On cotype and summing properties in banach spaces

Given a complex Banach space X and 2 ≤ q < ∞, we show that X has weak cotype q if and only if there is a constant c > 0 such that ∑_k ∥ x_k∥ ≤ cn^1-1/q sup_ε±1 ∥ ∼_k ε_kx_k∥ holds for all n-dimensional subspaces E ⊂ X and all vectors (x_k)_k ⊂ E. Moreover, these conditions are equivalent to a decrease rate of order k^-1/q for the sequence of eigenvalues of operators on l_∞ 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 > 2. Surprisingly, this 'weak' analog also extends to the case q = 2. Moreover, we show if q > 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{l, log(x)} applied to (log n)^1/q.