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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.

On the Maurey–Pisier and Dvoretzky–Rogers Theorems