On the number of non permutatively equivalent sequences in a Banach space

Abstract

This paper contains results concerning the Borel reduction of the relation $E_0$ of eventual agreement between sequences of 0's and 1's, to the relation of permutative equivalence between basic sequences in a Banach space. For more clarity in this abstract, we state these results in terms of classification by real numbers. If $R$ is some (analytic) equivalence relation on a Polish space $X$, it is said that $R$ is classifiable (by real numbers) if there exists a Borel map $g$ from $X$ into the real line such that $x R x'$ if and only if $g(x)=g(x')$. If $R$ is not classifiable, there must be $2^{\omega}$ $R$-classes. It is conjectured that any separable Banach space such that isomorphism between its subspaces is classifiable must be isomorphic to $l_2$. We prove the following results: - the relation $\sim^{perm}$ of permutative equivalence between normalized basic sequences is analytic non Borel, - if $X$ is a Banach space with a Schauder basis $(e_n)$, such that $\sim^{perm}$ between normalized block-sequences of $X$ is classifiable, then $X$ is $c_0$ or $\ell_p$ saturated for some $1 \leq p <+\infty$, - if $(e_n)$ is shrinking unconditional, and $\sim^{perm}$ between normalized disjointly supported sequences in $X$, resp. in $X^*$, are classifiable, then $(e_n)$ is equivalent to the unit vector basis of $c_0$ or $\ell_p$, - if $(e_n)$ is unconditional, then either $X$ is isomorphic to $l_2$, or $X$ contains $2^{\omega}$ subspaces or $2^{\omega}$ quotients which are spanned by pairwise non permutatively equivalent normalized unconditional bases.