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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Complex symmetric weighted shifts


Authors: Sen Zhu and Chun Guang Li
Journal: Trans. Amer. Math. Soc. 365 (2013), 511-530
MSC (2010): Primary 47B37, 47A05; Secondary 47A66
Published electronically: July 25, 2012
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Abstract: An operator $ T$ on a complex Hilbert space $ \mathcal {H}$ is said to be complex symmetric if there exists a conjugate-linear, isometric involution $ C:\mathcal {H}\longrightarrow \mathcal {H}$ so that $ CTC=T^*$. In this paper, it is completely determined when a scalar (unilateral or bilateral) weighted shift is complex symmetric. In particular, we give a canonical decomposition of weighted shifts with complex symmetry. Also we characterize those weighted shifts for which complex symmetry is invariant under generalized Aluthge transforms. As an application, we give a negative answer to a question of S. Garcia.


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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, Shanghai 200433, People’s Republic of China
Email: senzhu@163.com

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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05642-X
PII: S 0002-9947(2012)05642-X
Keywords: Complex symmetric operator, weighted shift, nilpotent operator, Aluthge transform, generalized Aluthge transform
Received by editor(s): April 8, 2011
Received by editor(s) in revised form: May 27, 2011
Published electronically: July 25, 2012
Additional Notes: This work was partially supported by NNSF of China (11101177, 11026038, 10971079), China Postdoctoral Science Foundation (2011M500064) and Shanghai Postdoctoral Scientific Program (12R21410500)
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.