Group inverse for anti-triangular block operator matrices

TL;DR

This paper investigates the existence and representation of the group inverse for certain anti-triangular block operator matrices, extending previous results in the mathematical literature under specific conditions.

Contribution

It provides new conditions and explicit representations for the group inverse of anti-triangular block matrices with identical subblocks, generalizing prior work.

Findings

Established the existence of the group inverse under the condition $FEF^{}$.

Derived explicit formulas for the group inverse of the matrices considered.

Extended previous results to a broader class of block matrices.

Abstract

We present the existence of the group inverse and its representation for the block operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ under the condition $FEF^{\pi}=0$. The group inverse for the anti-triangular block matrices with two identical subblocks under the same condition is thereby investigated. These extend the results of Zou, Chen and Mosi\'c (Studia Scient. Math. Hungar., 54(2017), 489--508), and Cao, Zhang and Ge (J. Appl. Math. Comput., 46(2014), 169--179).