An extension and refinement of the theorems of Douglas and Sebestyén for unbounded operators

For a closed densely defined operator T T from a Hilbert space H {H} to a Hilbert space K K, necessary and sufficient conditions are established for the factorization of T T with a bounded nonnegative operator X X on K 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 ² B^*B, 0, is equivalent to the factorization A = C B A=CB with ‖ C ‖ ≤ λ \|C\|. The main results give necessary and sufficient conditions for the existence of an intermediate selfadjoint operator H ≥ 0 H 0, such that A ∗ A ≤ λ H ≤