A new characterization of Browder’s theorem

Karmouni, Mohammed; Tajmouati, Abdelaziz

doi:10.2298/fil1814865k

Cited by 6 publications

(5 citation statements)

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“…Our next theorem gives new characterization of some Browder's theorem type classes in terms of spectra introduced and studied in this paper. This theorem is an improvement of some recent results dressed in [19,22]. The next theorem gives a sufficient condition for an operator T ∈ L(X) to have the W-SVEP.…”

Section: Weak Svep and Applicationssupporting

confidence: 51%

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On the $g_{z}$-Kato decomposition and generalization of Koliha Drazin invertibility

Aznay¹,

Ouahab²,

Zariouh³

2022

Preprint

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In [24], Koliha proved that T ∈ L(X) (X is a complex Banach space) is generalized Drazin invertible operator equivalent to there exists an operator S commuting with T such that ST S = S and σ(T 2 S − T ) ⊂ {0} which is equivalent to say that 0 ∈ acc σ(T ). Later, in [14,34] the authors extended the class of generalized Drazin invertible operators and they also extended the class of pseudo-Fredholm operators introduced by Mbekhta [27] and other classes of semi-Fredholm operators. As a continuation of these works, we introduce and study the class of gz-invertible (resp., gz-Kato) operators which generalizes the class of generalized Drazin invertible operators (resp., the class of generalized Kato-meromorphic operators introduced by Živković-Zlatanović and Duggal in [35]). Among other results, we prove that T is gz-invertible if and only if T is gz-Kato with p(T ) = q(T ) < ∞ which is equivalent to there exists an operator S commuting with T such that ST S = S and acc σ(T 2 S − T ) ⊂ {0} which in turn is equivalent to say that 0 ∈ acc (acc σ(T )). As application and using the concept of the Weak SVEP introduced at the end of this paper, we give new characterizations of Browder-type theorems.closed T -invariant subspaces andan upper or a lower (resp., upper and lower) semi-B-Fredholm then T it is called semi-B-Fredholm (resp., B-Fredholm) and its index is defined by ind(T ) := α(T [mT ] ) − β(T [mT ] ). T is said to be an upper semi-B-Weyl (resp., a lower semi-B-Weyl, B-Weyl, left Drazin invertible, right Drazin invertible, Drazin invertible) if T is an upper semi-B-Fredholm with ind(T ) ≤ 0 (resp., T is a lower semi-B-Fredholm with ind(T ) ≥ 0, T is a B-Fredholm with ind(T ) = 0, T is an upper semi-B-Fredholm 0 2020

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Section: Weak Svep and Applicationssupporting

confidence: 51%

“…The other implications are already done in [1]. The assertions (b) and (c) (in which some implications are already done in [1,6,19,22]) go similarly with (a). on T all other statements and under the assumption int σ gz w (T ) = ∅, all statements imply the statement (i).…”

Section: Weak Svep and Applicationsmentioning

confidence: 87%

On the $g_{z}$-Kato decomposition and generalization of Koliha Drazin invertibility

Aznay¹,

Ouahab²,

Zariouh³

2022

Preprint

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“…(ii) ⇔ (vi) By Corollary 2.12 we know that σ gDR (T ) = σ gDRW (T ) if and only if intσ * (T ) ⊂ σ w (T ), where σ * = σ b or σ ub or σ lb or σ bb or σ lsbb or σ usbb or σ gD or σ gDQ or σ llsbb or σ gDR or σ gDRQ or σ gDRJ or σ gDM Q or σ. By [12,Theorem 2.8] (ii) ⇒ (iii) As intσ * (T ) ⊂ σ usbw (T ), by Corollary 2.3 we have σ bb (T ) = σ usbw (T ). Therefore, by [18, Theorem 1] and [18,Theorem 5] we have σ gDM (T ) = σ gDM W + (T ).…”

mentioning

confidence: 87%

Generalized Drazin-Meromorphic Invertible Operators and Browder's Type Theorems

Gupta¹,

Kumar²

2020

Preprint

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“…Recall that an operator T satisfies Browder's theorem if σ b (T ) = σ w (T ) and generalized Browder's theorem if σ bb (T ) = σ bw (T ). Amouch et al [7] and Karmouni and Tajmouati [12] gave a new characterization of Browder's theorem using spectra arised from Fredholm theory and Drazin invertibilty. Motivated by them, we give a new characterization of operators satisfying generalized Browder's theorem.…”

