An extension and refinement of the theorems of Douglas and Sebestyén for unbounded operators
Abstract
For a closed densely defined operator T T from a Hilbert space H {\mathfrak {H}} to a Hilbert space K \mathfrak {K} , necessary and sufficient conditions are established for the factorization of T T with a bounded nonnegative operator X X on K \mathfrak {K} . This result yields a new extension and a refinement of a well-known theorem of R.G. Douglas [Proc. Amer. Math. Soc. 17 (1966), pp. 413–415], which shows that the operator inequality A ∗ A ≤ λ 2 B ∗ B , λ ≥ 0 A^*A\leq \lambda ^2 B^*B, \lambda \geq 0 , is equivalent to the factorization A = C B A=CB with ‖ C ‖ ≤ λ \|C\|\leq \lambda . The main results give necessary and sufficient conditions for the existence of an intermediate selfadjoint operator H ≥ 0 H\geq 0 , such that A ∗ A ≤ λ H ≤
Fields of Study & Subjects
Advanced Banach Space Theory
Holomorphic and Operator Theory
Mathematical Inequalities and Applications
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 3
- Pages
- 1133–1142
- Publication date
- 14 January 2026
- Language
- EN