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Marjan Sheibani

Publications and source records attributed to Marjan Sheibani.

At least 19 recordsLinked to original sources
arXiv2025

Generalized right group inverse in Banach *-algebras

In this paper, we introduce the concept of the generalized right group inverse within the context of a *-Banach algebra. This represents a natural extension of the generalized (weak) group inverse. Notably, this generalized inverse is characterized by integrating the right group inverse with the concept of quasinilpotency. We provide various characterizations and representations of the generalized right group inverse. Furthermore, we explore the relationship between the generalized right group inverse and the generalized right EP-inverse. The properties of the generalized (weak) group inverse in a Banach *-algebra are also extended to a more general framework.

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arXiv2025

The $m$-generalized right group inverses in Banach algebras

In this paper, we introduce the concept of the m-generalized right group inverse. This serves as a natural extension of both the m-weak group inverse and the generalized group inverse. We characterize this new generalized inverse using the m-generalized right group decomposition and a polar-like property. Additionally, we present the representation of the m-generalized right group inverse using the generalized right core inverse, leading to new insights and properties for both the m-weak group inverse and the generalized group inverse.

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arXiv2025

New properties of weighted generalized core-EP inverse in Banach algebras

We characterize the generalized weighted core-EP inverse via the canonical decomposition, utilizing a weighted core-EP invertible element and a quasinilpotent. We then offer a polar-like characterization for the generalized weighted core-EP invertible element. The representations of the generalized weighted core-EP inverse by leveraging the weighted generalized Drazin inverse are thereby presented. These lead to new properties for the weighted core-EP inverse.

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arXiv2023

On generalized core-EP invertibility in a Banach algebra

We present new properties of generalized core-EP inverse in a Banach *-algebra. We characterize this new generalized inverse by using involved annihilators. The generalized core-EP inverse for products is obtained. The core-EP orders for Banach *-algebra elements are thereby investigated. As applications, new properties of the core-EP inverse for block complex matrices are given.

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arXiv2023

G-Drazin inverse and group inverse for the anti-triangular block-operator matrices

We present the generalized Drazin inverse for certain anti-triangular operator matrices. Let $E,F,EF^π\in \mathcal{B}(X)^d$. If $EFEF^π=0$ and $F^2EF^π=0$, we prove that $M=\left( \begin{array}{cc} E&I F&0 \end{array} \right)$ has g-Drazin inverse and its explicit representation is established. Moreover, necessary and sufficient conditions are given for the existence of the group inverse of $M$ under the condition $FEF^π=0$. The group inverse for the anti-triangular block-operator matrices with two identical subblocks is thereby investigated. These extend the results of Zhang and Mosić (Filomat, 32(2018), 5907--5917) and Zou, Chen and Mosić (Studia Scient. Math. Hungar., 54(2017), 489--508).

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arXiv2022

The g-Drazin inverses of anti-triangular block operator matrices

An element $a$ in a Banach algebra $\mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $ab=ba, b=bab$ and $a-a^2b \in \mathcal{A}^{qnil}$. In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix $\left( \begin{array}{cc} E&I F&0 \end{array} \right)$. We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear $\&$ Multilinear Algebra, 14(1983), 195--198].

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arXiv2022

The Core Inverse in a Banach Algebra with Involution

We present new additive results for the core inverse in a Banach algebra with involution. We obtain necessary and sufficient conditions under which the sum of two core invertible elements in a Banach algebra with involution is core invertible. Then we apply our results to block complex matrices and obtain certain conditions under which a block complex is core invertible. These generalize many known results, e.g.,~\cite[Theorem 4.3]{XCZ}, ~\cite[Theorem 2.5]{XS}.

