The minimal and maximal operator ideals associated to \((n+1)\)-tensor norms of Michor's type
type: Item
instance of: scholarly article
title: The minimal and maximal operator ideals associated to \((n+1)\)-tensor norms of Michor's type
zbMATH Open document ID: 1416.46069
author: Yanyan Li
published in: Positivity
publication date: 2018-10-04T00:00:00Z
review text: Extending the classical case $n=1$, the author continues his work on the description of multilinear operators $T: \prod_{j=1}^n E_j \to E_{n+1}$, defined by certain $(n+1)$-fold tensor norms. To get some idea, let us briefly describe a prototypical result first discovered in \textit{A. Grothendieck}'s Résumé [Bol. Soc. Mat. São Paulo 8, 1--79 (1956; Zbl 0074.32303)]. Grothendieck defined an operator $T:E_1 \to E_2$ to be integral whenever the linearization of the bilinear form $(x, y') \to y'(Tx)$ gives a bounded functional on the injective tensor product $E_1 \otimes_\varepsilon E_2'$. Each embedding $L_\infty(\mu) \hookrightarrow L_1(\mu)$, where $\mu$ is some probability measure, is integral, and (in a sense conversely) $T:E_1 \to E_2$ is integral if and only if $T$, considered as an operator with values in $E_2''$, factorizes through such an embedding. These ideas were taken up by many authors, among them Kwapień, Lapresté, Maurey, Pietsch, and Saphar, in order to classify some Banach operator ideals of fundamental importance in terms of factorization through their prototypical examples.\par Let $ \mathbf{r} = (r_0, r_1, \ldots, r_{n}, r_{n+1})$ be such that $1 \leq r_0 \leq \infty$, $1 < r_j < \infty$, $1\leq j \leq n+1$, and $s_{\mathbf{r}} := 1/r_0 - \sum_{j=1}^{n+1}1/r_j$. Moreover, given $z \in \bigotimes_{j=1}^n E_j$, let $$\alpha_{\mathbf{r}}(z) = \inf \big(\sum_{k=1}^h|\lambda_k|^{r_0}\big)^{1/r_0} \prod_{j=1}^{n+1}\sup_{\|x_k'\|\leq 1} \big(\sum_{k=1}^h|x_k'(x_k^j)|^{r_j'} \big)^{1/r_j'}\, $$ where the infimum is taken over all representation $z = \sum_{k=1}^h \lambda_k \bigotimes_{j=1}^{n+1} x_k^j $. For $s_{\mathbf{r}}=1$, this gives an $(n+1)$-tensor norm, and the relevant factorization theorems of $\alpha_{\mathbf{r}}$-integral multilinear operators were proved in [\textit{J. A. López Molina}, Glasg. Math. J. 54, No. 3, 665--692 (2012; Zbl 1255.46037)], and earlier in [\textit{J. A. López Molina}, Positivity 11, No. 1, 95--117 (2007; Zbl 1128.46028)…
reviewed by: Andreas Defant
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 47J22
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 6946893
zbMATH Keywords: multilinear tensor products
zbMATH Keywords: multilinear nuclear operators
zbMATH Keywords: multilinear integral operators
zbMATH Keywords: ultraproducts
MaRDI profile type: Publication
full work available at URL: s11117-018-0563-8
OpenAlex ID: W2791151565
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DOI: 10.1007/S11117-018-0563-8
description: scientific article; zbMATH DE number 6946893