On Pseudo Drazin Inverse

The generalized inverse is a very popular research field which has been studied by many researchers.It is widely applied in many fields,such as the differential equations,cybernetics,Markov chains,even artificial intelligence.In this thesis,we study the pseudo Drazin invertibility of the operator matrices,anti-triangular operator matrices and the sum of two elements in a Banach algebra.Using different matrix decompositions,we obtain a series of spectral conditions and generalize the relevant conclusions in Drazin inverse.Some numerical examples are also given to illustrate our results.This thesis includes the following sections:In Chapter 1,we introduce the development of generalized inverse,the background of pseudo Drazin inverse.The main results in the thesis are also stated.In Chapter 2,we introduce some concepts and known results on pseudo Drazin inverse.In Chapter 3,we study the pseudo Drazin invertibility of operator matrices in a Banach space under certain conditions.And we obtain many spectral conditions by using matrix decompositions.Some numerical examples are also given to show our results.In Chapter 4,we investigate the pseudo Drazin invertibility of the sum of two elements in a Banach algebra.Using different matrix decompositions,we derive some additive properties and then apply them to operator matrices.In addition,we try to derive some extensions of our conditions and give related examples to illustrate the results.In Chapter 5,we study the pseudo Drazin invertibility of anti-triangular operator matrices in a Banach algebra.We obtain certain conditions under which the anti-triangular operator matrices have pseudo Drazin inverse.Some examples are also given to show our results.In the final chapter,we summarize the results of this thesis and give some research prospects of pseudo Drazin inverse.