The Uniform Boundedness And Convergence For The Core Inverses Of Linear Operators In Banach Space

Generalized inverse was first put forward by I.Fredholm in 1903.Nowadays,generalized inverse theory has lots of applications in many fields,such as numerical analysis,optimization,mathematical statistics and linear equation.Core inverse is an important generalized inverse.One of the main topic of this thesis is the relationship between uniform boundedness and convergence for the core inverses of linear operators in Banach spaces,another one is the relationship between the stable perturbation and the continuity of the core inverses,and the last one is the expression of the core inverses.The thesis is organized as follows.In Chapter 1,we give some introduction and preliminaries which will be used in the thesis.In Chapter 2,we study the relationship between uniform boundedness and convergence for the core inverses of linear operators in Banach spaces.Firstly,in the case of the existence of Tn(?),we prove that {Tn(?)} is a Cauchy sequence.It implies that the equivalence between uniform boundedness and convergence of the core inverses.Based on this,we prove that T(?)is exists.Finally,we use the stable perturbation to give the expression of T(?).In Chapter 3,we investigate the relationship between the stable perturbation and the continuity for the core inverses of linear operators in Banach spaces.Firstly,using the equivalent conditions for stable perturbation,we prove that the stable perturbation of the core inverses is equivalent to the continuity of the core inverses under the condition T(?)is exists.As applications,we give the continuity characterization for the core inverses of finite rank operators with a direct and brief method.Finally,we prove that the perturbed operator is core invertible and have the simplest possible expression if and only if Tn=TT(?)Tn.We also derive the expression for the core inverses of the perturbed operator under the null space-preserving perturbation.