A PROBLEM ON SPREADING MODELS

It is proved that if a Banach space X has a basis (e(n)) satisfying ev ery spreading model of a normalized block basis of (e(n)) is 1-equival ent to the unit vector basis of l(1) (respectively, c(0)) then X conta ins l(1) (respectively, c(0)) Furthermore, Tsirelson's space T is show n to have the property that every infinite dimensional subspace contai ns a sequence having spreading model 1-equivalent to the unit vector b asis of l(1). An equivalent norm is constructed on T so that \\s(1) s(2)\\ < 2 whenever (s(n)) is a spreading model of a normalized basic sequence in T. (C) 1998 Academic Press.