# Article

 Title: Extended Weyl type theorems (English) Author: Berkani, M. Author: Zariouh, H. Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 134 Issue: 4 Year: 2009 Pages: 369-378 Summary lang: English . Category: math . Summary: An operator $T$ acting on a Banach space $X$ possesses property $({\rm gw})$ if $\sigma _a(T)\setminus \sigma _{{\rm SBF}_+^-}(T)= E(T),$ where $\sigma _a(T)$ is the approximate point spectrum of $T$, $\sigma _{{\rm SBF} _+^-}(T)$ is the essential semi-B-Fredholm spectrum of $T$ and $E(T)$ is the set of all isolated eigenvalues of $T.$ In this paper we introduce and study two new properties $({\rm b})$ and $({\rm gb})$ in connection with Weyl type theorems, which are analogous respectively to Browder's theorem and generalized Browder's theorem. \endgraf Among other, we prove that if $T$ is a bounded linear operator acting on a Banach space $X$, then property $({\rm gw})$ holds for $T$ if and only if property $({\rm gb})$ holds for $T$ and $E(T)=\Pi (T),$ where $\Pi (T)$ is the set of all poles of the resolvent of $T.$ (English) Keyword: B-Fredholm operator Keyword: Browder's theorem Keyword: generalized Browder's theorem Keyword: property $({\rm b})$ Keyword: property $({\rm gb})$ MSC: 47A10 MSC: 47A11 MSC: 47A53 idZBL: Zbl 1211.47011 idMR: MR2597232 . Date available: 2010-07-20T18:10:24Z Last updated: 2013-09-20 Stable URL: http://hdl.handle.net/10338.dmlcz/140669 . Reference: [1] Amouch, M., Berkani, M.: On the property $({\rm gw})$.Mediterr. J. Math. 5 (2008), 371-378. MR 2465582, 10.1007/s00009-008-0156-z Reference: [2] Amouch, M., Zguitti, H.: On the equivalence of Browder's and generalized Browder's theorem.Glasgow Math. J. 48 (2006), 179-185. Zbl 1097.47012, MR 2224938, 10.1017/S0017089505002971 Reference: [3] Aiena, P., P. Peña: Variation on Weyl's theorem.J. Math. Anal. Appl. 324 (2006), 566-579. MR 2262492, 10.1016/j.jmaa.2005.11.027 Reference: [4] Barnes, B. A.: Riesz points and Weyl's theorem.Integral Equations Oper. Theory 34 (1999), 187-196. Zbl 0948.47002, MR 1694707, 10.1007/BF01236471 Reference: [5] Berkani, M.: B-Weyl spectrum and poles of the resolvent.J. Math. Anal. Applications 272 (2002), 596-603. Zbl 1043.47004, MR 1930862, 10.1016/S0022-247X(02)00179-8 Reference: [6] Berkani, M.: On the equivalence of Weyl theorem and generalized Weyl theorem.Acta Mathematica Sinica, English series 23 (2007), 103-110. Zbl 1116.47015, MR 2275483, 10.1007/s10114-005-0720-4 Reference: [7] Berkani, M.: Index of B-Fredholm operators and generalization of a Weyl theorem.Proc. Amer. Math. Soc. 130 (2002), 1717-1723. Zbl 0996.47015, MR 1887019, 10.1090/S0002-9939-01-06291-8 Reference: [8] Berkani, M., Koliha, J. J.: Weyl type theorems for bounded linear operators.Acta Sci. Math. (Szeged) 69 (2003), 359-376. Zbl 1050.47014, MR 1991673 Reference: [9] Berkani, M., Sarih, M.: On semi B-Fredholm operators.Glasgow Math. J. 43 (2001), 457-465. Zbl 0995.47008, MR 1878588, 10.1017/S0017089501030075 Reference: [10] Coburn, L. A.: Weyl's theorem for nonnormal operators.Michigan Math. J. 13 (1966), 285-288. Zbl 0173.42904, MR 0201969, 10.1307/mmj/1031732778 Reference: [11] Djordjević, S. V., Han, Y. M.: Browder's theorems and spectral continuity.Glasgow Math. J. 42 (2000), 479-486. Zbl 0979.47004, MR 1793814, 10.1017/S0017089500030147 Reference: [12] Heuser, H.: Functionl Analysis.John Wiley, New York (1982). Reference: [13] Radjavi, H., Rosenthal, P.: Invariant Subspaces.Springer, Berlin (1973). Zbl 0269.47003, MR 0367682 Reference: [14] Rakočević, V.: Operators obeying a-Weyl's theorem.Rev. Roumaine Math. Pures Appl. 34 (1989), 915-919. MR 1030982 Reference: [15] Rakočević, V.: On a class of operators.Mat. Vesnik. 37 (1985), 423-426. MR 0836891 Reference: [16] Taylor, A. E.: Theorems on ascent, descent, nullity and defect of linear operators.Math. Ann. 163 (1966), 18-49. Zbl 0138.07602, MR 0190759, 10.1007/BF02052483 .

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