Weyl spectra and Weyl's theorem

Cao, Xiaohong; Guo, Maozheng; Meng, Bo

doi:10.1016/j.jmaa.2003.09.026

Cited by 14 publications

(3 citation statements)

References 17 publications

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“…The commuting perturbational results established in Corollary 3.9, in particular, improve many recent results of [11,14,17,18,38] For a-Weyl's theorem, some other commuting perturbational theorems for it have been proved in [18,21,35].…”

Section: Some Applicationssupporting

confidence: 74%

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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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Section: Some Applicationssupporting

confidence: 74%

“…For a-Weyl's theorem, some other commuting perturbational theorems for it have been proved in [18,21,35].…”

Section: Some Applicationsmentioning

confidence: 99%

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

2013

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“…The study on Weyl type theorems can well reflect the structural characteristics of spectrums. Hence the research of Weyl type theorem has attracted much attention and got many good results in resent years [5][6][7] . In this paper, we mainly study a-Browder􀆳s theorem and a-Weyl􀆳s theorem for bounded linear operators and operator functions by means of the property of the topological uniform descent.…”

Section: Introductionmentioning

confidence: 99%

Topological Uniform Descent and Judgement of A-Weyl's Theorem

SUN,

WANG,

CAO

2023

Wuhan Univ. J. Nat. Sci.

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In this paper, a-Browder's theorem and a-Weyl's theorem for bounded linear operators are studied by means of the property of the topological uniform descent. The sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space holding a-Browder's theorem and a-Weyl's theorem are established. As a consequence of the main result, the new judgements of a-Browder's theorem and a-Weyl's theorem for operator function are discussed.

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Weyl Type Theorems Under Compact Perturbations

Jia

Feng

2017

Mediterr. J. Math.

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Weyl spectra and Weyl's theorem

Cited by 14 publications

References 17 publications

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

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

Topological Uniform Descent and Judgement of A-Weyl's Theorem

Weyl Type Theorems Under Compact Perturbations

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