## Abstract

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_{k}x_{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}.

Original language | English (US) |
---|---|

Pages (from-to) | 331-356 |

Number of pages | 26 |

Journal | Illinois Journal of Mathematics |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - 2002 |

## ASJC Scopus subject areas

- Mathematics(all)