The representation of the Drazin inverse of anti-triangular operator matrices based on resolvent expansions

This paper deals with the anti-triangular operator matrix M=ABC0 with A2=AA2=A and CAπB=0CAπB=0. Using the resolvent expansion technique, we obtain the explicit representation of the Drazin inverse of M  , in terms of its entries and the Drazin inverses of the entries and their compositions. The result extends the main results in Bu et al. (2011) [2] and Castro-González and Dopazo (2005) [6] and the results on group inverses in Bu et al. (2008) [3] and Liu and Yang (2012) [14]. As an application, a new additive result is given of the Drazin inverse for two matrices P,Q∈Cn×nP,Q∈Cn×n with P2=PP2=P and PQ2=0PQ2=0.