Linear spaces of nilpotent matrices

We consider several questions on spaces of nilpotent matrices. We present sufficient conditions for triangularizability and give examples of irreducible spaces. We give a necessary and sufficient condition, in terms of the trace, for all linear combinations of a given set of operators to be nilpotent. We also consider the question of the dimension of a space L of nilpotents on Fn. In particular, we give a simple new proof of a theorem due to M. Gerstenhaber concerning the maximal dimension of such spaces.