We study generalized inverses on semigroups by means of Green’s relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse and Moore-Penrose inverse) belong to this class. There exist many specific generalized inverses in the literature, such as the group inverse, the Moore-Penro...

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. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A09; 15A23

In this paper we get the formula for the condition number of the W -weighted Drazin inverse solution of a linear system WAWx = b, where A is a bounded linear operator between Hilbert spaces X and Y , W is a bounded linear operator between Hilbert spaces Y and X, x is an unknown vector in the range of (AW ) and b is a vector in the range of (WA). AMS Mathematics Subject Classification (2000): 47...

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

Let X be a Banach space and T be a bounded linear operator from X to itself (T ∈ B(X).) An operator D ∈ B(X) is a Drazin inverse of T if TD = DT , D = TD and T k = T D for some nonnegative integer k. In this paper we look at the Jörgens algebra, an algebra of operators on a dual system, and characterise when an operator in that algebra has a Drazin inverse that is also in the algebra. This resu...

We study semi-iterative methods for an approximate solution of a linear equation x = Tx+c with a bounded linear operator T acting on a Banach space X, considering separately the cases when λ = 1 is an isolated and accumulation spectral point for T . We give necessary and sufficient conditions for the convergence of a semi-iterative method, utilizing the generalized Drazin inverse introduced by ...

In this article, we introduce two new generalized inverses for rectangular matrices called $W$-weighted generalized-Drazin--Moore--Penrose (GDMP) and generalized-Drazin-reflexive (GDR) inverses. The first inverse can be seen as a generalization of the recently introduced GDMP square matrix to matrix. second class contains inverse. We then exploit their various properties establish that proposed...

In this paper, we investigate additive properties of generalized Drazin inverse for bounded linear operators. As an application present new conditions under which a 2 ? operator matrix has g-Drazin inverse. These extend the main results Dana and Yousefi (Int. J. Appl. Comput. Math., 4(2018), page 9), Yang Liu (J. 235(2011), 1412-1417) Sun et al. (Filomat, 30(2016), 3377-3388).

the index of matrix a in cn.n is equivalent to the dimension of largest jor-dan block corresponding to the zero eigenvalue of a. in this paper, indicialequations and normal equations for solving inconsistent singular linear systemof equations are investigated.

In this paper, a class of Hessenberg matrices is presented for adoption as test matrices. The Moore-Penrose inverse and the Drazin inverse for each member of this class are determined explicitly.