an:07190872
Zbl 1450.46007
Ara??jo, Gustavo; Santos, Joedson
On the Maurey-Pisier and Dvoretzky-Rogers theorems
EN
Bull. Braz. Math. Soc. (N.S.) 51, No. 1, 1-9 (2020).
00448506
2020
j
46B07 46A32 47H60
absolutely summing operators; Maurey-Pisier theorem; Dvoretzky-Rogers theorem
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 \(1 \leq b < \infty\).
\begin{itemize} \item[(i)] If \(b \ge (\text{cot} X)^\ast\), the conjugate index, then
\[
\inf \{ a : \text{\(\operatorname{id}_X\) is absolutely \((a,b)\)-summing} \} = \infty.
\]
\item[(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}.
\]
\end{itemize}
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 Kwapie?? 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 [\textit{G. Bennett}, Duke Math. J. 44, 603--639 (1977; Zbl 0389.47015)].
Andreas Defant (Oldenburg)
Zbl 0389.47015; Zbl 0036.36303