On the structure of the spreading models of a Banach space

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain ${\mathrm{\ell }}_1$ but none of them is isomorphic to ${\mathrm{\ell }}_1$. We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$. In certain cases this ensures that $X$ admits, for each $\alpha < \omega_1$,aspreadingmodel$(\tilde x_i^\alpha)_i$suchthatif$\alpha < \beta$then$(\tilde x_i^\alpha)_i$isdominatedby(andnotequivalentto)$(\tilde x_i^\beta)_i$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.