相关论文: Remark on common properties of the products $ac$ a…

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…

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 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…

Let R be a unit-regular ring, and let a,b,c in R satisfy aba=aca. If ac and ba are group invertible, we prove that ac is similar to ba. Furthermore, if ac and ba are Drazin invertible, then their Drazin inverses are similar. For any n\times…

Let A be a Banach algebra, and let a; b; c 2 A satisfying a(ba)^2 = abaca = acaba = (ac)^2a: We prove that 1 - ba\in A^d if and only if 1 - ac \in A^d. In this case, (1-ac)^d =1-a(1-ba)^{\pi}(1-\alpha(1+ba))^{-1}bac (1+ac)+a((1-ba)^d)bac.…

We establish several fundamental properties of one-sided (generalized) Drazin inverses in Banach algebras, including intertwining properties and reverse order laws. In particular, we introduce the concepts of one-sided strongly…

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…

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 =…

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 note, we study common local spectral properties for bounded linear operators $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $$A(BA)^2=ABACA=ACABA=(AC)^2A.$$ We prove that $AC$ and $BA$ share the single valued…

In \cite{C, LCC}, it is proven that if an element $ab$ in a ring is (generalized) Drazin invertible, then so is $ba$. In this paper, we give a new and short proof of it in an effective manner. In particular, we show that if $ab$ is strongly…

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…

Let $X,Y$ be Banach spaces, $A:X \longrightarrow Y$ and $B,C:Y \longrightarrow X$ be bounded linear operators satisfying operator equation $ABA=ACA$. Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common…

Let $R$ be a ring with identity $1$. Jacobson's lemma states that for any $a,b\in R$, if $1-ab$ is invertible then so is $1-ba$. Jacobson's lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse,…

Given a (not necessarily continuous) homomorphism between Banach algebras $\T\colon\A\to\B$, an element $a\in\A$ will be said to be B-Fredholm (respectively generalized B-Fredholm) relative to $\T$, if $\T(a)\in \B$ is Drazin invertible…

Let $X$ and $Y$ be Banach spaces, $A\,:\,X\rightarrow Y$ and $B,\,C\,:\,Y\rightarrow X$ be bounded linear operators. We prove that if $A(BA)^2=ABACA=ACABA=(AC)^2A,$ then $$\sigma_{*}(AC)\setminus\{0\}=\sigma_{*}(BA)\setminus\{0\}$$ where…

Given an idempotent $p$ in a Banach algebra and following the study in \cite{P50} of p-invertibility, we consider here left p-invertibility, right p-invertibility and p-invertibility in the Calkin Algebra $\mathcal{C}(X),$ where $X$ is a…

In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. Under the weakly commutative condition of $ab = \lambda ba$, we show that $a-b$ is Drazin invertible if and…

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 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.