On the Drazin inverse of the sum of two operators and its application to operator matrices

Abstract

Given two bounded linear operators F , G on a Banach space X such that G 2 F = G F 2 = 0, we derive an explicit expression for the Drazin inverse of F + G. For this purpose, firstly, we obtain a formula for the resolvent of an auxiliary operator matrix in the form M = F I G F G . From the provided representation of (F + G) D several special cases are considered. In particular, we recover the case G F = 0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207-217] for matrices and by Djordjević and Wei [D.S. Djordjević, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (1) (2002) 115-126] for operators. Finally, we apply our results to obtain representations for the Drazin inverse of operator matrices in the form M = A B C D which are extensions of some cases given in the literature.

The Drazin inverse of the sum of two matrices and its applications

“…Castro-González [6] derived a result under the conditions P D Q = 0, PQ D = 0 and Q π PQP π = 0, and Mosić and Djordjević [7] extended these results to the W -weight Drazin inverse on a Banach space. In 2008, Castro-González et al [8] extended these results to the case P 2 Q = 0 and PQ 2 = 0, and some similar results were extended to a Banach algebra by Castro-González and Martínez-Serrano [9]. For idempotent matrices P and Q on a Hilbert space, their Drazin inverse of sum and difference were established by Deng [10], with PQP = 0 or PQP = PQ or PQP = P satisfied.…”

“…This problem was first proposed by Campbell and Meyer [2], and is quite complicated. To the best of our knowledge, there was no explicit formula for the Drazin inverse of M. Some special cases have been considered, which can be found in [18,8,19,20,3,[21][22][23][24][25][26][27][28][29][30].…”

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…”

Additive results for the group inverse in an algebra with applications to block operators

“…A related result in the setting of operators in a Banach space and Drazin inverses was given in [6]. Theorem 2.1.…”