相关论文: Additive properties of G-Drazin inverse of linear …

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 $ab=\lambda a^{\pi}bab^{\pi}$ then $a+b\in \mathcal{A}^{d}$. The explicit representation of $(a+b)^d$…

Let $R$ be an associative ring with an identity and suppose that $a,b,c,d \in R$ satisfy $bdb = bac,dbd = acd$. If $ac$ has generalized Drazin ( respectively, pseudo Drazin, Drazin) inverse, we prove that $bd$ has generalized Drazin…

The objective of this paper is to study the existence of the generalized Drazin inverse of the sum $a+b$ in a Banach algebra and present explicit expressions for the generalized Drazin inverse of this sum, under new conditions.

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a $2\times2$ operator matrix under certain circumstances, which generalizes and unifies several results in the literature.

Given a complex Banach space $X$, let $\mathcal{B}(X)$ be the collection of all bounded linear operators on $X.$ For $A,B\in\mathcal{B}(X)$ we define $A,B$ are $A$-weakly commutative if there exists $C\in\mathcal{B}(X)$ satisfying…

In this paper, we begin by introducing some necessary and sufficient conditions for generalized $n$-strong Drazin invertibility (g$n$s-invertibility) and pseudo $n$-strong Drazin invertibility (p$n$s-invertibility) of an element in a Banach…

We study the generalized Drazin invertibility as well as the Drazin and ordinary invertbility of an operator matrix (A C \\ 0 B) acting on a Banach or on a Hilbert space. As a consequence some recent results are extended.

Let $\mathcal{A}$ be a complex Banach algebra. An element $a\in \mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $$b=bab, ab=ba, a-a^2b\in \mathcal{A}^{qnil}.$$ Let $a,b\in \mathcal{A}$ have g-Drazin inverses.…

In this paper, some Drazin inverse representations of the linear combinations of two idempotents in Banach algebra are obtained.

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…

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.…

Clines formula for the well known generalized inverses such as Drazin inverse, generalized Drazin inverse is extended to the case when $a(ba)^2=abaca=acaba=(ac)^2a$ . Applications are given to some interesting Banach space operators.

In this paper, we give a generalized Cline's formula for the generalized Drazin inverse. Let $R$ be a ring, and let $a,b,c,d\in R$ satisfying $$\begin{array}{c} (ac)^2 = (db)(ac), (db)^2 = (ac)(db);\\ b(ac)a = b(db)a, c(ac)d =…

In a Banach algebra, we introduce a new type of generalized inverse called g$\pi$-Hirano inverse. Firstly, several existence criteria and the equivalent definition of this inverse are investigated. Then, we discuss the relationship between…

We extend the notion of generalized Drazin-Riesz inverse introduced for bounded linear operators in \cite{Ziv} to elements in a complex unital semi-simple Banach algebra. Several characterizations and properties of generalized Drazin-Riesz…

Let $A$ be a right linear operator on a two-sided quaternionic Banach space $X$. The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin…

We present some formulae for the Drazin inverse of difference and product of idempotents in a ring. A number of results of bounded linear operators in Banach spaces are extended to the ring case.

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…

We study properties of pseudo Drazin inverse in a Banach algebra with unity 1. If $ab=ba$ and $a,b$ are pseudo Drazin invertible, we prove that $a+b$ is pseudo Drazin invertible if and only if $1+a^\ddag b$ is pseudo Drazin invertible.…

The famous Drazin inverse and generalized Drazin inverse were introduced by Drazin in 1958 and Koliha in 1996, respectively. In the present paper, the author introduces the concepts of left and right (generalized) Drazin inverses, which are…