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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
DOI: https://doi.org/10.1090/S0002-9947-2012-05642-X
Published electronically: July 25, 2012
MathSciNet review: 2984066
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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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  • 1. A. Aluthge, On $ p$-hyponormal operators for $ 0<p<1$, Integral Equations Operator Theory 13 (1990), no. 3, 307-315. MR 1047771 (91a:47025)
  • 2. A. Aluthge, Some generalized theorems on $ p$-hyponormal operators, Integral Equations Operator Theory 24 (1996), no. 4, 497-501. MR 1382022 (97a:47032)
  • 3. L. Balayan and S. R. Garcia, Unitary equivalence to a complex symmetric matrix: geometric criteria, Oper. Matrices 4 (2010), no. 1, 53-76. MR 2655004 (2011f:15068)
  • 4. J. A. Cima, S. R. Garcia, W. T. Ross, and Warren R. Wogen, Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity, Indiana Univ. Math. J. 59 (2010), no. 2, 595-620. MR 2648079
  • 5. S. R. Garcia, Aluthge transforms of complex symmetric operators, Integral Equations Operator Theory 60 (2008), no. 3, 357-367. MR 2392831 (2008m:47052)
  • 6. S. R. Garcia, Open questions about complex symmetric operators, Tech. report, Southeastern Analysis Meeting (SEAM 25), University of South Florida, Mar. 2009.
  • 7. S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285-1315 (electronic). MR 2187654 (2006j:47036)
  • 8. S. R. Garcia and M. Putinar, Complex symmetric operators and applications. II, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3913-3931 (electronic). MR 2302518 (2008b:47005)
  • 9. S. R. Garcia and M. Putinar, Interpolation and complex symmetry, Tohoku Math. J. (2)60 (2008), no. 3, 423-440. MR 2453732 (2009k:47048)
  • 10. S. R. Garcia and W. R. Wogen, Complex symmetric partial isometries, J. Funct. Anal. 257 (2009), no. 4, 1251-1260. MR 2535469 (2011g:47005)
  • 11. S. R. Garcia and W. R. Wogen, Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), no. 11, 6065-6077. MR 2661508 (2011g:47086)
  • 12. T. M. Gilbreath and W. R. Wogen, Remarks on the structure of complex symmetric operators, Integral Equations Operator Theory 59 (2007), no. 4, 585-590. MR 2370050 (2009f:47037)
  • 13. J. Guyker, A structure theorem for operators with closed range, Bull. Austral. Math. Soc. 18 (1978), no. 2, 169-186. MR 499972 (80b:47020)
  • 14. P. R. Halmos and L. J. Wallen, Powers of partial isometries, J. Math. Mech. 19 (1969/1970), 657-663. MR 0251574 (40:4801)
  • 15. D. A. Herrero, Approximation of Hilbert space operators. Vol. 1, second edition, Pitman Research Notes in Mathematics Series, vol. 224, Longman Scientific & Technical, Harlow, 1989. MR 1088255 (91k:47002)
  • 16. A. L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I. 1974, pp. 49-128. MR 0361899 (50:14341)
  • 17. X. H. Wang and Z. S. Gao, A note on Aluthge transforms of complex symmetric operators and applications, Integral Equations Operator Theory 65 (2009), no. 4, 573-580. MR 2576310 (2011c:47003)
  • 18. S. Zhu, C. G. Li, and Y. Q. Ji, The class of complex symmetric operators is not norm closed, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1705-1708. MR 2869154

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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: https://doi.org/10.1090/S0002-9947-2012-05642-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.

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