Related papers: Expressions for the g-Drazin inverse in a Banach a…
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.…
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 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…
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 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…
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…
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…
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…
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.
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…
In this paper, some Drazin inverse representations of the linear combinations of two idempotents in Banach algebra are obtained.
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.
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.…
We study the Drazin inverses of the sum and product of two elements in a ring. For Drazin invertible elements $a$ and $b$ such that $a^2b=aba$ and $b^2a=bab$, it is shown that $ab$ is Drazin invertible and that $a+b$ is Drazin invertible if…
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.
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.
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…
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 present new presentations of group inverse for the sum of two group invertible elements in a Banach algebra. We then apply these results to block complex matrices. The group invertibility of certain block complex matrices…