tailieunhanh - Representations for generalized Drazin inverse of operator matrices over a Banach space

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a 2 × 2 operator matrix under certain circumstances, which generalizes and unifies several results in the literature. | Turk J Math (2016) 40: 428 – 437 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Representations for generalized Drazin inverse of operator matrices over a Banach space Daochang ZHANG1,2,∗ College of Sciences, Northeast Dianli University, Jilin, . China 2 School of Mathematics, Jilin University, Changchun, . China 1 Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a 2 × 2 operator matrix under certain circumstances, which generalizes and unifies several results in the literature. Key words: Generalized Drazin inverse, operator matrix, Banach space 1. Introduction The concept of the generalized Drazin inverse (GD-inverse) in a Banach algebra was introduced by Koliha [21]. Let B be a complex unital Banach algebra. An element a of B is generalized Drazin invertible in the case that there is an element b ∈ B satisfying ab = ba, bab = b, and a − a2 b is quasinilpotent. Such b , if it exists, is unique; it is called a generalized Drazin inverse of a and will be denoted by ad . Then the spectral idempotent aπ of a corresponding to 0 is given by aπ = 1 − aad . The GD-inverse was extensively investigated for matrices over complex Banach algebras and matrices of bounded linear operators over complex Banach spaces. The GD-inverse of the operator matrix has various applications in singular differential equations and singular difference equations, Markov chains and iterative methods, and so on (see [1, 2, 3, 4, 8, 10, 14, 16, 26, 27, 28]). The generalized Drazin inverse is a generalization of Drazin inverses and group inverses. The study of representations for the Drazin inverse of block matrices essentially originated from finding the general expressions for the solutions to singular systems of differential equations [4, 5, 6]. .