Finite rank operators in closed maximal triangular algebras II

Abstract

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