Equivalence of Conditions for Convergence of Iterative Methods for Singular Linear Systems

Dedicated to Professor Ivo Marek on the occasion of his sixtieth birthday. Equivalence is shown between different conditions for convergence of iterative methods for consistent singular systems of linear equations on Banach spaces. These systems appear in many applications, such as Markov chains and Markov processes. The conditions considered relate the range and null spaces of different operators. Let E be a Banach space and let A be a bounded linear operator on E. For the solution of equations of the form Au = f, (1) it is customary to consider a splitting A = M −N, where M is bounded and invertible, and iterative methods of the form u k+1 = T u k + M −1 f, k = 0, 1, · · · , (2) where T = M −1 N. In this article we consider solutions of (1) where A is a singular operator and f is such that a solution exists, i.e., consistent 1