arXiv subjects

Yanxun Ren

Publications and source records attributed to Yanxun Ren.

3 recordsLinked to original sources

arXiv2022

A new extension of generalized Drazin inverse in Banach algebras

In this paper, we introduce and study a new generalized inverse, called ag-Drazin inverses in a Banach algebra $\mathcal{A}$ with unit $1$. An element $a\in\mathcal{A}$ is ag-Drazin invertible if there exists $x\in\mathcal{A}$ such that $ax=xa, \, xax=x \ {\rm and} \ a-axa\in\mathcal{A}^{acc}$, where $\mathcal{A}^{acc}\triangleq\{a\in\mathcal{A}: a-λ1 \ {\rm is \ generalized \ Drazin\ invertible} \ {\rm for \ all} \ λ\in\mathbb{C}\backslash\{0\}\}.$ Using idempotent elements, we characterize this inverse and give some its representations. Also, we prove that $a\in\mathcal{A}$ is ag-Drazin invertible if and only if $0$ is not an accumulation point of $σ_{d}(a)$, where $σ_{d}(a)$ is the generalized Drazin spectrum of $a$.

math.FA↗

arXiv2021

Extensions of Jacobson's lemma for generalized inverses in a ring

Let $R$ be an associative ring with unit $1$, and $a, b, c\in R$ satisfy $a(ba)^{2}=abaca=acaba=(ac)^{2}a$, this paper proves that $1-ac$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively) if and only if $1-ba$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively). In particular, we obtain new common spectral properties for $ac$ and $ba$ in Banach algebras. As applications, new extension of Jacobson's lemma for B-Fredholm elements and generalized Fredholm elements in rings is established.

math.RA↗

arXiv2021

Remark on common properties of the products $ac$ and $ba$

In this paper, we discuss the common properties for the products $ac$ and $ba$ in various categories under the condition $a(ba)^{2}=abaca=acaba=(ac)^{2}a$. We prove that generalized Jacobson's lemma and Cline's formula are suitable for generalized n-strong Drazin invertible in rings, and generalized Jacobson's lemma is suitable for left and right Fredholm operator on Banach spaces.

math.RA↗