New additive results for the generalized Drazin inverse

Abstract

In this paper, we investigate additive properties of generalized Drazin inverse of two Drazin invertible linear operators in Banach spaces. Under the commutative condition of P Q = Q P, we give explicit representations of the generalized Drazin inverse (P + Q ) d in term of P , P d , Q and Q d . We consider some applications of our results to the perturbation of the Drazin inverse and analyze a number of special cases.

On the group inverse of linear combinations of two group invertible matrices

“…Recently, some formulas for the Drazin inverse of a sum of two matrices (or two bounded operators in a Banach space) under some conditions were given (see [4,5,7,8,9,10,11,14] and references therein). Let us remark that the group inverse…”

Some additive results for the generalized Drazin inverse in a Banach algebra

“…It is well-known that a ∈ A d can be represented in the following matrix form, pAp is the inverse of a 11 in sub-algebra pAp. In this note, we generalize the results from [2] to the Banach algebra case.…”

Generalized Drazin invertibility of the product and sum of two elements in a Banach algebra and its applications

“…For instance, in [10], for two commutative generalized Drazin invertible elements a, b in a Banach algebra, Koliha gave the expression of are weaker than ab = 0 in Banach algebras. In [6], Deng and Wei derived necessary and sufficient conditions for the existence of (P + Q) d under the condition P Q = QP , where P, Q are bounded linear operators. Moreover, the expression of (P + Q) d was given.…”

“…Moreover, the expression of (P + Q) d was given. In [3], Cvetković-Ilić et al extended the result of [6] to Banach algebras.…”