characterizes when one bounded operator’s range lies in another’s: range inclusion, a positive-operator inequality, and bounded factorization are equivalent.

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Douglas lemma: A theorem in operator theory characterizing range inclusion through operator inequalities and bounded factorization. This broader name is commonly used for the same range-factorization result.
Bounded linear operator: A linear map between normed spaces whose outputs are bounded in norm by a constant times input norms. The theorem concerns bounded operators between Hilbert spaces.
Operator equation: An equation in which unknowns or given quantities are linear operators. Factorization asks whether an operator equation AX=B has a bounded solution X.
Ronald G. Douglas: An American mathematician known for work in operator theory and functional analysis. Douglas published the influential range-inclusion and factorization formulation.
Range inclusion: A relation in which every output of one map is also an output of another. It is the theorem’s geometric condition, involving the ranges of two operators.
Hilbert space: A complete inner-product space whose norm is induced by its inner product. Inner products make adjoints and operator positivity available in the theorem.
Least-squares problem: An optimization problem that minimizes the norm of a residual between observed and modeled data. Range factorization helps establish bounded solution operators for least-squares equations.
C. A. McCarthy: An American mathematician whose work included operator theory and functional analysis. McCarthy is associated with the theorem’s historical naming and development.
Loewner order: A partial order on self-adjoint operators defined by positivity of their difference. The theorem expresses range containment as an inequality between positive operators.
Adjoint operator: The operator defined by transferring a bounded operator across the inner products of two Hilbert spaces. Adjoints turn the range condition into the product inequality in the theorem.