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 $E,F,EF^{\pi}\in \mathcal{B}(X)^d$. If $EFEF^{\pi}=0$ and $F^2EF^{\pi}=0$, we prove that $M=\left( \begin{array}{cc} E&I F&0 \end{array} \right)$ 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 $FEF^{\pi}=0$. The group inverse for the anti-triangular block-operator matrices with two identical subblocks is thereby investigated. These extend the results of Zhang and Mosi\'c (Filomat, 32(2018), 5907--5917) and Zou, Chen and Mosi\'c (Studia Scient. Math. Hungar., 54(2017), 489--508).