On the convergence of semiiterative methods to the Drazin inverse solution of linear equations in Banach spaces
N. Castro González · 1995
We consider general semiiterative methods (SIMs) to find approximate solutions of singular linear equations of the type x = Tx + c, where T is a bounded linear operator on a complex Banach space X such that its resolvent has a pole of order ν1 at the point 1. Necessary and sufficient conditions for the convergence of SIMs to a solution of x = Tx+ c, where c belongs to the subspace range R(I − T) ν1 ν1, are established. If c ∈ R(I − T) sufficient conditions for the convergence to the Drazin inverse solution are described. For the class of normal operators in a Hilbert space, we analyze the convergence to the minimal norm solution and to the least squares minimal norm solution. 1.