We provide representations for the generalized Drazin inverse of an anti-triangular matrix of the form ( arraycc a&b\\ 1&0 array ) in a Banach algebra A, under the condition that $ab=ba$. Specifically, we present the representation of Drazin inverse for these types of anti-triangular matrices in Banach algebras.
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Huanyin Chen (2024) studied this question.
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