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 $i{d}_E$ is absolutely $\left(q,1\right)$-summing is precisely $\cot E$. In the same direction, the Dvoretzky--Rogers Theorem asserts $i{d}_E$ 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 \textit{strings of coincidences} that characterize a family of coincidences between classes of summing operators. We illustrate the usefulness of this result by extending classical result of Diestel, Jarchow and Tonge and the coincidence result of Kwapień.