Consistent invertibility and perturbations of property $(R)$

Ren, Yanxun; Jiang, Lining; Kong, Yingying

doi:10.5486/pmd.2022.9181

Cited by 3 publications

(1 citation statement)

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“…In [5], Cao et al investigated the CI spectrum on a Banach space, and applied the CI spectrum to characterize those operators T satisfying Weyl type theorems. The relation between CI spectrum and Weyl type theorems also can be seen in [27,28]. Illuminated by results above we answer the question affirmably that Jacobson's lemma still holds for CI spectrum.…”

Section: Proof (1) Let λ = 0 and Asupporting

confidence: 63%

Jacobson's lemma for the generalized $n$-strong Drazin inverses in rings and in operator algebras

Ren¹,

Jiang²

2022

Glas. Mat. Ser. III

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In this paper, we extend Jacobson's lemma for Drazin inverses to the generalized $n$-strong Drazin inverses in a ring, and prove that $1-ac$ is generalized $n$-strong Drazin invertible if and only if $1-ba$ is generalized $n$-strong Drazin invertible, provided that $a(ba)^{2}=abaca=acaba=(ac)^{2}a$. In addition, Jacobson's lemma for the left and right Fredholm operators, and furthermore, for consistent in invertibility spectral property and consistent in Fredholm and index spectral property are investigated.

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Section: Proof (1) Let λ = 0 and Asupporting

confidence: 63%

Jacobson's lemma for the generalized $n$-strong Drazin inverses in rings and in operator algebras

Ren¹,

Jiang²

2022

Glas. Mat. Ser. III

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Left and right-Drazin inverses in rings and operator algebras

Ren

Jiang

2022

J. Algebra Appl.

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The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson’s lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated.

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B-Fredholm elements in primitive C*-algebras

Kong

Jiang

2022

Open Mathematics

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Let A {\mathcal{A}} be a unital primitive C ∗ \ast -algebra. This article studies the properties of the B-Fredholm elements, the B-Weyl elements and the B-Browder elements in A {\mathcal{A}} . Particularly, this article describes the B-Fredholm element as the sum of a Fredholm element and a nilpotent element. In addition, the socle of A {\mathcal{A}} is characterized by the B-Fredholm elements.

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Consistent invertibility and perturbations of property $(R)$

Cited by 3 publications

References 0 publications

Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

Left and right-Drazin inverses in rings and operator algebras

B-Fredholm elements in primitive C*-algebras

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