Section: Introductionmentioning

confidence: 99%

“…Since σ gDM φ + (T ) ⊂ σ uf (T ), T has SVEP at every λ / ∈ σ uf (T ). Therefore, by [12,Theorem 2.8] we have σ uf = σ ub (T ). Thus, by Lemma 2.9 σ usbf = σ usbb (T ).…”

mentioning

confidence: 99%

A new characterization of generalized Browder’s theorem and a cline’s formula for generalized Drazin-eromorphic inverses

Gupta

Kumar

2019

Filomat

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In this paper, we give a new characterization of generalized Browder's theorem by considering equality between the generalized Drazin-meromorphic Weyl spectrum and the generalized Drazinmeromorphic spectrum. Also, we generalize Cline's formula to the case of generalized Drazinmeromorphic invertibility under the assumption that A k B k A k = A k+1 for some positive integer k. Mathematics Subject Classification: 47A10, 47A53. Keywords: SVEP, generalized Drazin-meromorphic invertible, meromorphic operators, operator equation.semi-Fredholm, Fredholm, lower semi-Weyl , upper semi-Weyl and Weyl spectra are defined byA bounded linear operator T is said to be bounded below if it is injective and R(T ) is closed. For T ∈ B(X) the ascent denoted by p(T ) is the smallest non negative integer p such that kerT p = kerT p+1 . If no such integer exists we set p(T ) = ∞. For T ∈ B(X) the descent denoted by q(T ) is the smallest non negative integer q such that R(T q ) = R(T q+1 ). If no such integer exists we set q(T ) = ∞. By [1, Theorem 1.20] if both p(T ) and q(T ) are finite then p(T ) = q(T ). An operatorcalled upper semi-Browder if it is an upper semi-Fredholm and p(T ) < ∞. An operator T ∈ B(X) is called lower semi-Browder if it is an lower semi-Fredholm and q(T ) < ∞. We say that an operator T ∈ B(X) is Browder if it is upper semi-Browder and lower semi-Browder. The lower semi-Browder, upper semi-Browder and Browder spectra are defined by σ lb (T ) : = {λ ∈ C : λI − T is not lower semi-Browder}, σ ub (T ) : = {λ ∈ C : λI − T is not upper semi-Browder}, σ b (T ) : = {λ ∈ C : λI − T is not Browder}, respectively. Clearly, every Browder operator is Drazin invertible. An operator T ∈ B(X) is said to possess the single-valued extension property (SVEP) at λ 0 ∈ C if for every neighbourhood V of λ 0 the only analytic function f : V → X which satisfies the equation (λI − T )f (λ) = 0 is the function f = 0. If an operator T has SVEP at every λ ∈ C, then T is said to have SVEP. Morever, the set of all points λ ∈ C such that T does not have SVEP at λ is an open set contained in interior of σ(T ). Therefore, if T has SVEP at each point of an open punctured disc D \ {λ 0 } centered at λ 0 , T also has SVEP at λ 0 . p(λI − T ) < ∞ ⇒ T has SVEP at λ and q(λI − T ) < ∞ ⇒ T * has SVEP at λ. An operator T ∈ B(X) is called Riesz if λI −T is Browder for all λ ∈ C\{0}. An operator T ∈ B(X) is called meromorphic if λI − T is Drazin invertible for all λ ∈ C \ {0}. Clearly, every Riesz operator is meromorphic. A subspace M of X is said to be T -invariant if T (M) ⊂ M. For a T -invariant subspace M of X we define T M : M → M by T M (x) = T (x), x ∈ M. We say T is completely reduced by the pair (M, N) (denoted by (M, N) ∈ Red(T )) if M and N are two closed T -invariant subspaces of X such that X = M ⊕ N. An operator T ∈ B(X) is called semi-regular if R(T ) is closed and ker(T ) ⊂ R(T n ) for every n ∈ N. An operator T ∈ B(X) is called nilpotent if T n = 0 for some n ∈ N and called quasi-nilpotent if ||T n || 1 n → 0, i.e λI − T is invertible for ...

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A new characterization of Browder’s theorem

Abstract: We give a new characterization of Browder's theorem using spectra originated from Drazin-Fredholm theory.

Cited by 6 publications

References 20 publications

On the $g_{z}$-Kato decomposition and generalization of Koliha Drazin invertibility

On the $g_{z}$-Kato decomposition and generalization of Koliha Drazin invertibility

Generalized Drazin-Meromorphic Invertible Operators and Browder's Type Theorems

A new characterization of generalized Browder’s theorem and a cline’s formula for generalized Drazin-eromorphic inverses

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