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$ (\mathcal{C}_{p}, \alpha)$-hyponormal operators and trace-class self-commutators with trace zero


Author: Vasile Lauric
Journal: Proc. Amer. Math. Soc. 137 (2009), 945-953
MSC (2000): Primary 47B20
DOI: https://doi.org/10.1090/S0002-9939-08-09731-1
Published electronically: October 28, 2008
MathSciNet review: 2457434
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Abstract: We define the class of $ ({\mathcal{C}}_{p}, \alpha)$-hyponormal operators and study the inclusion between such classes under various hypotheses for $ p$ and $ \alpha$, and then obtain some sufficient conditions for the self-commutator of the Aluthge transform $ \tilde T=\vert T\vert^{\frac{1}{2}} U \vert T\vert^{\frac{1}{2}}$ of $ (\mathcal{C}_{p},\alpha)$-hyponormal operators to be in the trace-class and have trace zero.


References [Enhancements On Off] (What's this?)

  • 1. M. Chō, M. Itoh, and S. Ōshiro, Weyl's theorem holds for $ p$-hyponormal operators, Glasgow Math. J. 39 (1997), 217-220. MR 1460636 (98e:47038)
  • 2. R. Curto, P. Muhly and D. Xia, A trace estimate for $ p$-hyponormal operators, Integral Equations and Operator Theory 6 (1983), 507-514. MR 708409 (85b:47029)
  • 3. T. Furuta, $ A\ge B\ge 0$ assures $ (B^{r} A^{p} B^{r})^{1/q}\ge B^{(p+2r)/q}$ for $ r\ge 0, p\ge 0, q\ge 1$ with $ (1+2r)q\ge p+2r$, Proc. Amer. Math. Soc. 101 (1987), 85-88. MR 897075 (89b:47028)
  • 4. D. Hadwin and E. Nordgren, Extensions of the Berger-Shaw theorem, Proc. Amer. Math. Soc. 102 (1988), 517-525. MR 928971 (89e:47026)
  • 5. D. Jocić, Integral representation formula for generalized normal derivations, Proc. Amer. Math. Soc. 127(8) (1999), 2303-2314. MR 1486737 (99j:47026)
  • 6. I. B. Jung, E. Ko and C. Pearcy, Spectral pictures of Aluthge transforms of operators, Integral Equations and Operator Theory 40 (2001), 52-60. MR 1829514 (2002b:47007)
  • 7. V. Lauric and C. M. Pearcy, Trace-class commutators with trace zero, Acta. Sci. Math. (Szeged) 66 (2000), 341-349. MR 1768871 (2001g:47038)
  • 8. J. Ringrose, Compact non-self-adjoint operators, Van Nostrand Reinhold Company, London (1971).
  • 9. R. Schatten, Norm ideals of completely continuous operators, Ergeb. Math. Grenzgeb. 27, Springer-Verlag, Berlin (1960). MR 0119112 (22:9878)
  • 10. J. G. Stampfli, Compact perturbations, normal eigenvalues and a problem of Salinas, J. London Math. Soc. (2) 9 (1974/1975), 165-175. MR 0365196 (51:1449)

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Additional Information

Vasile Lauric
Affiliation: Department of Mathematics, Florida A&M University, Tallahassee, Florida 32307

DOI: https://doi.org/10.1090/S0002-9939-08-09731-1
Keywords: $\alpha $-commutators, trace zero, $(\mathcal {C}_{p}, \alpha )$-hyponormal operators, Weyl spectrum of area zero, Aluthge transform
Received by editor(s): February 7, 2008
Published electronically: October 28, 2008
Dedicated: This paper is dedicated to the memory of my grandparents.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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