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Finite rank operators in closed maximal triangular algebras II


Authors: Zhe Dong and Shijie Lu
Journal: Proc. Amer. Math. Soc. 131 (2003), 1515-1525
MSC (2000): Primary 47L75
Published electronically: October 1, 2002
MathSciNet review: 1949882
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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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Additional Information

Zhe Dong
Affiliation: Institute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
Address at time of publication: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: dzhe8@mail.china.com

Shijie Lu
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-02-06748-5
Keywords: Closed maximal triangular algebra, finite rank operator, \, $w^{*}$-closure
Received by editor(s): December 12, 2000
Received by editor(s) in revised form: December 16, 2001
Published electronically: October 1, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society