On stability of triangular factorization of positive operators

Abstract

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