Researches On Generalized Inverses Of Banach Algebraic Elements And Metric Generalized Inverses

The researches of this thesis mainly divide into two parts: The first part is con-cerned with some problems about the generalized inverses of Banach algebraic elements, we mainly study some kinds of (p,q)-generalized inverses, these results are included in Chapter2and Chapter3; The second one is devoted to the nonlinear operator generalized inverses in Banach spaces, we mainly investigate the homogeneous operator generalized inverse, the Moore-Penrose metric generalize inverse and its applications, these contain Chapter4and Chapter5. Our main results are as follows:Let A be a complex Banach algebra with the unit1. Let α∈A. Let p, q∈A be idempotent elements. In Chapter2, we first define a new generalized inverse αp,q(2,l) with prescribed idempotents p, q for the element a. Then, some new characterizations and explicit representations for these generalized inverses, such as αp,q,(2) αp,q(1,2) and αp,q(2,l) are presented; Let δα∈A, let p’, q’∈A be idempotent elements. Set α=α+δα∈A. In Chapter3, we first study the stable perturbations and expressions for the generalized inverses of perturbed element α, then, by using the gap function and idempotent elements, we investigate the error estimate for and when the elements p, q and α all have some small perturbations.Let X, Y be Banach spaces. Let T,δT: X�'Y be bounded linear operators from X to Y. Set T=T+δT. In Chapter4, we first characterize the existence of a homogeneous generalized inverse Th of the bounded linear operator T. Then, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse Th and quasi-linear projector generalized inverse TH of T. By using the property of Th. we also give a representation: TM=(Ix-πN(T))ThπR(T);In Chapter5, by means of smooth geometric assumptions of Bauach spaces, we first give some equivalent conditions for the Moore-Penrose metric generalized inverse of perturbed operator TM to have the simplest expression TM(IY+δTTM)-1. Then, by means of gap function, we obtain some error estimations of‖TN-TM‖in the Banach space Lp(Ω,μ). Simple results related to the existence and perturbed bound of TM are also given in a general reflexive and strictly convex Banach space. Finally, by using the Moore-Penrose metric generalized inverse, we give a further study on the stability of the best approximate solutions of the operator equation Tx=b. and also get the perturbed bound of where b＝b+δb∈Y.