B-Browder spectra and localized SVEP

Carpintero, Carlos; García, Orlando; Rosas, Ennis; Sanabria, José

doi:10.1007/s12215-008-0017-4

Cited by 8 publications

(6 citation statements)

References 13 publications

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“…From this equality, and by hypothesis, if λ = 0 we have that 0 < p(T n ) = q(T n ) < ∞. By Lemmas 2 and 3 in [10] and [15,Proposition 38…”

Section: Spectral Properties and Restrictionsmentioning

confidence: 84%

“…From this equality, and by hypothesis, λ = 0 implies that 0 < p(T n ) = q(T n ) < ∞. By Lemmas 2 and 3 in [10] and [15,Proposition 38.6], 0 < p(T ) = q(T ) < ∞, a contradiction. Now, being λ = 0, by Lemma 1.4…”

Section: Spectral Properties and Restrictionsmentioning

confidence: 85%

“…If R(T n ) is closed, we have: ) is closed for all k ≥ 0. Also, by Lemma 2 in [10], p(T ) < ∞ because p(T n ) = 0. Thus, if 0 ∈ σ ap (T ) then 0 is a left pole of the resolvent of T , a contradiction.…”

Section: Relations Between the Spectra Of T And T Nmentioning

confidence: 93%

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Spectral properties and restrictions of bounded linear operators

Carpintero¹,

Rosas²,

Rodríguez³

et al. 2015

Ann. Funct. Anal.

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“…From this equality, and by hypothesis, if λ = 0 we have that 0 < p(T n ) = q(T n ) < ∞. By Lemmas 2 and 3 in [10] and [15,Proposition 38…”

Section: Spectral Properties and Restrictionsmentioning

confidence: 84%

Section: Spectral Properties and Restrictionsmentioning

confidence: 85%

See 1 more Smart Citation

Spectral properties and restrictions of bounded linear operators

Carpintero¹,

Rosas²,

Rodríguez³

et al. 2015

Ann. Funct. Anal.

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“…For T ∈ B(X), let us define the lef t Drazin spectrum, the right Drazin spectrum, the Drazin spectrum, the lef t essentially Drazin spectrum, and the right essentially Drazin spectrum of T as follows respectively: These spectra have been extensively studied by several authors, see e.g [2,7,8,9,22,24,25,33].…”

Section: Introductionmentioning

confidence: 99%

“…LD (T ) = {λ ∈ C : T − λI is not a left essentially Drazin invertible operator}; σ e RD (T ) = {λ ∈ C : T − λI is not a right essentially Drazin invertible operator}. These spectra have been extensively studied by several authors, see e.g [2,7,8,9,22,24,25,33].…”

Section: Introductionmentioning

confidence: 99%

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

2013

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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 ...

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Weyl Type Theorems and Restrictions

et al. 2013

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Berkani and Kachad [18], [19], and Sanabria et al. [32], introduced and studied strong variations of Weyl type Theorems. In this paper, we study the behavior of these strong variations of Weyl type theorems for an operator T on a proper closed and Tinvariant subspace W ⊆ X such that T n (X) ⊆ W for some n ≥ 1, where T ∈ L(X) and X is an infinite-dimensional complex Banach space. The main purpose of this paper is to prove that for these subspaces (which generalize the case T n (X) closed for some n ≥ 0), these strong variations of Weyl type theorems are preserved from T to its restriction on W and vice-versa. As consequence of our results, we give sufficient conditions for which these strong variations of Weyl type Theorems are equivalent for two given operators. Also, some applications to multiplication operators acting on the boundary variation space BV [0, 1] are given. Keywordsnew Weyl-type theorems • strong variations of Weyl type theorems • restrictions of operators • spectral properties • multiplication operators Mathematics Subject Classification (2010) 47A10 • 47A11 • 47A53 • 47A55

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B-Browder spectra and localized SVEP

Cited by 8 publications

References 13 publications

Spectral properties and restrictions of bounded linear operators

Spectral properties and restrictions of bounded linear operators

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

Weyl Type Theorems and Restrictions

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