Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-02-06748-5
Published electronically: October 1, 2002
MathSciNet review: 1949882
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. W.B.Arveson, A density theorem for operator algebras, Duke Math. J. 34(1967), 635-647. MR 36:4345
  • 2. E.Christensen, Derivations of nest algebras, Math.Ann. 229(1977), 155-161. MR 56:6420
  • 3. K.R.Davidson, Nest Algebras, Pitman Research Notes in Mathematics Series 191, Longman Scientific and Technical, Burnt Mill Harlow, Essex,UK,1988. MR 90f:47062
  • 4. Z.Dong and S.J.Lu, Finite rank operators in closed maximal triangular algebras, Northeastern Math.J.(Chinese) (2)16(2000), 167-173. MR 2002b:47139
  • 5. J.A.Erdos, Some results on triangular operator algebras, Amer.J. Math. 89 (1967), 85-93. MR 35:800
  • 6. J.A.Erdos, Operators of finite rank in nest algebras, J.London Math.Soc. 43(1968), 391-397. MR 37:5721
  • 7. J.A.Erdos and S.C.Power, Weakly closed ideals of nest algebras, J.Operator Theory 7(1982), 219-230. MR 84a:47056
  • 8. R.V.Kadison and I.M.Singer, Triangular operator algebras, Amer. J.Math. 82(1960), 227-259. MR 22:12409
  • 9. E.C.Lance, Cohomology and perturbations of nest algebras, Proc. London Math. Soc. (3)43(1981), 334-356. MR 83b:47053
  • 10. F.Y.Lu, Some questions on nest algebras and maximal triangular algebras, Ph.D.thesis, Zhejiang University (Chinese), 1997.
  • 11. J.L.Orr, On the closure of triangular algebras, Amer.J.Math. 112(1990), 481-497. MR 92c:47053
  • 12. Y.T.Poon, Maximal triangular subalgebras need not be closed, Proc.Amer. Math. Soc. (2)111(1991), 475-479. MR 91f:46082
  • 13. H.Radjavi and P.Rosenthal, Invariant subspaces, Springer-Verlag, Berlin, Heidelberg, New York, 1973. MR 51:3924
  • 14. J.R.Ringrose, On some algebras of operators, Proc.London Math.Soc. (3) 15 (1965), 61-83. MR 30:1405
  • 15. J.R.Ringrose, On some algebras of operators II, Proc.London Math. Soc. (3)16(1966), 385-402. MR 33:4703
  • 16. P.Rosenthal, Weakly closed maximal triangular algebras are hyperreducible, Proc. Amer. Math.Soc. 24(1970), 220. MR 40:1784

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47L75

Retrieve articles in all journals with MSC (2000): 47L75


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

American Mathematical Society