Applied Mathematics and Computation 284 (2016) 162–168 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On the generalized Drazin inverse in Banach algebras in terms of the generalized Schur complement J. Robles ∗ , M.F. Martínez-Serrano , E. Dopazo ETS Ingenieros Informáticos, Universidad Politécnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain a r t i c l e i n f o Keywords: Generalized Drazin inverse Generalized Schur complement Block matrix a b s t r a c t We present new results on existence and representation of the generalized Drazin inverse for elements in Banach algebras, involving the generalized Schur complement. They extend cases studied in the literature where the generalized Schur complement is assumed to be nonsingular, or zero, or group invertible. The results are applied to obtain formulas for cases of special interest. © 2016 Elsevier Inc. All rights reserved. 1. Introduction The Drazin inverse plays an important role in various fields like Markov chains, singular differential and difference equa- tions, and iterative methods [1,2]. The concept of Drazin inverse, initially introduced for complex matrices, was extended to the context of Banach algebras by Koliha [14]. Harte [11] gave an alternative definition of a generalized Drazin inverse in a ring. Both concepts were proved to be equivalent in the case the ring is actually a complex Banach algebra with unit. Let A be a complex unital Banach algebra with unit 1. We write σ (a) for the spectrum of a ∈ A. The sets of invert- ible elements, nilpotent elements, and quasinilpotent elements (σ (a) = {0}) of A will be denoted by A −1 , A nil , and A qnil , respectively. An element b ∈ A is called the generalized Drazin inverse of a ∈ A if ab = ba, ab 2 = b, a − a 2 b ∈ A qnil . (1.1) If such element b exists, it is denoted by a d , and a is said generalized Drazin invertible. We denote by A d the set of general- ized Drazin invertible elements of A. We recall that a ∈ A d if and only if 0 is not an accumulation point of σ (a) [14]. In this case, either 0 ∈ σ (a) or 0 is an isolated spectral point of a. It is well-known that a d is unique whenever it exists. The spectral idempotent of a ∈ A d corresponding to the set {0} is denoted by a π and it is given by a π = 1 − aa d . We note that a d = a −1 if a ∈ A −1 , and a d = 0 if a ∈ A qnil . The ordinary Drazin inverse is the generalized Drazin inverse for which a − a 2 b ∈ A nil in (1.1). The group inverse of a, denoted by a # , is a special case of the Drazin inverse for which the condition a − a 2 b ∈ A nil in (1.1) is replaced by a = a 2 b. The set A # consists of elements a ∈ A such that a # exists. A problem of great interest in the theory of Drazin inverses is concerned with the Drazin inverse of block matrices. This topic has been studied intensively in contexts varying from complex matrices, to linear bounded operators, and to general Banach algebras. Some authors have developed formulas in terms of the generalized Schur complement under specific con- ditions. The cases in which the generalized Schur complement is nonsingular or equal to zero have been studied in [12,16,18] for complex matrices, and in [15] for elements in a Banach algebra. The case where the generalized Schur complement is ∗ Corresponding author. Tel.: +34 913366943. E-mail addresses: jrobles@fi.upm.es, juan.robles@upm.es (J. Robles), fmartinez@fi.upm.es (M.F. Martínez-Serrano), edopazo@fi.upm.es (E. Dopazo). http://dx.doi.org/10.1016/j.amc.2016.02.057 0096-3003/© 2016 Elsevier Inc. All rights reserved.