Group Inverse and Generalized Drazin Inverse of Block Matrices in a Banach Algebra

Let A be a complex unital Banach algebra with unit 1. The sets of all invertible and quasinilpotent elements (σ(a) = {0}) of A will be denoted by A and A, respectively. The group inverse of a ∈ A is the unique element a ∈ A which satisfies aaa = a, aaa = a, aa = aa. If the group inverse of a exists, a is group invertible. Denote by A the set of all group invertible elements of A. The generalized Drazin inverse of a ∈ A (or Koliha–Drazin inverse of a) is the unique element a ∈ A which satisfies aaa = a, aa = aa, a− aa ∈ A. Recall that a = 1− aa is the spectral idempotent of a corresponding to the set {0} [9]. We use A to denote the set of all generalized Drazin invertible elements 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 Banach algebras [4]. Lemma 1.1 ([4, Example 4.5]). Let a, b ∈ A and let ab = 0. Then