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WEYL TYPE-THEOREMS FOR DIRECT SUMS

  • Berkani, Mohammed ;
  • Zariouh, Hassan
  • Received : 2011.06.01
  • Published : 2012.09.30

Abstract

The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum $S{\oplus}T$, where S and T are bounded linear operators acting on a Banach space X. Among other results, we prove that if both T and S possesses property ($gb$) and if ${\Pi}(T){\subset}{\sigma}_a(S)$, ${\PI}(S){\subset}{\sigma}_a(T)$, then $S{\oplus}T$ possesses property ($gb$) if and only if ${\sigma}_{SBF^-_+}(S{\oplus}T)={\sigma}_{SBF^-_+}(S){\cup}{\sigma}_{SBF^-_+}(T)$. Moreover, we prove that if T and S both satisfies generalized Browder's theorem, then $S{\oplus}T$ satis es generalized Browder's theorem if and only if ${\sigma}_{BW}(S{\oplus}T)={\sigma}_{BW}(S){\cup}{\sigma}_{BW}(T)$.

Keywords

property (gb);property (b);property (gw);direct sums;essential semi-B-Fredholm spectrum

References

  1. P. Aiena and P. Pena, A variation on Weyl's theorem, J. Math. Anal. Appl. 324 (2006), no. 1, 566-579. https://doi.org/10.1016/j.jmaa.2005.11.027
  2. M. Amouch and M. Berkani, On the property (gw), Mediterr. J. Math. 5 (2008), no. 3, 371-378. https://doi.org/10.1007/s00009-008-0156-z
  3. M. Berkani, On a class of quasi-Fredholm operators, Integr. Equ. Oper. Theory 34 (1999), no. 2, 244-249. https://doi.org/10.1007/BF01236475
  4. M. Berkani, Index of B-Fredholm operators and generalization of a-Weyl's theorem, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1717-1723. https://doi.org/10.1090/S0002-9939-01-06291-8
  5. M. Berkani, N. Castro, and S. V. Djordjevic, Single valued extension property and generalized Weyl's theorem, Math. Bohem. 131 (2006), no. 1, 29-38.
  6. M. Berkani and J. J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 359-376.
  7. M. Berkani and M. Sarih, On semi B-Fredholm operators, Glasg. Math. J. 43 (2001), no. 3, 457-465.
  8. M. Berkani and H. Zariouh, Extended Weyl type theorems, Math. Bohem. 134 (2009), no. 4, 369-378.
  9. S. Clary, Equality of spectra of quasisimilar hyponormal operators, Proc. Amer. Math. Soc. 53 (1975), no. 1, 88-90. https://doi.org/10.1090/S0002-9939-1975-0390824-7
  10. L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285-288. https://doi.org/10.1307/mmj/1031732778
  11. S. V. Djordjevic and Y. M. Han, A note on Weyl's theorem for operator matrices, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2543-2547. https://doi.org/10.1090/S0002-9939-02-06808-9
  12. B. P. Duggal and C. S. Kubrusly, Weyl's theorem for direct sums, Studia Sci. Math. Hungar. 44 (2007), no. 2, 275-290.
  13. A. Gupta and N. Kashyap, Generalized a-Weyl's theorem for direct sums, Mat. Vesnik 62 (2010), no. 4, 265-270.
  14. Y. M. Han and S. V. Djordjevic, a-Weyl's theorem for operator matrices, Proc. Amer. Math. Soc. 130 (2002), no. 3, 715-722. https://doi.org/10.1090/S0002-9939-01-06110-X
  15. R. E. Harte and W. Y. Lee, Another note on Weyl's theorem, Trans. Amer. Math. Soc. 349 (1997), no. 5, 2115-2124. https://doi.org/10.1090/S0002-9947-97-01881-3
  16. H. Heuser, Functional Analysis, John Wiley & Sons Inc, New York, 1982.
  17. K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, Clarendon, Oxford, 2000.
  18. D. C. Lay, Spectral analysis using ascent, descent, nullity and defect, Math. Ann. 184 (1970), 197-214. https://doi.org/10.1007/BF01351564
  19. W. Y. Lee, Weyl spectra of operator matrices, Proc. Amer. Math. Soc. 129 (2001), no. 1, 131-138. https://doi.org/10.1090/S0002-9939-00-05846-9
  20. V. Rakocevic, On a class of operators, Mat. Vesnik 37 (1985), no. 4, 423-426.
  21. V. Rakocevic, Operators obeying a-Weyl's theorem, Rev. Roumaine Math. Pures Appl. 34 (1989), no. 10, 915-919.
  22. S. Roch and B. Silbermann, Continuity of generalized inverses in Banach algebras, Studia Math. 136 (1999), no. 3, 197-227.