# Article

 Title: Single valued extension property and generalized Weyl’s theorem (English) Author: Berkani, M. Author: Castro, N. Author: Djordjević, S. V. Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 131 Issue: 1 Year: 2006 Pages: 29-38 Summary lang: English . Category: math . Summary: Let $T$ be an operator acting on a Banach space $X$, let $\sigma (T)$ and $\sigma _{BW}(T)$ be respectively the spectrum and the B-Weyl spectrum of $T$. We say that $T$ satisfies the generalized Weyl’s theorem if $\sigma _{BW}(T)= \sigma (T) \setminus E(T)$, where $E(T)$ is the set of all isolated eigenvalues of $T$. The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$, extending an earlier result of J. K. Finch and a recent result of Aiena and Monsalve. Our second goal is to give necessary and sufficient conditions under which an operator having the single valued extension property satisfies the generalized Weyl’s theorem. (English) Keyword: single valued extension property Keyword: B-Weyl spectrum Keyword: generalized Weyl’s theorem MSC: 47A10 MSC: 47A53 MSC: 47A55 idZBL: Zbl 1114.47015 idMR: MR2211001 . Date available: 2009-09-24T22:23:51Z Last updated: 2012-06-18 Stable URL: http://hdl.handle.net/10338.dmlcz/134080 . Reference: [1] Aiena, P., Monsalve, O.: Operators which do not have single valued extension property.J. Math. Anal. Appl. 250 (2000), 435–448. MR 1786074 Reference: [2] Berkani, M.: On a class of quasi-Fredholm operators.Int. Equ. Oper. Theory 34 (1999), 244–249. Zbl 0939.47010, MR 1694711 Reference: [3] Berkani, M.: Restriction of an operator to the range of its powers.Studia Math. 140 (2000), 163–175. Zbl 0978.47011, MR 1784630 Reference: [4] Berkani, M.: Index of B-Fredholm operators and generalization of a Weyl’s Theorem.Proc. Amer. Math. Soc. 130 (2002), 1717–1723. MR 1887019 Reference: [5] Berkani, M., Sarih, M.: An Atkinson type theorem for B-Fredholm operators.Studia Math. 148 (2001), 251–257. MR 1880725 Reference: [6] Berkani, M., Koliha, J. J.: Weyl type theorems for bounded linear operators.Acta Sci. Math. (Szeged) 69 (2003), 359–376. MR 1991673 Reference: [7] Berkani, M.: B-Weyl spectrum and poles of the resolvent.J. Math. Anal. Appl. 272 (2002), 596–603. Zbl 1043.47004, MR 1930862 Reference: [8] Finch, J. K.: The single valued extension property on a Banach space.Pac. J. Math. 58 (1975), 61–69. Zbl 0315.47002, MR 0374985 Reference: [9] Grabiner, S.: Uniform ascent and descent of bounded operators.J. Math. Soc. Japan 34 (1982), 317–337. Zbl 0477.47013, MR 0651274 Reference: [10] Jeon, I. H.: Weyl’s theorem for operators with a growth condition and Dunford’s property $(C)$.Indian J. Pure Appl. Math. 33 (2002), 403–407. MR 1894635 Reference: [11] Kordula, V., Müller, V.: On the axiomatic theory of the spectrum.Stud. Math. 119 (1996), 109–128. MR 1391471 Reference: [12] Lay, D. C.: Spectral analysis using ascent, descent, nullity and defect.Math. Ann. 184 (1970), 197–214. Zbl 0177.17102, MR 0259644 Reference: [13] Mbekhta, M., Müller V.: On the axiomatic theory of the spectrum, II.Stud. Math. 119 (1996), 129–147. MR 1391472 Reference: [14] Roch, S., Silbermann, B.: Continuity of generalized inverses in Banach algebras.Stud. Math. 136 (1999), 197–227. MR 1724245 Reference: [15] Schmoeger, C.: On isolated points of the spectrum of a bounded linear operator.Proc. Am. Math. Soc. 117 (1993), 715–719. Zbl 0780.47019, MR 1111438 Reference: [16] Weyl, H.: Über beschränkte quadratische Formen, deren Differenz vollstetig ist.Rend. Circ. Mat. Palermo 27 (1909), 373–392. .

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