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

We derive a very short expression for the group inverse of a(1) + ... + a(n) when a(1), ... , a(n) are elements in an algebra having group inverse and satisfying a(i)a(j) = 0 for i < j. We apply this formula in order to find the group inverse of 2 x 2 block operators under some conditions. (C) 2011 Taylor & Francis X. Liu and T. Zhu are supported by Guangxi Science Foundation (0640016, 0832084). Benítez López, J.; Liu, X.; Zhu, T. (2011). Additive results for the group inverse in an algebra with applications to block operators. Linear and Multilinear Algebra. 59(3):279-289. https://doi.org/10.1080/03081080903410262 Benítez, J. (2008). Moore–Penrose inverses and commuting elements of C∗-algebras. Journal of Mathematical Analysis and Applications, 345(2), 766-770. doi:10.1016/j.jmaa.2008.04.062 Benítez, J, Liu, X and Zhu, T.Nonsingularity and group invertibility of linear combinations of two k-potent matrices, Linear Multilinear Algebra (accepted) Castro-González, N., Dopazo, E., & Martínez-Serrano, M. F. (2009). On the Drazin inverse of the sum of two operators and its application to operator matrices. Journal of Mathematical Analysis and Applications, 350(1), 207-215. doi:10.1016/j.jmaa.2008.09.035 González, N. C., & Koliha, J. J. (2004). New additive results for the g-Drazin inverse. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 134(6), 1085-1097. doi:10.1017/s0308210500003632 Cvetković-Ilić, D. S., Djordjević, D. S., & Wei, Y. (2006). Additive results for the generalized Drazin inverse in a Banach algebra. Linear Algebra and its Applications, 418(1), 53-61. doi:10.1016/j.laa.2006.01.015 Deng, C. Y. (2009). The Drazin inverses of sum and difference of idempotents. Linear Algebra and its Applications, 430(4), 1282-1291. doi:10.1016/j.laa.2008.10.017 Deng, C, Cvetković-Ilić, DS and We