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 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.