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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The class of complex symmetric operators is not norm closed


Authors: Sen Zhu, Chun Guang Li and You Qing Ji
Journal: Proc. Amer. Math. Soc. 140 (2012), 1705-1708
MSC (2010): Primary 47A05; Secondary 47B99
Published electronically: September 15, 2011
MathSciNet review: 2869154
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Abstract: An operator $ T\in\mathcal{B(H)}$ is complex symmetric if there exists a conjugate-linear, isometric involution $ C:\mathcal{H}\longrightarrow\mathcal{H}$ so that $ CTC=T^*$. In this paper, a class of complex symmetric operators on finite dimensional Hilbert spaces is constructed. As an application, it is shown that Kakutani's unilateral weighted shift operator is not complex symmetric; however, it is a norm limit of complex symmetric operators. This gives a negative answer to a question of S. Garcia and W. Wogen: that is, whether or not the class of complex symmetric operators is norm closed.


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Additional Information

Sen Zhu
Affiliation: Department of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Address at time of publication: School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, People’s Republic of China
Email: zhusen@jlu.edu.cn

Chun Guang Li
Affiliation: Institute of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: licg09@mails.jlu.edu.cn

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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11345-5
PII: S 0002-9939(2011)11345-5
Keywords: Complex symmetric operators, Kakutani’s shift
Received by editor(s): January 20, 2011
Published electronically: September 15, 2011
Additional Notes: This work was supported by NNSF of China (11026038, 10971079, 11101177) and the Basic Research Foundation of Jilin University (201001001, 201103194).
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.