(untitled) (Communications in Mathematical Research, 2022)
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.