On the convergence of semiiterative methods to the Drazin inverse solution of linear equations in Banach spaces (Q1915566)

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 \(\nu_1\) 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 \({\mathcal R}(I - T)^{\nu_1}\), are established. If \(c \notin {\mathcal R} (I - T)^{\nu_1}\) 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.