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

We consider general semiiterative methods (\textbf{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 \textbf{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.