The generalized Drazin inverse of anti-triangular block-operator matrices

Abstract

In this paper we investigate the g-Drazin invertibility of an anti-triangular block-operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ with $F^{\pi}EF^d=0$ and $F^{\pi}EF^iE=0$ for all $i\in {\mathbb N}$. This generalizes the main results of [Guo, Zou and Chen, Hacet. J. Math. Stat., 49(3)(2020), 1134-1149] and [Chen and Sheibani, Appl. Math. Comput., 463(2024), 128368 (12 pp)] to a wider case.

Keywords

• g-Drazin inverse
• block-operator matrix
• bounded linear operator
• spectral idempotent

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