Components of topological uniform descent resolvent set and local spectral theory

Jiang, Qiaofen; Zhong, Huaijie; Zhang, Shifang

doi:10.1016/j.laa.2012.10.004

Cited by 9 publications

(12 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As an application we get that a bounded linear operator T is meromorphic, that is its non-zero spectral points are poles of its resolvent, if and only if σ BΦ (T ) ⊂ {0} and this is exactly when σ T UD (T ) ⊂ {0}. This result was obtained earlier (see [9] and [18]). Q. Jiang, H. Zhong and S. Zhang in [18,Corollary 3.3] proved it by using the local constancy of the mappings λ → K(λI Theorem 2.6] and results about SVEP established in [17], but our method of proof is rather different and more direct.…”

Section: Introductionsupporting

confidence: 66%

Topological Uniform Descent, Quasi-Fredholmness and Operators Originated from Semi-B-Fredholm Theory

Živković-Zlatanović

Berkani

2019

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we get some results regarding boundaries and connected hulls of the corresponding spectra. In particular, we prove that a bounded linear operator T acting on a Banach space, having topological uniform descent, is a BR operator if and only if 0 is not an accumulation point of the associated spectrum σR(T ) = {λ ∈ C : T − λI / ∈ R}, where R denote any of the following classes: upper semi-Weyl operators, Weyl operators, upper semi-Fredholm operators, Fredholm operators, operators with finite (essential) descent and BR the B-regularity associated to R as in [6]. Under the stronger hypothesis of quasi-Fredholmness of T, we obtain a similar characterisation for T being a BR operator for much larger families of sets R.2010 Mathematics subject classification: 47A53, 47A10.

show abstract

Section: Introductionsupporting

confidence: 66%

Topological Uniform Descent, Quasi-Fredholmness and Operators Originated from Semi-B-Fredholm Theory

Živković-Zlatanović

Berkani

2019

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

show abstract

“…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%

“…In [29], Jiang, Zhong and Zhang obtained a classification of the components of eventual topological unif orm descent resolvent set ρ ud (T ) := C\σ ud (T ). As an application of the classification, they show that σ ud (T ) = ∅ precisely when T is algebraic.…”

Section: Resultsmentioning

confidence: 99%

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

2013

Self Cite

View full text Add to dashboard Cite

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 ...

show abstract

“…(ii) ⇔ (iv) Every semi B-Fredholm operator has topological uniform ascent ( [20]), so, by [31,Corollary 2.8],…”

Section: Definementioning

confidence: 99%

Property (gab) through localized SVEP

Aiena

Triolo

2015

Moroccan Journal of Pure and Applied Analysis

View full text Add to dashboard Cite

In this article we study the property (gab) for a bounded linear operator T ∈ L(X)\ud on a Banach space X which is a stronger variant of Browder’s theorem. We shall give several\ud characterizations of property (gab). These characterizations are obtained by using typical tools\ud from local spectral theory. We also show that property (gab) holds for large classes of operators\ud and prove the stability of property (gab) under some commuting perturbations

show abstract

Components of topological uniform descent resolvent set and local spectral theory

Cited by 9 publications

References 15 publications

Topological Uniform Descent, Quasi-Fredholmness and Operators Originated from Semi-B-Fredholm Theory

Topological Uniform Descent, Quasi-Fredholmness and Operators Originated from Semi-B-Fredholm Theory

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

Property (gab) through localized SVEP

Contact Info

Product

Resources

About