A problem on spreading models

Abstract

It is proved that if a Banach space $X$ has a basis $(e_n)$ satisfying every spreading model of a normalized block basis of $(e_n)$ is 1-equivalent to the unit vector basis of ${\ell }_1$ (respectively, $c_0$) then $X$ contains ${\ell }_1$ (respectively, $c_0$). Furthermore Tsirelson's space $T$ is shown to have the property that every infinite dimensional subspace contains a sequence having spreading model 1-equivalent to the unit vector basis of ${\ell }_1$. An equivalent norm is constructed on $T$ so that $\Vert s_1+s_2\Vert <2$ whenever $(s_n)$ is a spreading model of a normalized basic sequence in $T$.

References in corpus (3)

• Complexity of weakly null sequences
• Distorting Mixed Tsirelson Spaces
• On asymptotic properties of Banach spaces under renormings