On stability of triangular factorization of positive operators

Let f=\ Fₛ>0 be a nest and C a bounded positive operator in a Hilbert space F. The representation C=V^*V provided V Fₛ⊂ Fₛ is a triangular factorization (TF) of C w.r.t. f. The factorization is stable if C^αα→∞→ C and C^α=Vα\,*V^α implies V^α→ V. If C is positive definite (isomorphism), then TF is stable. The paper deals with the case of positive but not positive definite C. We impose some assumptions on C^α and C which provide the stability of TF.