The Drazin inverses of sum and difference of idempotents

In this note, the Drazin inverses of sum and difference of idempotents on a Hilbert space are established under some conditions.     

Characterizations and representations of the group inverse involving idempotents

This paper is to present some results on the group invertibility of products and differences of idempotents. In addition, some formulae for the group inverse of sums, differences and products of idempotents are established by using some given idempotents.     

The Drazin inverse of the linear combinations of two idempotents in the Banach algebra

In this paper, some Drazin inverse representations of the linear combinations of two idempotents in a Banach algebra are obtained. Moreover, we present counter-examples to and establish the corrected versions of two theorems by Cvetkovi?-Ili? and Deng.     

Properties of m-EP operators

In this paper, we study the characteristics of m-EP operators and the properties of the particular case of m-EP operators that are Drazin invertible. The relations between the Moore–Penrose inverses and Drazin inverses are built by m-EP and many closely equivalent relations are investigated by using appropriate idempotents.     

两个幂等算子线性组合的Drazin逆

利用空间分解的技巧,在条件PQP=QPQ下,得到两个幂等算子P和Q的多线性组合aP+bQ+cPQ+dQP+ePQP的Drazin逆的表达式.     

任意域上的代数中两个幂等元线性组合的Drazin可逆性（英文）

对域上的代数中两个幂等元P和q,满足一定的条件PqP=s,其中s∈{O,P,q,Pq,qP,qPq},本文得到了它们的线性组合ip＋jq均是Drazin可逆的,并给出了其Drazin逆的表示.     

有足够幂等元的群分次环上的Morita对偶

首先定义了具有足够幂等元的群分次环的Ｓｍ ａｓｈ 积并讨论其性质,其次,借助Ｓｍ ａｓｈ 积给出了这类分环上的分次Ｍｏｒｉｔａ对偶的刻划．     

The Drazin inverses of products and differences of orthogonal projections

In this note, the Drazin inverses of products and differences of orthogonal projections on a Hilbert space are established.     

Congruences on *-Regular Semigroups

In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup. This revised version was published online in June 2006 with corrections to the Cover Date.     

Formulae for the generalized Drazin inverse of a block matrix in terms of Banachiewicz–Schur forms

We introduce new expressions for the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra under some conditions. We generalized some recent results for Drazin inverse and group inverse of complex matrices.     

The Drazin invertibility of the difference and the sum of two idempotent operators

In this note, some equivalents are established of the Drazin invertibility of differences and sums of idempotent operators on a Hilbert space.     

On Drazin inverse of operator matrices

In this short paper, we offer (another) formula for the Drazin inverse of an operator matrix for which certain products of the entries vanish. We also give formula for the Drazin inverse of the sum of two operators under special conditions.     

Some additive results on Drazin inverse

Some additive perturbation results for Drazin inverses are given. In particular, a formula is given for the Drazin inverse of a sum of two matrices, when one of the products of these matrices vanishes. Some special applications of this are also considered.     

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.     

The Drazin inverse in an arbitrary semiring

In this article, we investigate the Drazin invertibility for the elements of an arbitrary semiring. We give necessary and sufficient conditions for the existence and expressions of the Drazin inverse of an element in an arbitrary semiring. Moreover, we consider the product paq under some additional necessary conditions for which the Drazin inverse of the product paq exists.     

Additive results for the Drazin inverse of block matrices and applications

In this paper, we consider the Drazin inverse of a sum of two matrices and derive additive formulas under conditions weaker than those used in some recent papers on the subject. As a corollary we get the main results from the paper of Yang and Liu [H. Yang, X. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Appl. Math. 235 (2011) 1412-1417]. As an application we give some new representations for the Drazin inverse of a block matrix.     

Some additive results on Drazin inverses

In this paper, some additive results on Drazin inverse of a sum of Drazin invertible elements are derived. Some converse results are also presented.     

Fredholm Spectra and Weyl Type Theorems for Drazin Invertible Operators

Mediterranean Journal of Mathematics - In this paper we investigate the relationship between some spectra originating from Fredholm theory of a Drazin invertible operator and its Drazin inverse, if...     

A NOTE ON THE REPRESENTATIONS FOR THE GENERALIZED DRAZIN INVERSE OF BLOCK MATRICES

We present some representations for the generalized Drazin inverse of a block matrix x =[abcd]in a Banach algebra A in terms of a~d and(bc)~d under certain conditions,extending some recent result related to the generalized Drazin inverse of an anti-triangular operator matrix.Also,several particular cases of this result are considered.     

关于幂等元之差的可逆性

本文研究在一个有单位元的环中两个幂等元之差的可逆性问题,利用幂等元的性质,得到了两个幂等元之差可逆的几个充分必要条件,并给出了在矩阵环中的几个应用.