The Iterative Methods For Computing The Generalized Inverse Of Operators On Banach Space

In this note, we mainly study the iterative methods for computing the generalizedinverse of the bounded linear operators between Banach spaces. The paper is organizedas follows:In chapter 1, we give some required preparative knowledge including general sym-bols, definitions and basic lemmas. Also, the main results of the note are presented.In chapter 2, we present the iterative methods for computing the generalized inverseA_T1S⁽²⁾ over Banach spaces, and also for computing the generalized Drazin inverses ofBanach algebra elements, reveal the intrinsic relationship between the convergence ofsuch iterations and the existence of A_T1S⁽²⁾ ( a^d). Moreover, the necessary and su?cientconditions for iterative convergence are given, and the error bounds of the iterativemethods are presented. Finally, the choice of the scalar in the iteration is considered.And we extend the results of [3] and [4] to infinite dimensional spaces.In chapter 3, existence equivalent conditions, expressions and characterizations forthe weighted group inverse A_W^# of operator A are given by using the technique of blockoperator matrix, three iterative methods for computing A_W^# are established, and thenecessary and su?cient conditions for iterative convergence to A_W^# are presented.In chapter 4, a splitting and its relative iteration for calculating the W-weightedDrazin inverse of linear operators on Banach space are presented, and the necessary andsu?cient conditions for the convergence of the iteration is given.