Finite rank operators in closed maximal triangular algebras II

In this paper, we discuss finite rank operators in a closed maximal triangular algebra ${\mathcal{S}}$. Based on the following result that each finite rank operator of ${\mathcal{S}}$ can be written as a finite sum of rank one operators each belonging to ${\mathcal{S}}$, we proved that $({\mathcal{S}}\cap{\mathcal{F(H)}})^{w^{*}}=\{T\in{\mathcal{B(H)}}: TN\subseteq N_{\sim}, \forall N\in\mathcal{N}\}$, where $N_{\sim}=N$, if $dim N\ominus N_{-}\leq 1$; and $N_{\sim}=N_{-}$, if $dim N\ominus N_{-}=\infty$. We also proved that the Erdos Density Theorem holds in ${\mathcal{S}}$ if and only if ${\mathcal{S}}$ is strongly reducible.