Finite rank operators in closed maximal triangular algebras II

Dong, Zhe; Lu, Shijie

doi:10.1090/S0002-9939-02-06748-5

Finite rank operators in closed maximal triangular algebras II
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by Zhe Dong and Shijie Lu PDF

Proc. Amer. Math. Soc. 131 (2003), 1515-1525 Request permission

Abstract:

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.

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