Spectra originating from semi-B-Fredholm theory and commuting perturbations

Abstract

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 ...

PROPERTY (Bgw) AND PERTURBATIONS

“…(i) This follows from Theorem 2.7 and [6, Theorem 2.6]. (ii) We conclude from[28, Theorem 2.8] that σ SBF − + (T + K) = σ SBF − + (T ). By assumption we have σ a (T + K) = σ a (T ), so…”

“…In this section we are interested to study the stability of property (Bgw) under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators and by Riesz operators commuting with T . We begin with this lemma: Lemma 3.1 [28] Let T ∈ B(X) and let N ∈ B(X) be a nilpotent operator commuting with T . Then E 0 (T + N ) = E 0 (T ).…”

Property (gab) through localized SVEP

“…Proof. We have σ d (T ) = σ d (T + K) and σ ld (T ) = σ ld (T + K), see [35,Theorem 2.11], or [13,Theorem 2.8]. By Lemma 3.14 we then obtain Π(T ) = Π(T + K) and Π a (T ) = Π a (T + K).…”

Property $(aBw)$ and perturbations

“…T is upper semi-B-Fredholm and ind(T ) ≤ 0). It is established in [24] that if T ∈ B(X) is B-Weyl (resp. upper semi-B-Weyl) and F ∈ B(X) is an operator satisfying F n ∈ F(X) for some n ∈ N that commutes with T , then T + F also is B-Weyl (resp.…”