WEYL TYPE-THEOREMS FOR DIRECT SUMS

Title & Authors
WEYL TYPE-THEOREMS FOR DIRECT SUMS
Berkani, Mohammed; Zariouh, Hassan;

Abstract
The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum $\small{S{\oplus}T}$, 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 ($\small{gb}$) and if $\small{{\Pi}(T){\subset}{\sigma}_a(S)}$, $\small{{\PI}(S){\subset}{\sigma}_a(T)}$, then $\small{S{\oplus}T}$ possesses property ($\small{gb}$) if and only if \${\sigma}_{SBF^-_+}(S{\oplus}T)
Keywords
property (gb);property (b);property (gw);direct sums;essential semi-B-Fredholm spectrum;
Language
English
Cited by
References
1.
P. Aiena and P. Pena, A variation on Weyl's theorem, J. Math. Anal. Appl. 324 (2006), no. 1, 566-579.

2.
M. Amouch and M. Berkani, On the property (gw), Mediterr. J. Math. 5 (2008), no. 3, 371-378.

3.
M. Berkani, On a class of quasi-Fredholm operators, Integr. Equ. Oper. Theory 34 (1999), no. 2, 244-249.

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.

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.

10.
L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285-288.

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.

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.

15.
R. E. Harte and W. Y. Lee, Another note on Weyl's theorem, Trans. Amer. Math. Soc. 349 (1997), no. 5, 2115-2124.

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.

19.
W. Y. Lee, Weyl spectra of operator matrices, Proc. Amer. Math. Soc. 129 (2001), no. 1, 131-138.

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.