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. Necessary and sucient conditions for the existence of thegroup inverse of an anti-triangular block matrix in Banach algebras arepresented. Using these results, we give the formulae for the generalizedDrazin inverse of a block matrix. 1. IntroductionLet A be a complex unital Banach algebra with unit 1. The sets of allinvertible and quasinilpotent elements (σ(a) = {0}) of A will be denoted byA −1 and A qnil , respectively.The group inverse of a ∈ A is the unique element a # ∈ A which satisesa # aa # = a # , aa # a = a, aa # = a # a.If the group inverse of a exists, a is group invertible. Denote by A # the set ofall group invertible elements of A.The generalized Drazin inverse of a ∈ A (or Koliha–Drazin inverse of a) isthe unique element a d ∈ A which satisesa d aa d = a d , aa d = a d a, a−a 2 a d ∈ A qnil .Recall that a π = 1− aa d is the spectral idempotent of a corresponding to theset {0} [9]. We use A d to denote the set of all generalized Drazin invertibleelements of A.We state the following result which is proved for matrices [8, Theorem 2.1],for bounded linear operators [7, Theorem 2.3] and for elements of Banachalgebras [4].Lemma1.1([4, Example 4.5]). Let a,b ∈ A

GROUP INVERSE AND GENERALIZED DRAZIN INVERSE OF BLOCK MATRICES IN A BANACH ALGEBRA