On the Maurey–Pisier and Dvoretzky–Rogers Theorems

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 ( q,1) -summing is precisely E . In the same direction, the Dvoretzky–Rogers Theorem asserts id_E fails to be absolutely ( p,p) -summing, for all p≥ 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 ( q,p) -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.