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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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