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

Given two bounded linear operators F,G on a Banach space X such that G2F=GF2=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 . From the provided representation of D(F+G) several special cases are considered. In particular, we recover the case GF=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 which are extensions of some cases given in the literature.