Let A be a bounded linear operator on a Banach space such that the resolvent of A is rational. If 0 is in the spectrum of A, then it is well known that A is Drazin invertible. We investigate spectral properties of the Drazin inverse of A. For example we show that the Drazin inverse of A is a polynomial in A.

A new result on the Drazin inverse of 2× 2 block matrix M = [ A B D C ] , where A and C are square matrices are presented, extended in the case when D = 0, the well known representation for the Drazin inverse of M , given by Hartwig, Meyer and Rose in 1977, respectively. Using that new result, an explicit representation for the Drazin inverse of a modified matrix P + RS and its generalized matr...

If a and b are elements of an algebra, then we show that ab is Drazin invertible if and only if ba is Drazin invertible. With this result we investigate products of bounded linear operators on Banach spaces. 1. Drazin inverses Throughout this section A is a real or complex algebra with identity e 6= 0. We denote the group of invertible elements of A by A. We call an element a ∈ A relatively reg...

In the linear system Ax = b the points x are sometimes constrained to lie in a given subspace S of column space of A. Drazin inverse for any singular or nonsingular matrix, exist and is unique. In this paper, the singular consistent or inconsistent constrained linear systems are introduced and the effect of Drazin inverse in solving such systems is investigated. Constrained linear system arise ...

in the linear system ax = b the points x are sometimes constrained to lie in a given subspace s of column space of a. drazin inverse for any singular or nonsingular matrix, exist and is unique. in this paper, the singular consistent or inconsistent constrained linear systems are introduced and the effect of drazin inverse in solving such systems is investigated. constrained linear system arise ...

If k = 1, then X is called the group inverse of A, and is denoted by X = Ag. The Drazin inverse is very useful in various applications (see, e.g. [1]–[4]; applications in singular differential and difference equations, Markov chains and iterative methods). In 1980, Cline and Greville [5] extended the Drazin inverse of square matrix to rectangular matrix, which can be generalized to the quaterni...

If a ring element α = 1 − ab is Drazin invertible, so is β = 1 − ba. We show that the Drazin inverse of β is expressible by a simple formula (in generalization of “Jacobson’s Lemma”) in terms of the Drazin inverse and the spectral idempotent of α. The commutation rules αa = a β and b α = β b are extended to the Drazin inverses, and the spectral idempotents of α and β are shown to be isomorphic....

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in C∗-algebras is consider...

REPRESENTATIONS FOR THE DRAZIN INVERSE OF BLOCK CYCLIC MATRICES M. CATRAL AND P. VAN DEN DRIESSCHE Abstract. A formula for the Drazin inverse of a block k-cyclic (k ≥ 2) matrix A with nonzeros only in blocks Ai,i+1, for i = 1, . . . , k (mod k) is presented in terms of the Drazin inverse of a smaller order product of the nonzero blocks of A, namely Bi = Ai,i+1 · · ·Ai−1,i for some i. Bounds on ...

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