Property (gb) and perturbations

Rashid, M. H. M.

doi:10.1016/j.jmaa.2011.05.001

Cited by 9 publications

(14 citation statements)

References 30 publications

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“…As an immediate consequence of Theorem 2.11 (that is, main result), we obtain the following corollary which, in particular, is a corrected version of [37,Theorem 3.15] and also provide positive answers to Questions 1.1 and 1.2. As an immediate consequence of Theorem 2.11, we also obtain the following corollary which, in particular, provide a positive answer to Question 3.2.…”

Section: Some Applicationsmentioning

confidence: 59%

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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 Applicationsmentioning

confidence: 59%

“…Rashid claimed in [37,Theorem 3.15] that if T ∈ B(X) and Q is a quasi-nilpotent operator that commute with T , then (in [37], σ U SBW is denoted as…”

Section: Some Applicationsmentioning

confidence: 99%

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

2013

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“…Recently in [25], property (gb) and perturbations were extensively studied by Rashid. According to [20], an operator T ∈ B(X ) is said to satisfy property (Bw) if ∆ g (T ) = σ(T )\σ BW (T ) = E 0 (T ).…”

Section: Definition 11 ([8])mentioning

confidence: 99%

Property (Sw) for bounded linear operators

Rashid

Prasad²

2015

Asian-European J. Math.

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“…According to [14], an operator T ∈ L (X ) is said to possess property (gb) if ∆ g a (T ) = π(T ), and is said to possess property (b) if ∆ a (T ) = π 0 (T ). It is shown in Theorem 2.3 of [14] that an operator possessing property (gb) possesses property (b) but the converse is not true in general, see also [26]. Following [4], we say an operator T ∈ L (X ) is said to be satisfies property (R) if π 0 a (T ) = E 0 (T ).…”

Section: Introduction and Preliminarymentioning

confidence: 99%

Properties (S) and (gS) for bounded linear operators

Rashid

2014

Filomat

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An operator T acting on a Banach space X obeys property (R) if π 0 a (T ) = E 0 (T ), where π 0 a (T ) is the set of all left poles of T of finite rank and E 0 (T ) is the set of all isolated eigenvalues of T of finite multiplicity. In this paper we introduce and study two new properties (S) and (gS) in connection with Weyl type theorems. Among other things, we prove that if T is a bounded linear operator acting on a Banach space, then T satisfies property (R) if and only if T satisfies property (S) and π 0 (T ) = π 0 a (T ), where π 0 (T ) is the set of poles of finite rank. Also we show if T satisfies Weyl theorem, then T satisfies property (S). Analogous results for property (gS) are given. Moreover, these properties are also studied in the frame of polaroid operator.A bounded linear operator T acting on a Banach space X is Weyl, T ∈ W (X ), if it is Fredholm

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Property (gb) and perturbations

Cited by 9 publications

References 30 publications

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

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

Property (Sw) for bounded linear operators

Properties (S) and (gS) for bounded linear operators

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