A Study On Zhou Inverses

Generalized inverse theory has been well developed in recent years after being studied by scholars from many directions,such as algebra,analysis,and computational mathematics.It has become an active research field in matrix and operator theory.Not only because of its own high theoretical value,but more importantly,it has a wide range of practical application background in mathematical statistics,system theory,finite Markov process,differential equation systems,population growth model,and optimization control.In this thesis,we mainly study the Zhou inverse on Banach algebra,especially the Zhou inverse of partitioning operator matrix,and construct the corresponding numerical examples.The main innovative works are as follows:Zhou invertibility of elements in Banach algebras is explored,and then Zhou invertibility of operator matrices on Banach spaces is studied.This thesis mainly includes several parts:Chapter 1: This chapter summarizes the research background of generalized inverse and the known research results of relevant scholars,as well as the main research content of this thesis,and introduces the basic concepts and common conclusions involved in this thesis.Chapter 2: In a Banach algebra,the Zhou invertibility of elements is studied,and the existing condition of the Zhou invertibility of the sum of two elements is obtained.The corresponding numerical examples are given to demonstrate the results,which provide the theoretical basis for the subsequent research.Chapter 3: On a Banach space,we investigate the existence of the Zhou inverse of two bounded operators difference,and extend the results to block operator matrix.What’s more,we study the Zhou inverse of the related partitioning operator matrix by using matrix decomposition and Perice decomposition.Chapter 4: On Banach space,the existence of the anti-triangualr partitioning operator matrix Zhou inverse is studied,and a new partitioning operator matrix with Zhou inverse is obtained.Finally,some problems related to this thesis are put forward,and the prospect of related research is forecasted.