In this paper, we define the generalized Kato spectrum of an operator, and obtain that the generalized Kato spectrum differs from the semi-regular spectrum on at most countably many points. We study the localized version of the single-valued extension property at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ 0 ∈ C in the case that λ 0 I − T admits a generalized Kato decomposition. From this characterization we shall deduce several results on cluster points of some distinguished parts of the spectrum.

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In this paper, we shall study the components of topological uniform descent resolvent set ρ ud (T) for a bounded linear operator T acting on a Banach space. We first show the constancy of certain subspace valued mappings on the components of ρ ud (T). Then using these results and, the equivalences of SVEP at a point λ for T and T * in the case that λI−T has topological uniform descent, we obtain a classification of these components. Finally, we give some applications of the classification. In particular, we give a sufficient condition on an operator T such that its topological uniform descent spectrum σ ud (T) coincides with some distinguished parts of its spectrum σ (T).

In [19], Burgos, Kaidi, Mbekhta and Oudghiri provided an affirmative answer to a question of Kaashoek and Lay and proved that an operator F is power finite rank if and only if σ dsc (T + F ) = σ dsc (T ) for every operator T commuting with F . Later, several authors extended this result to the essential descent spectrum, the left Drazin spectrum and the left essentially Drazin spectrum. In this paper, using the theory of operator with eventual topological uniform descent and the technique used in [19], we generalize this result to various spectra originated from seni-B-Fredholm theory. As immediate consequences, we give affirmative answers to several questions posed by Berkani, Amouch and Zariouh. Besides, we provide a general framework which allows us to derive in a unify way commuting perturbational results of Weyl-Browder type theorems and properties (generalized or not). These commuting perturbational results, in particular, improve many recent results of [11,14,17,18,38] by removing certain extra assumptions. 2010 Mathematics Subject Classification: primary 47A10, 47A11; secondary 47A53, 47A55Fredj generalized this result in [24] to the essential descent spectrum. Fredj, Burgos and Oudghiri extended this result in [25] to the left Drazin spectrum and the left essentially Drazin spectrum. The present paper is concern with commuting power finite rank perturbations of semi-B-Fredholm operators. As seen in Theorem 2.11 (i.e., main result), we generalize the previous results to various spectra originated from semi-B-Fredholm theory. The proof of our main result is mainly dependent upon the theory of operator with eventual topological uniform descent and the technique used in [19].Spectra originated from semi-B-Fredholm theory include, in particular, the upper semi-B-Weyl spectrum σ U SBW (resp. the B-Weyl spectrum σ BW ) which is closely related to generalized a-Weyl's theorem, generalized a-Browder's theorem, property (gw) and property (gb) (resp. generalized Weyl's theorem, generalized Browder's theorem, property (gaw) and property (gab)). Concerning the upper semi-B-Weyl spectrum σ U SBW , Berkani and Amouch posed in [11] the following question: Question 1.1. Let T ∈ B(X) and let N ∈ B(X) be a nilpotent operator commuting with T . Do we always have σ U SBW (T + N) = σ U SBW (T ) ? Similarly, for the B-Weyl spectrum σ BW , Berkani and Zariouh posed in [17] the following question: Question 1.2. Let T ∈ B(X) and let N ∈ B(X) be a nilpotent operator commuting with T . Do we always have σ BW (T + N) = σ BW (T ) ?Recently, Amouch, Zguitti, Berkani and Zariouh have given partial answers in [5,7,11,14] to Question 1.1. As immediate consequences of our main result (see Theorem 2.11), we provide positive answers to Questions 1.1 and 1.2 and some other questions posed by Berkani and Zariouh (see Corollaries 3.1, 3.3 and 3.8). Besides, we provide a general framework which allows us to derive in a unify way commuting perturbational results of Weyl-Browder type theorems and properties (generalized or not). These commuting ...