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 $E,F,EF^{\pi}\in \mathcal{B}(X)^d$‎. ‎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)‎.