On cotype and summing properties in banach spaces

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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 ∥ xk∥ ≤ cn1-1/q supε±1 ∥ ∼k εkxk∥ holds for all n-dimensional subspaces E ⊂ X and all vectors (xk)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.

Original languageEnglish (US)
Pages (from-to)331-356
Number of pages26
JournalIllinois Journal of Mathematics
Issue number2
StatePublished - 2002

ASJC Scopus subject areas

  • Mathematics(all)

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