On Neumann Operators

We give a sufficient condition for the validity of the implication lim nª` T n xs0 « `ns0T n x converges, where T: XªX is a bounded linear operator, X is a Banach space, and xgX. From this condition we derive the results given by other authors. Moreover, we give some properties about the operators that verifies the above implication, which are called Neumann operators.