On stability of triangular factorization of positive operators

Let $\mathfrak f=\{\mathscr F_s\}_{s>0}$ be a nest and $C$ a bounded positive operator in a Hilbert space $\mathscr F$. The representation $C=V^*V$ provided $V\mathscr F_s\subset\mathscr F_s$ is a triangular factorization (TF) of $C$ w.r.t. $\mathfrak f$. The factorization is stable if $C^\alpha\underset{\alpha\to\infty}\to C$ and $C^\alpha=V^{\alpha\,*}V^\alpha$ implies $V^\alpha\to V$. 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 $C^\alpha$ and $C$ which provide the stability of TF.