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 is an element of B ( X )( d) . If EFEF pi = 0 and F (2) EF pi = 0, we prove that M =(E (F) I (0) )has g-Drazin inverse and F 0 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 Mosic<acute accent> (Filomat, 32(2018), 5907-5917) and Zou, Chen and Mosic<acute accent> (Studia Scient. Math. Hungar., 54(2017), 489-508).