This is an announcement for the paper “On the Maurey--Pisier and Dvoretzky--Rogers theorems” by Gustavo Araújohttps://arxiv.org/search/math?searchtype=author&query=Ara%C3%BAjo%2C+G, Joedson Santoshttps://arxiv.org/search/math?searchtype=author&query=Santos%2C+J.
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 $p\geq1$. 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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1811.09183