On the Maurey–Pisier and Dvoretzky–Rogers Theorems

Abstract

A famous theorem due to Maurey and Pisier asserts that for an infinite dimensional Banach space E, the infumum of the q such that the identity map $id_{E}$ is absolutely $\left( q,1\right)$-summing is precisely $\cot E$. In the same direction, the Dvoretzky–Rogers Theorem asserts $id_{E}$ fails to be absolutely $\left( p,p\right)$-summing, for all $p\ge 1$. In this note, among other results, we unify both theorems by charactering the parameters q and p for which the identity map is absolutely $\left( q,p\right)$-summing. We also provide a result that we call strings of coincidences that characterize a family of coincidences between classes of summing operators. We illustrate the usefulness of this result by extending a classical result of Diestel, Jarchow and Tonge and the coincidence result of Kwapień.