The aim of this paper is to establish an explicit representation the generalized Drazin inverse $(a+b)^d$ under condition $$ab^2=0, ba^2=0, a^{\pi}b^{\pi}(ba)^2=0.$$ Furthermore, we apply our results give some for a $2\times 2$ block operator matrix. These extend on Bu, Feng and Bai [Appl. Math. Comput. 218, 10226-10237, 2012] Dopazo Martinez-Serano [Linear Algebra Appl. 432, 1896-1904, 2010].

In this paper we offer new representations for Drazin inverse of block matrix, which recover some representations from current literature on this subject. 2000 Mathematics Subject Classification: 15A09

In this paper, we investigate a perturbation of the Drazin inverse AD of a closed linear operator A; the main tool for obtaining the estimates is the gap between subspaces and operators. By (X) we denote the set of all closed linear operators acting on a linear subspace of X to X , where X is a complex Banach space. We write (A), (A), (A), ρ(A), σ(A), and R(λ,A) for the domain, nullspace, range...

Given an n × n singular matrix A with Ind(A) = k, an index splitting of A is one of the form A = U − V , where R(U) = R(Ak) and N(U) = N(Ak). This splitting, introduced by the first author, generalizes the proper splitting proposed by Berman and Plemmons. Regarding singular systems Au = f , the first author has shown that the iterations u(i+1) = U#V u(i) + U#f converge to ADf , the Drazin inver...

In this paper , some results about the index of matrix and Drazin inverse are given. We then explain applications of them in solving singular linear system of equations.

Given a bounded operator A on a Banach space X with Drazin inverse A and index r, we study the class of group invertible bounded operators B such that / + A (B — A) is invertible and 1Z(B) f]N(A) = {0}. We show that they can be written with respect to the decomposition X = lZ(A) 0 Af(A) as a matrix operator, B = ( _ _ __i _ ), where B\ and B\ + B\9#?1 a r e invertible. ^ if21 t>2\tf\ i*\2 Sever...