The G-Drazin Inverse of an Operator Matrix over Banach Spaces

Farzaneh Tayebi; Nahid Ashrafi; Rahman Bahmani; Marjan Sheibani Abdolyousefi

doi:10.5666/KMJ.2024.64.2.205

Article

Kyungpook Mathematical Journal 2024; 64(2): 205-218

Published online June 30, 2024 https://doi.org/10.5666/KMJ.2024.64.2.205

The G-Drazin Inverse of an Operator Matrix over Banach Spaces

Farzaneh Tayebi, Nahid Ashrafi and Rahman Bahmani, Marjan Sheibani Abdolyousefi*

Department of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran
e-mail : ftayebis@gmail.com, nashrafi@semnan.ac.ir and rbahmani@semnan.ac.ir

Farzanegan Campmus, Semnan University, Semnan, Iran
e-mail : m.sheibani@semnan.ac.ir

Received: June 29, 2023; Revised: January 22, 2024; Accepted: January 22, 2024

Abstract

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Abstract

Let A be a Banach algebra. An element a∈A has generalized Drazin inverse if there exists b∈A such that b=bab,ab=ba,a−a2b∈Aqnil.New additive results for the generalized Drazin inverse of an operator over a Banach space are presented. we extend the main results of a paper of Shakoor, Yang and Ali from 2013 and of Wang, Huang and Chen from 2017. Appling these results to 2×2 operator matrices we also generalize results of a paper of Deng, Cvetković-Ilić and Wei from 2010.

Keywords: generalized Drazin inverse, additive property, operator matrix, spectral idempotent