G-Drazin inverse and group inverse for the anti-triangular block-operator matrices

Abstract

We present the generalized Drazin inverse for certain anti-triangular operator matrices. Let $E,F,E{F}^{\pi }\in \mathcal{B}(X{)}^d$. If $EFE{F}^{\pi }=0$ and $F^{2}E{F}^{\pi }=0$, we prove that $M=\left(\begin{array}{ccc}E& IF& 0\end{array}\right)$ has g-Drazin inverse and its explicit representation is established. Moreover, necessary and sufficient conditions are given for the existence of the group inverse of $M$ under the condition $FE{F}^{\pi }=0$. The group inverse for the anti-triangular block-operator matrices with two identical subblocks is thereby investigated. These extend the results of Zhang and Mosić (Filomat, 32(2018), 5907--5917) and Zou, Chen and Mosić (Studia Scient. Math. Hungar., 54(2017), 489--508).

arXiv admin note: substantial text overlap with arXiv:2203.09086