Higher Order Spreading Models

We introduce the higher order spreading models associated to a Banach space $X$. Their definition is based on $\ff$-sequences $(x_s)_{s\in\ff}$ with $\ff$ 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 $(\mathcal{SM}_ξ(X))_{ξ<ω_1}$. Each $\mathcal{SM}_ξ(X)$ contains all spreading models generated by $\ff$-sequences $(x_s)_{s\in\ff}$ with order of $\ff$ equal to $ξ$. We also provide a study of the fundamental properties of the hierarchy.

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