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≤ λ ^2 B^*B, λ ≥ 0 , is equivalent to the factorization A = C B A=CB with ‖ C ‖ ≤ λ \|C\|≤ λ . 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 ≤