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arXiv2022

Pseudo core invertibility in Banach *-algebras and its applications

We present new additive results for the pseudo core inverse in a Banach algebra with involution. The necessary and sufficient conditions under which the sum of two pseudo core invertible elements in Banach *-algebra is pseudo core invertible are obtained. As an application, the pseudo core invertibility for block complex matrices is investigated. These extend the main results of pseudo core invertibility of Gao and Chen [Comm. Algebra, 46(2018), 38--50].

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arXiv2022

Group inverse for anti-triangular block operator matrices

We present the existence of the group inverse and its representation for the block operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ under the condition $FEF^π=0$. The group inverse for the anti-triangular block matrices with two identical subblocks under the same condition is thereby investigated. These extend the results of Zou, Chen and Mosić (Studia Scient. Math. Hungar., 54(2017), 489--508), and Cao, Zhang and Ge (J. Appl. Math. Comput., 46(2014), 169--179).

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arXiv2022

The g-Drazin invertibility in a Banach algebra

We present necessary and sufficient conditions under which the anti-triangular matrix $\left( \begin{array}{cc} a&b 1&0 \end{array} \right)$ over a Banach algebra has g-Drazin inverse. New additive results for g-Drazin inverse are obtained. Then we apply our results to $2\times 2$ operator matrices and generalize many known results, e.g.,~\cite[Theorem 2.2]{D}, ~\cite[Theorem 2.1]{YL} and \cite[Theorem 4.1]{Y}.

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arXiv2022

Group invertibility of the sum in rings and its applications

We present new additive results for the group invertibility in a ring. Then we apply our results to block operator matrices over Banach spaces and derive the existence of group inverses of $2\times 2$ block operator matrices. These generalize many known results, e.g., Benitez, Liu and Zhu(Linear Multilinear Algebra, {\bf 59}(2011), 279--289) and Zhou, Chen and Zhu(Comm. Algebra, {\bf 48}(2020),676-690).

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arXiv2022

The group inverse of the sum in a Banach algebra

In this paper, we present new necessary and sufficient conditions under which the sum of two group invertible elements in a Banach algebra has group inverse. We then apply these results to block operator matrices over Banach spaces. The group inverses of certain operator block matrices are thereby obtained. Additionally, this paper extends the results obtained in \cite{L}.

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arXiv2021

Generalized Cline's formula and Jacobson's lemma in a ring

We present new generalized Cline's formula and Jacobson's lemma for the g-Drazin inverse in a ring. These extend many known results, e.g., Chen and Abdolyousefi (Generalized Jacobson's Lemma in a Banach algebra, Comm. Algebra, {\bf 49}(2021), 3263--3272), Yan and Zeng (The generalized inverses of the products of two elements in a ring, Turk. J. Math., {\bf 44}(2020), 1744--1756).

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arXiv2020

Jacobson's Lemma for the generalized n-strongly Drazin inverse

Let $n\in {\Bbb N}$. An element $a\in R$ has generalized n-strongly Drazin inverse if there exists $x\in R$ such that $xax=x, x\in comm^2(a), a^n-ax\in R^{qnil}.$ For any $a,b\in R$, we prove that $1-ab$ has generalized n-strongly Drazin inverse if and only if $1-ba$ has generalized n-strongly Drazin inverse. Extensions in Banach algebra are also obtained.

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arXiv2019

Expressions for the g-Drazin inverse in a Banach algebra

We explore the generalized Drazin inverse in a Banach algebra. Let $\mathcal{A}$ be a Banach algebra, and let $a,b\in \mathcal{A}^{d}$. If $ab=λa^πbab^π$ then $a+b\in \mathcal{A}^{d}$. The explicit representation of $(a+b)^d$ is also presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., {\bf 2015}, 156934.8] are extended.

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arXiv2019

Additive properties of G-Drazin inverse of linear operators

In this paper, we investigate additive properties of generalized Drazin inverse for linear operators in Banach spaces. Under new polynomial conditions on generalized Drazin invertible operators a and b, we prove their sum has generalized Drazin inverse and give explicit representations of the generalized inverse $(a+b)^d$.

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