Finite rank operators in closed maximal triangular algebras II
HTML articles powered by AMS MathViewer
- 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.References
- William B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967), 635–647. MR 221293
- Erik Christensen, Derivations of nest algebras, Math. Ann. 229 (1977), no. 2, 155–161. MR 448110, DOI 10.1007/BF01351601
- Kenneth R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Triangular forms for operator algebras on Hilbert space. MR 972978
- Zhe Dong and Shi-jie Lu, Finite rank operators in closed maximal triangular algebras, Northeast. Math. J. 16 (2000), no. 2, 167–172. MR 1805700
- J. A. Erdos, Some results on triangular operator algebras, Amer. J. Math. 89 (1967), 85–93. MR 209905, DOI 10.2307/2373098
- J. A. Erdos, Operators of finite rank in nest algebras, J. London Math. Soc. 43 (1968), 391–397. MR 230156, DOI 10.1112/jlms/s1-43.1.391
- J. A. Erdos and S. C. Power, Weakly closed ideals of nest algebras, J. Operator Theory 7 (1982), no. 2, 219–235. MR 658610
- Richard V. Kadison and I. M. Singer, Triangular operator algebras. Fundamentals and hyperreducible theory, Amer. J. Math. 82 (1960), 227–259. MR 121675, DOI 10.2307/2372733
- E. Christopher Lance, Cohomology and perturbations of nest algebras, Proc. London Math. Soc. (3) 43 (1981), no. 2, 334–356. MR 628281, DOI 10.1112/plms/s3-43.2.334
- F.Y.Lu, Some questions on nest algebras and maximal triangular algebras, Ph.D.thesis, Zhejiang University (Chinese), 1997.
- John Lindsay Orr, On the closure of triangular algebras, Amer. J. Math. 112 (1990), no. 3, 481–497. MR 1055655, DOI 10.2307/2374753
- Yiu Tung Poon, Maximal triangular subalgebras need not be closed, Proc. Amer. Math. Soc. 111 (1991), no. 2, 475–479. MR 1034888, DOI 10.1090/S0002-9939-1991-1034888-5
- Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6
- J. R. Ringrose, On some algebras of operators, Proc. London Math. Soc. (3) 15 (1965), 61–83. MR 171174, DOI 10.1112/plms/s3-15.1.61
- J. R. Ringrose, On some algebras of operators. II, Proc. London Math. Soc. (3) 16 (1966), 385–402. MR 196516, DOI 10.1112/plms/s3-16.1.385
- Peter Rosenthal, Weakly closed maximal triangular algebras are hyperreducible, Proc. Amer. Math. Soc. 24 (1970), 220. MR 248532, DOI 10.1090/S0002-9939-1970-0248532-4
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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1515-1525
- MSC (2000): Primary 47L75
- DOI: https://doi.org/10.1090/S0002-9939-02-06748-5
- MathSciNet review: 1949882