Higher Order Spreading Models

We introduce the higher order spreading models associated to a Banach space $X$. Their definition is based on $\text{ff}$-sequences $(x_s{)}_{s\in \text{ff}}$ with $\text{ff}$ a regular thin family and the plegma families. We show that the higher order spreading models of a Banach space $X$ form an increasing transfinite hierarchy $({\mathcal{S}\mathcal{M}}_{\xi }(X){)}_{\xi <{\omega }_1}$. Each ${\mathcal{S}\mathcal{M}}_{\xi }(X)$ contains all spreading models generated by $\text{ff}$-sequences $(x_s{)}_{s\in \text{ff}}$ with order of $\text{ff}$ equal to $\xi$. We also provide a study of the fundamental properties of the hierarchy.