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

DOI:
https://doi.org/10.1090/S0002-9939-2011-11345-5

Published electronically:
September 15, 2011

MathSciNet review:
2869154

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Abstract: An operator is complex symmetric if there exists a conjugate-linear, isometric involution so that . 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:
https://doi.org/10.1090/S0002-9939-2011-11345-5

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.