Generalized invertibility in two semigroups of Banach algebras

Motivated by the results involving Drazin inverses of Patricio and Puystjens, we investigate the relevant results for pseudo Drazin invertibility and generalized Drazin invertibility in two semigroups of Banach algebras. Given a Banach algebra A and e2 = e E A, we firstly establish the relation between pseudo Drazin invertibility (resp., generalized Drazin invertibility) of elements in eAe and eAe + 1 - e. Then this result leads to a remarkable behavior of pseudo Drazin invertibility (resp., generalized Drazin invertibility) between the operators in the semigroup AA-g(Y )AA- + IY - AA- and the semigroup A=Ag(X)A=A+ IX -A=A, where A-, A= E g(Y, X) are inner inverses of A E g(X, Y ).