The Representation And Approximation Theorems For The W-weighted Drazin Inverse Of Linear Operator

Operator theory is an important branch in functional analysis,and have wide applications in many fields,such as integral equations,differential equations,numerical analysis and so on.Generally,an operator is singular,as the different circumstances as were concerned,the various generalized inverses were defined.Especially,the Drazin inverse is useful in backward projection problems,such as the recovery of past states of a system from a given state,if the system can be modeled by a linear operator with finite ascent and descent.In recent years,with the study for Drazin inverse of matrices ,the theory for the Drazin inverse of linear operator in Banach space and Hilbert space had developed as well.In this paper,we further studied the W-Weighted Drazin inverse in Banach space and Hilbert space.First of all,we present the new presentation theory of W-weighted Drazin inverse in Banach space.And then,by the property of Hilbert space,we present another presentation theory of W-weighted Drazin inverse.Finally,we employ the spectral theory present the approximation theorem for the new W-weighted Drazin inverse in Hilbert space.The iterative methods such as Euler-Knopp method, Newton method,Newton interpolation method,and Hermite method are considered, and the iterative error bounds for each method were given.