On cotype and summing properties in Banach spaces

Abstract

Given a complex Banach space $X$ and $2\le q \lt \infty$, we show that $X$ has weak cotype $q$ if and only if there is a constant $c \gt 0$ such that \[ \sum_k \| x_k \| \leq c n^{1-{1/q}} \sup_{\varepsilon_k \pm 1} \left\| \sum_k \varepsilon_k x_k \right\| \] holds for all $n$-dimensional subspaces $E\subset X$ and all vectors $(x_k)_k \subset E$. Moreover, these conditions are equivalent to a decrease rate of order $k^{-1/q}$ for the sequence of eigenvalues of operators on $\ell_\infty$ factoring through $X$. This is an analog of Talagrand's theorem on the equivalence of the cotype $q$ property and the absolutely $(q,1)$-summing property for Banach spaces in the range $q \gt 2$. Surprisingly, this `weak' analog also extends to the case $q=2$. Moreover, we show if $q \gt 2$ and $X$ has weak cotype $q$, then the cotype $q$ constant with $n$ vectors can be estimated by any iterates of the function $L(x)=\max\{1,\log(x)\}$ applied to $(\log n)^{1/q}$.