Cotype and summing properties in Banach spaces

Abstract

It is well known in Banach space theory that for a finite dimensional space $E$ there exists a constant $c_E$, such that for all sequences $(x_k)_k \subset E$ one has \[ \summ_k \noo x_k \rrm \kl c_E \pl \sup_{\eps_k \pm 1} \noo \summ_k \eps_k x_k \rrm \pl .\] Moreover, if $E$ is of dimension $n$ the constant $c_E$ ranges between $\sqrt{n}$ and $n$. This implies that absolute convergence and unconditional convergence only coincide in finite dimensional spaces. We will characterize Banach spaces $X$, where the constant $c_E \sim \sqrt{n}$ for all finite dimensional subspaces. More generally, we prove that an estimate $c_E \kll c n^{1-\frac{1}{q}}$holds for all $n \in \nz$ and all $n$-dimensional subspaces $E$ of $X$ if and only if the eigenvalues of every operator factoring through $\ell_{\infty}$ decrease of order $k^{-\frac{1}{q}}$ if and only if $X$ is of weak cotype $q$, introduced by Pisier and Mascioni. We emphasize that in contrast to Talagrand's equivalence theorem on cotype $q$ and absolutely $(q,1)$-summing spaces this extendsto the case $q=2$. If $q>2$ and one of the conditions above is satisfied one has \[ \kla \summ_k \noo x_k \rrm^q \mer^{\frac{1}{q}} \kl C^{1+l}\pl (1+{\rm log}_2)^{(l)}((1 +{\rm log}_2 n)^{\frac{1}{q}}) \pl \ez \noo \summ_k \eps_k x_k \rrm \] for all $n,l \in \nz$ and $(x_k)_k \subset E$, $E$ a $n$ dimensional subspace of $X$. In the case $q=2$ the same holds if we replace the expected value by the supremum.