Research Of The Generalized N-strong Drazin Inverses In Rings And Banach Algebras

In this thesis,as a kind of classical generalized inverse,Drazin inverse plays an important role in many research fields.Strong Drazin inverse,Hirano inverse,n-strong Drazin inverse and generalized n-strong Drazin inverse derived from Drazin inverse also promote the development of generalized inverse theory.This paper focuses on generalized n-strong Drazin inverse in rings and Banach algebras.The main contents of this paper are as follows:In Chapter 2,we study the properties of the n-strong Drazin inverse in rings.Firstly,the relation between n-strong Drazin inverse and strong Drazin inverse,as well as the equivalent characterization of n-strong Drazin inverse in rings are studied.Secondly,n-strong Drazin invertibility of the product and difference of idempotents in a ring is investigated.Finally,the n-strong Drazin invertibility of the product paq are considered,where a is regular and p,q are arbitrary elements in a ring.In Chapter 3,we consider the generalized n-strong Drazin inverse in Banach algebras.Firstly,the equivalent characterization of generalized n-strong Drazin inverses in Banach algebras is given,which is a generalization of the n-strong Drazin inverse in rings.Then,the necessary and sufficient conditions for the reverse order law of generalized n-strong Drazin inverses in Banach algebras are considered.