Small Essential Spectral Radius Perturbations of Operators With Topological Uniform Descent

Zeng, Qingping; Zhong, Huaijie; Wu, Zhenying

doi:10.1017/s0004972711002632

Cited by 4 publications

(5 citation statements)

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“…Especially, operators which have topological uniform descent for n ≥ 0 are precisely the semi-regular operators studied by Mbekhta in [23]. Discussions of operators with eventual topological uniform descent may be found in [6,11,18,20,28]. Lemma 2.10.…”

Section: Resultsmentioning

confidence: 99%

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A note on property (gb) and perturbations

Zeng,

Zhong

2012

Preprint

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An operator T ∈ B(X) defined on a Banach space X satisfies property (gb) if the complement in the approximate point spectrum σ a (T ) of the upper semi-B-Weyl spectrum σ SBF − + (T ) coincides with the set Π(T ) of all poles of the resolvent of T . In this note we continue to study property (gb) and the stability of it, for a bounded linear operator T acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators commuting with T . Two counterexamples show that property (gb) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.

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Section: Resultsmentioning

confidence: 99%

“…Since T is a-polaroid, λ ∈ Π a (T ) = Π(T ). Thus by [28,Theorem 3.12], λ ∈ Π(T +Q). Therefore Π a (T +Q) ⊆ Π(T +Q), and this completes the proof.…”

Section: Resultsmentioning

confidence: 99%

A note on property (gb) and perturbations

Zeng,

Zhong

2012

Preprint

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“…Especially, operators which have topological uniform descent for n ≥ 0 are precisely the semi-regular operators studied by Mbekhta in [32]. Discussions of operators with eventual topological uniform descent may be found in [12,20,27,28,29,40].…”

Section: Resultsmentioning

confidence: 99%

“…for sufficiently large m and n. Proof. Parts (c) and (d) follow directly from [40, Theorems 3.8 and 3.12 and Remark 4.5].When d = 0 (that is, T is not semi-regular), parts (a) and (b) follow also directly from [40, Theorems 3.8 and 3.12 and Remark 4.5].When d = 0 (that is, T is semi-regular), then by[40, Theorems 3.8] we have that V = T + S is essentially semi-regular. So, there exist p ∈ N such that V has topological uniform descent for n ≥ p. If p = 0 (that is, V is not semi-regular), then parts(a) and (b) follow directly from [40, Theorem 3.12 and Remark 4.5].…”

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confidence: 89%

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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“…One of the most important results for operators with eventual topological uniform descent is Grabiner's punctured neighbourhood theorem [12,Theorem 4.7]. Discussions of operators with eventual topological uniform descent may also be found in [6,9,12,17,18,26].…”

Section: Property (Abw) and Weyl Type Theoremsmentioning

confidence: 99%

Property $(aBw)$ and perturbations

Zeng¹

2015

Ann. Funct. Anal.

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A bounded linear operator T acting on a Banach space X satisfies property (aBw), a strong version of a-Weyl's theorem, if the complement in the approximate point spectrum σ a (T ) of the upper semi-B-Weyl spectrum σ U SBW (T ) is the set of all isolated points of approximate point spectrum which are eigenvalues of finite multiplicity. In this paper we investigate the property (aBw) in connection with Weyl type theorems. In particular, we show that T satisfies property (aBw) if and only if T satisfies a-Weyl's theorem and σ U SBW (T ) = σ U SW (T ), where σ U SW (T ) is the upper semi-Weyl spectrum of T . The preservation of property (aBw) is also studied under commuting nilpotent, quasi-nilpotent, power finite rank or Riesz perturbations. The theoretical results are illustrated by some concrete examples.

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Small Essential Spectral Radius Perturbations of Operators With Topological Uniform Descent

Cited by 4 publications

References 17 publications

A note on property (gb) and perturbations

A note on property (gb) and perturbations

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

Property $(aBw)$ and perturbations

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