Representation for the generalized Drazin inverse of block matrices in Banach algebras Dijana Mosi´ c and Dragan S. Djordjevi´ c ∗ Abstract Several representations of the generalized Drazin inverse of a block matrix with a group invertible generalized Schur complement in Ba- nach algebra are presented. Key words and phrases : generalized Drazin inverse, Schur comple- ment, block matrix. 2010 Mathematics subject classification : 46H05, 47A05, 15A09. 1 Introduction The Drazin inverse has applications in a number of areas such as control the- ory, Markov chains, singular differential and difference equations, iterative methods in numerical linear algebra, etc. Representations for the Drazin in- verse of block matrices under certain conditions where given in the literature [1, 3, 4, 5, 6, 7, 9, 14, 17]. In this paper, we present formulas for the generalized Drazin inverse of block matrix with generalized Schur complement being group invertible in Banach algebra. Moreover, necessary and sufficient conditions for the exis- tence as well as the expressions for the group inverse of triangular matrices are obtained. Let A be a complex unital Banach algebra with unit 1. For a ∈A, we use σ(a) and ρ(a), respectively, to denote the spectrum and the resolvent set of a. The sets of all nilpotent and quasinilpotent elements (σ(a)= {0}) of A will be denoted by A nil and A qnil , respectively. The generalized Drazin inverse of a ∈A (or Koliha–Drazin inverse of a) is the element b ∈A which satisfies bab = b, ab = ba, a - a 2 b ∈A qnil . * The authors are supported by the Ministry of Education and Science, Republic of Serbia, grant no. 174007. 1