On stability of triangular factorization of positive operators

Abstract: Let $\mathfrak f={\mathscr F_s}_{s&gt;0}$ be a nest and $C$ a bounded positive operator in a Hilbert space $\mathscr F$. The representation $C=V<sup>*V$ 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 $C<sup>\alpha=V<sup>{\alpha\,*}V<sup>\alpha$ 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 $C<sup>\alpha$ and $C$ which provide the stability of TF.