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A $ C^*$-algebra approach to complex symmetric operators


Authors: Kunyu Guo, Youqing Ji and Sen Zhu
Journal: Trans. Amer. Math. Soc. 367 (2015), 6903-6942
MSC (2010): Primary 47C10, 47A58; Secondary 47B37, 47A45
DOI: https://doi.org/10.1090/S0002-9947-2015-06215-1
Published electronically: February 26, 2015
MathSciNet review: 3378818
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Abstract: In this paper, certain connections between complex symmetric operators and anti-automorphisms of singly generated $ C^*$-algebras are established. This provides a $ C^*$-algebra approach to the norm closure problem for complex symmetric operators. For $ T\in \mathcal {B(H)}$ satisfying $ C^*(T)\cap \mathcal {K(H)}=\{0\}$, we give several characterizations for $ T$ to be a norm limit of complex symmetric operators. As applications, we give concrete characterizations for weighted shifts with nonzero weights to be norm limits of complex symmetric operators. In particular, we prove a conjecture of Garcia and Poore. On the other hand, it is proved that an essentially normal operator is a norm limit of complex symmetric operators if and only if it is complex symmetric. We obtain a canonical decomposition for essentially normal operators which are complex symmetric.


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Additional Information

Kunyu Guo
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
Email: kyguo@fudan.edu.cn

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

Sen Zhu
Affiliation: Department of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: senzhu@163.com

DOI: https://doi.org/10.1090/S0002-9947-2015-06215-1
Keywords: Complex symmetric operators, approximation, $C^*$-algebras, anti-automorphisms, weighted shifts, essentially normal operators
Received by editor(s): April 14, 2013
Received by editor(s) in revised form: June 7, 2013
Published electronically: February 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.