Finite order spreading models

Extending the classical notion of spreading model, the kk-spreading models of a Banach space are introduced, for every k∈Nk∈N. The definition, which is based on the kk-sequences and plegma families, reveals a new class of spreading sequences associated to a Banach space. Most of the results of the classical theory are stated and proved in the higher order setting. Moreover, new phenomena like the universality of the class of the 2-spreading models of c0c0 and the composition property are established. As consequence, a problem concerning the structure of the kk-iterated spreading models is solved.