Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices.

Xiaoji Liu,Xiaolan Qin

doi:10.1155/2015/156934

Abstract

We investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a, b ∈ A. As an application we give some new representations for the generalized Drazin inverse of an operator matrix.

Highlights

Let A be a complex Banach algebra with unite 1
The sets of all nilpotent and quasinilpotent elements (σ(a) = {0}) of A will be denoted by Anil and Aqnil, respectively
The set of all generalized Drazin invertible elements of A is denoted by Ad

Summary

Introduction

Let A be a complex Banach algebra with unite 1. The sets of all nilpotent and quasinilpotent elements (σ(a) = {0}) of A will be denoted by Anil and Aqnil, respectively. The generalized Drazin inverse of a ∈ A (introduced by Koliha in [1]) is the element b ∈ A which satisfies xax = x, ax = xa, a − a2x ∈ Aqnil. The set of all generalized Drazin invertible elements of A is denoted by Ad. For interesting properties of the generalized. Relative to p = aad, where a1 is invertible in the algebra pAp, ad is its inverse in pAp, and a2 is quasinilpotent in the algebra pAp. the generalized Drazin inverse of a can be expressed as ad = [a01d 00]p. We will apply these formulas to provide some representations for the generalized Drazin inverse of the operator matrix M =

Main Results

Applications

Conflict of Interests