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)

 
 

 

Operators with closed numerical ranges in nest algebras


Authors: Youqing Ji and Bin Liang
Journal: Proc. Amer. Math. Soc. 146 (2018), 2563-2575
MSC (2010): Primary 47L35, 47A12; Secondary 47A55
DOI: https://doi.org/10.1090/proc/13948
Published electronically: March 12, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the present paper, we continue our research on numerical ranges of operators. With newly developed techniques, we show that

Let $ \mathcal {N}$ be a nest on a Hilbert space $ \mathcal {H}$ and $ T\in \mathcal {T} (\mathcal {N})$, where $ \mathcal {T} (\mathcal {N})$ denotes the nest algebra associated with $ \mathcal {N}$. Then for given $ \varepsilon >0$, there exists a compact operator $ K$ with $ \Vert K\Vert<\varepsilon $ such that $ T+K \in \mathcal {T} (\mathcal {N})$ and the numerical range of $ T+K$ is closed.

As applications, we show that the statement of the above type holds for the class of Cowen-Douglas operators, the class of nilpotent operators and the class of quasinilpotent operators.


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

  • [1] William Arveson, A short course on spectral theory, Graduate Texts in Mathematics, vol. 209, Springer-Verlag, New York, 2002. MR 1865513
  • [2] Mohamed Barraa and Vladimir Müller, On the essential numerical range, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 285-298. MR 2160367
  • [3] Jean-Christophe Bourin, Compressions and pinchings, J. Operator Theory 50 (2003), no. 2, 211-220. MR 2050126
  • [4] John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
  • [5] 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
  • [6] P. A. Fillmore, J. G. Stampfli, and J. P. Williams, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged) 33 (1972), 179-192. MR 0322534
  • [7] Karl E. Gustafson and Duggirala K. M. Rao, Numerical range, Universitext, Springer-Verlag, New York, 1997. The field of values of linear operators and matrices. MR 1417493
  • [8] Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982. Encyclopedia of Mathematics and its Applications, 17. MR 675952
  • [9] Felix Hausdorff, Der Wertvorrat einer Bilinearform, Math. Z. 3 (1919), no. 1, 314-316 (German). MR 1544350, https://doi.org/10.1007/BF01292610
  • [10] Domingo A. Herrero, Approximation of Hilbert space operators. Vol. 1, 2nd ed., Pitman Research Notes in Mathematics Series, vol. 224, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1088255
  • [11] Domingo A. Herrero, Spectral pictures of operators in the Cowen-Douglas class $ {\mathcal{B}}_n(\Omega)$ and its closure, J. Operator Theory 18 (1987), no. 2, 213-222. MR 915506
  • [12] Domingo A. Herrero, The diagonal entries in the formula ``quasitriangular $ -$ compact $ =$ triangular'' and restrictions of quasitriangularity, Trans. Amer. Math. Soc. 298 (1986), no. 1, 1-42. MR 857432, https://doi.org/10.2307/2000608
  • [13] Y.Q. Ji, B. Liang, On operators with closed numerical ranges, Ann. Funct. Anal., advance publication, 6 December 2017. doi:10.1215/20088752-2017-0051.
  • [14] Chunlan Jiang and Zongyao Wang, Structure of Hilbert space operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR 2221863
  • [15] John S. Lancaster, The boundary of the numerical range, Proc. Amer. Math. Soc. 49 (1975), 393-398. MR 0372644, https://doi.org/10.2307/2040652
  • [16] Otto Toeplitz, Das algebraische Analogon zu einem Satze von Fejér, Math. Z. 2 (1918), no. 1-2, 187-197 (German). MR 1544315, https://doi.org/10.1007/BF01212904
  • [17] Kehe Zhu, Operators in Cowen-Douglas classes, Illinois J. Math. 44 (2000), no. 4, 767-783. MR 1804320

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47L35, 47A12, 47A55

Retrieve articles in all journals with MSC (2010): 47L35, 47A12, 47A55


Additional Information

Youqing Ji
Affiliation: Department of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: jiyq@jlu.edu.cn

Bin Liang
Affiliation: Department of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: liangbinmath@163.com

DOI: https://doi.org/10.1090/proc/13948
Keywords: Nest algebra, numerical range, Cowen-Douglas operator, quasinilpotent operator.
Received by editor(s): August 26, 2017
Received by editor(s) in revised form: September 2, 2017
Published electronically: March 12, 2018
Additional Notes: The first author was supported by National Natural Science Foundation of China (no. 11271150, no. 11531003).
The second author was supported by National Natural Science Foundation of China (no. 11671167).
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society