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On the Maurey-Pisier and Dvoretzky-Rogers theorems. (English) Zbl 1450.46007
Given $$2 \leq q < \infty$$, a Banach space $$E$$ has cotype $$q$$ whenever there is a constant $$C >0$$ such that for each choice of finitely many vectors $$x_1, \ldots, x_n \in X$$ we have $\Big( \sum_{k=1}^n \|x_k\|^q \Big)^{1/q} \leq C \Big( \int_0^1 \Big\| \sum_{k=1}^n r_k(t) x_k \Big\|^2\Big)^{1/2}\,,$ where $$r_k:[0,1] \to \mathbb{R}$$ denotes the $$k$$-th Rademacher function. Given $$1 \leq p \leq q < \infty$$, a (linear) operator $$T: E \to F$$ in Banach spaces is absolutely $$(q,p)$$-summing whenever there is a constant $$C >0$$ such that, for each choice of finitely many vectors $$x_1, \dots, x_n\in X$$, we have $\Big( \sum_{k=1}^n \|x_k\|^q \Big)^{1/q} \leq C \sup_{\|x^\ast\|_{X^\ast} \leq 1} \Big( \sum_{k=1}^n| x^\ast(x_k) |^p\Big)^{1/p}\,.$ The Maurey-Pisier theorem shows that for every infinite dimensional Banach space $$X$$ the best $$2 \leq a < \infty$$ such that $$X$$ has cotype $$a$$, denoted by $$\operatorname{cot} X$$, equals the best $$2 \leq a < \infty$$ such that the identity $$\operatorname{id}_X$$ on $$X$$ is $$(a,1)$$-summing. Moreover, the Dovoretzky-Rogers theorem tells us that the identity $$\text{id}_X$$ is not absolutely $$(q,p)$$-summing for any infinite dimensional Banach space $$X$$ whenever $$1/p-1/q < 1/2$$, and that this estimate is even sharp. Mainly based on the work of Maurey and Pisier, the main result here is as follows: Let $$X$$ be an infinite dimensional Banach space and $$1 \leq b < \infty$$.
(i)
If $$b \ge (\text{cot} X)^\ast$$, the conjugate index, then $\inf \{ a : \text{$$\operatorname{id}_X$$ is absolutely $$(a,b)$$-summing} \} = \infty.$
(ii)
If $$b < (\operatorname{cot} X)^\ast$$, then $\inf \{ a : \text{$$\operatorname{id}_X$$ is absolutely $$(a,b)$$-summing} \} = \frac{b\,\operatorname{cot}X}{ b + \operatorname{cot}X -b\, \operatorname{cot}X}.$
Note that (i) completes the information given by the DR-theorem, and for $$b=1$$ both statements together form the MP-theorem. As an application the authors prove an extension of a well-known Grothendieck-type theorem of Kwapień for operators from $$\ell_1$$ into $$\ell_p$$ (Theorem 8) which, in fact, is part of a similar full characterization given much earlier in Proposition 5.2 of [G. Bennett, Duke Math. J. 44, 603–639 (1977; Zbl 0389.47015)].
##### MSC:
 46B07 Local theory of Banach spaces 46A32 Spaces of linear operators; topological tensor products; approximation properties 47H60 Multilinear and polynomial operators
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