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

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).