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THE RIESZ DECOMPOSITION THEOREM FOR SKEW SYMMETRIC OPERATORS

  • Zhu, Sen (Department of Mathematics Jilin University) ;
  • Zhao, Jiayin (Department of Mathematics Jilin University)
  • Received : 2014.06.01
  • Published : 2015.03.01

Abstract

An operator T on a complex Hilbert space $\mathcal{H}$ is called skew symmetric if T can be represented as a skew symmetric matrix relative to some orthonormal basis for $\mathcal{H}$. In this note, we explore the structure of skew symmetric operators with disconnected spectra. Using the classical Riesz decomposition theorem, we give a decomposition of certain skew symmetric operators with disconnected spectra. Several corollaries and illustrating examples are provided.

Keywords

References

  1. N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publications, Inc., New York, 1993.
  2. J. L. Boersema, The range of united K-theory, J. Funct. Anal. 235 (2006), no. 2, 701- 718.
  3. N. Chevrot, E. Fricain, and D. Timotin, The characteristic function of a complex sym- metric contraction, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2877-2886. https://doi.org/10.1090/S0002-9939-07-08803-X
  4. S. R. Garcia and D. E. Poore, On the closure of the complex symmetric operators: compact operators and weighted shifts. , J. Funct. Anal. 264 (2013), no. 3, 691-712. https://doi.org/10.1016/j.jfa.2012.11.009
  5. S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285-1315. https://doi.org/10.1090/S0002-9947-05-03742-6
  6. S. R. Garcia and M. Putinar, Complex symmetric operators and applications. II, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3913-3931. https://doi.org/10.1090/S0002-9947-07-04213-4
  7. S. R. Garcia and W. Ross, Recent progress on truncated Toeplitz operators, Blaschke products and their applications, 275-319, Fields Inst. Commun., 65, Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-5341-3_15
  8. S. R. Garcia and W. R. Wogen, Complex symmetric partial isometries, J. Funct. Anal. 257 (2009), no. 4, 1251-1260. https://doi.org/10.1016/j.jfa.2009.04.005
  9. 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. https://doi.org/10.1007/s00020-007-1528-7
  10. K. Guo, Y. Ji, and S. Zhu, A C∗-algebra approach to complex symmetric operators, Trans. Amer. Math. Soc. (to appear).
  11. K. Guo and S. Zhu, A canonical decomposition of complex symmetric operators, J. Operator Theory (to appear).
  12. D. A. Herrero, Approximation of Hilbert Space Operators. Vol. 1, second ed., Pitman Research Notes in Mathematics Series, vol. 224, Longman Scientific & Technical, Harlow, 1989.
  13. C. G. Li and T. T. Zhou, Skew symmetry of a class of operators, Banach J. Math. Anal. 8 (2014), no. 1, 279-294. https://doi.org/10.15352/bjma/1381782100
  14. C. G. Li and S. Zhu, Skew symmetric normal operators, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2755-2762. https://doi.org/10.1090/S0002-9939-2013-11759-4
  15. N. C. Phillips and M. G. Viola, A simple separable exact $C^∗$-algebra not anti-isomorphic to itself, Math. Ann. 355 (2013), no. 2, 783-799. https://doi.org/10.1007/s00208-011-0755-z
  16. H. Radjavi and P. Rosenthal, Invariant Subspaces, second ed., Dover Publications Inc., Mineola, NY, 2003.
  17. P. J. Stacey, Antisymmetries of the CAR algebra, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6439-6452, With an appendix by J. L. Boersema and N. C. Phillips. https://doi.org/10.1090/S0002-9947-2011-05263-3
  18. S. M. Zagorodnyuk, On a J-polar decomposition of a bounded operator and matrices of J-symmetric and J-skew-symmetric operators, Banach J. Math. Anal. 4 (2010), no. 2, 11-36. https://doi.org/10.15352/bjma/1297117238
  19. S. M. Zagorodnyuk, On the complex symmetric and skew-symmetric operators with a simple spectrum, Symmetry, Integrability and Geometry: Methods and Applications 7 (2011), 1-9.
  20. S. Zhu, Approximate unitary equivalence to skew symmetric operators, Complex Anal. Oper. Theory, 2014, doi: 10.1007/s11785-014-0369-z.
  21. S. Zhu, Skew symmetric weighted shifts, Banach J. Math. Anal. (to appear).
  22. S. Zhu and C. G. Li, Complex symmetry of a dense class of operators, Integr. Equat. Oper. Th. 73 (2012), no. 2, 255-272. https://doi.org/10.1007/s00020-012-1957-9
  23. S. Zhu and C. G. Li, Complex symmetric weighted shifts, Trans. Amer. Math. Soc. 365 (2013), no.1, 511-530. https://doi.org/10.1090/S0002-9947-2012-05642-X

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  1. ON SKEW SYMMETRIC OPERATORS WITH EIGENVALUES vol.52, pp.6, 2015, https://doi.org/10.4134/JKMS.2015.52.6.1271