Abstract: Let a,b be two generalized Drazin invertible elements  in a Banach algebra A with unity 1, a^d be the generalized  Drazin inverse of a. The generalized Drazin inverse of the sum and product of two elements in a Banach algebra was considered by means of the system of idempotents. The expressions for the generalized Drazin inverse of the sum and product of a and b were given under some conditions such as ab²=bab, b^πba²=b^πaba, a^db²=ba^db etc.

Key words: generalized Drazin inverse, quasi-nilpotent element, Banach algebra