On necessary conditions for convergence of stationary iterative methods for Hermitian semidefinite linear systems

In an earlier paper (Frommer et al. (2008) [5]) we gave sufficient conditions in terms of an energy seminorm for the convergence of stationary iterations for solving linear systems whose coefficient matrix is Hermitian and positive semidefinite. In this paper we show in which cases these conditions are also necessary, and show that they are not necessary in others.