The generalized Drazin inverse M d of a 2 Â 2 operator matrixare generalized Drazin invertible. Expressions for the generalized Drazin inverse M d of operator matrix M in terms of the individual blocks A, B, C, D, A d and D d are derived under some conditions.

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In this paper, we investigate additive properties of generalized Drazin inverse of two Drazin invertible linear operators in Banach spaces. Under the commutative condition of P Q = Q P, we give explicit representations of the generalized Drazin inverse (P + Q ) d in term of P , P d , Q and Q d . We consider some applications of our results to the perturbation of the Drazin inverse and analyze a number of special cases.

In this paper, several equivalent conditions for the Drazin invertibility of the sum, difference and the product of idempotents in Banach algebra are established.

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