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

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- by Kunyu Guo, Youqing Ji and Sen Zhu PDF
- Trans. Amer. Math. Soc.
**367**(2015), 6903-6942 Request permission

## 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.## References

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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
- Received by editor(s): April 14, 2013
- Received by editor(s) in revised form: June 7, 2013
- Published electronically: February 26, 2015
- © Copyright 2015
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3378818