The weighted g-Drazin inverse for operators

Abstract

The paper introduces and studies the

Perturbation of the generalized Drazin inverse

“…In this paper, we investigate the perturbation of the generalized Drazin invertible operators and derive explicit generalized Drazin inverse expressions for the perturbations under certain restrictions on the perturbing operators. It is natural to ask if we can extend our results to the W-weighted Drazin inverse [6,10,21], which will be the future research topic.…”

THE DRAZIN INVERSES OF PRODUCTS AND DIFFERENCES OF PROJECTIONS IN A C*-ALGEBRA

“…For a, b in an associative semigroup, if ba is Drazin invertible, then it is easy to verify that the element c = a((ba) D ) 2 b (see [5]) satisfies the definition of the Drazin inverse [3] The Drazin inverses of products and differences of projections in a C PROOF. For p, q ∈ P(H),…”

The gDMP inverse of Hilbert space operators

“…Then we can easily verify this result. [4]. For the weighted core-EP inverse, the situation is similar in the case of the operator W A, but it is a little different for AW [13].…”

“…We now present some special classes of outer inverses. For a fixed operator W ∈ B(Y, X)\{0}, an operator A ∈ B(X, Y ) is called Wg-Drazin invertible [4] if there exists a unique operator B ∈ B(X, Y ) (denoted by A d,w ) such that AW B = BW A, BW AW B = B and A − AW BW A is quasinilpotent.…”