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=A and CAπB=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×n with P2=P and PQ2=0. [ABSTRACT FROM AUTHOR]