The Pseudo Drazin inverses in Banach Algebras

Let A be a complex Banach algebra and J be the Jacobson radical of A . (1) We firstly show that a is generalized Drazin invertible in A if and only if a+ J is generalized Drazin invertible in A /J. Then we prove that a is pseudo Drazin invertible in A if and only if a+ J is Drazin invertible in A /J. As its application, the pseudo Drazin invertibility of elements in a Banach algebra is explored. (2) The pseudo Drazin order is introduced in A . We give the necessary and sufficient conditions under which elements in A have pseudo Drazin order, then we prove that the pseudo Drazin order is a pre-order. AMS subject classifications: 15A09, 16U90, 46H05