A new Browder type property

Jayanthi, N.; Vasanthakumar, P.

doi:10.12988/ijma.2020.91065

Cited by 3 publications

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References 4 publications

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“…As deduction, we give analogous result to Theorem 3.12 for the properties (U W E ) and (Z Ea ). And, we give in Remark IV a very crucial observation on the results of the paper [7].…”

mentioning

confidence: 88%

“…Furthermore, we specify the mistakes committed in each of them and we give the correct versions. We also give a global note on the paper [7].…”

mentioning

confidence: 99%

“…Remark IV: In the paper [7], the authors defined a property named (Bv) as follows: T ∈ L(X) satisfies property (Bv) if σ(T ) \ σ uw (T ) = σ(T ) \ σ ub (T ). Moreover, they studied this property in connection with different known Weyl type theorems and properties.…”

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confidence: 99%

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The Berkani's property and a note on some recent results

Zakariae¹,

Zariouh²

2020

Preprint

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In this paper, we continue the study of property (U WΠ) introduced in [5], in connection with other Weyl type theorems. Moreover, we give counterexamples to show that some recent results related to this property, which are announced and proved by P. Aiena and M. Kachad in [2] are false. Furthermore, we specify the mistakes committed in each of them and we give the correct versions. We also give a global note on the paper [7]. IntroductionWe continue the study of properties introduced in [4, 5], in connection with other properties and Weyl type theorems. Moreover, we prove by counterexample (see Remark I), that we do not expected that an operator satisfying property (U W Π ) satisfies property (Z Πa ), contrarily to what has been proved in [2, Thorem 2.5]. Furthermore, we give the correct version of [2, Thorem 2.5] by proving in Theorem 3.7, that if T ∈ L(X) is an operator satisfying property (U W Π ) and Π a (T )∩σ uw (T ) = ∅, then it satisfies property (Z Πa ). We also give the correct version of [2, Thorem 2.6] (see Remark II and Theorem 3.12). As deduction, we give analogous result to Theorem 3.12 for the properties (U W E ) and (Z Ea ). And, we give in Remark IV a very crucial observation on the results of the paper [7]. Terminology and preliminariesLet X denote an infinite dimensional complex Banach space, and we denote by L(X) the algebra of all bounded linear operators on X. For T ∈ L(X), we denote by α(T ) the dimension of the kernel N (T ) and by β(T ) the codimension of the range R(T ). By σ(T ) and σ a (T ), we denote the spectrum, the approximate spectrum of T, respectively. For an operator T ∈ L(X), the ascent p(T ) and the descent q(T ) of T are defined by p(T ) = inf{n ∈ N : N (T n ) = N (T n+1 )} and q(T ) = inf{n ∈ N : R(T n ) = R(T n+1 )}, respectively; the infimum over the empty set is taken ∞.In order to simplify, and to give a global view to the reader, we use the same symbols and notations used in [2]. For more details on the several classes and spectra originating from Fredholm theory and B-Fredholm theory, we refer the reader to [2,5,8].

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“…As deduction, we give analogous result to Theorem 3.12 for the properties (U W E ) and (Z Ea ). And, we give in Remark IV a very crucial observation on the results of the paper [7].…”

mentioning

confidence: 88%

“…Furthermore, we specify the mistakes committed in each of them and we give the correct versions. We also give a global note on the paper [7].…”

mentioning

confidence: 99%

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The Berkani's property and a note on some recent results

Zakariae¹,

Zariouh²

2020

Preprint

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The Berkani's property and a note on some recent results

Zakariae

Zariouh

2021

Linear and Multilinear Algebra

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Property (Bv) and Tensor Product

Aponte

Vasanthakumar²,

Jayanthi³

2022

Symmetry

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In this article, a-Browder’s classical theorem is considered through the property (Bv), and we show that if two operators are norm equivalent, then property (Bv) holds for one if and only if it holds for the other. The necessary conditions for the difference of the spectrum and essential approximate point spectrum of the tensor product of two operators coincide with the product of differences between the spectrum and the essential approximate point spectrum of its two factors are investigated. We also discuss the necessary conditions for the tensor product of two operators to verify the property (Bv) and simultaneously give the equivalence between their various spectra with their Browder spectrum, likewise with the Drazin spectrum.

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A new Browder type property

Cited by 3 publications

References 4 publications

The Berkani's property and a note on some recent results

The Berkani's property and a note on some recent results

The Berkani's property and a note on some recent results

Property (Bv) and Tensor Product

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