The minimal and maximal operator ideals associated to \((n+1)\)-tensor norms of Michor's type
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)]. In the quasi normed case $s_{\mathbf{r}} < 1$, the Minkowski gauge functional of the convex set of all $z$ with $\alpha_{\mathbf{r}}(z) \leq 1$ defines a norm $\alpha^C_{\mathbf{r}}$ on $\otimes_{j=1}^{n+1} E_j$, which for $n=1$ was first investigated by \textit{P. W. Michor} [Functors and categories of Banach spaces. Tensor products, operator ideals and functors on categories of Banach spaces. Cham: Springer (1978; Zbl 0369.46069)]. This leads to an $(n+1)$-fold tensor norm $\alpha^C_{\mathbf{r}}$ in normed spaces, and the main aim then is to characterize their associated $\alpha^C_{\mathbf{r}}$-integral operators $T: \prod_{j=1}^n E_j \to E_{n+1}$ through a natural factorization theorem. \par Like for some classical Banach operator ideals ($n=1$), the author in this interesting article in a first step isolates the prototypical examples of such $\alpha^C_{\mathbf{r}}$-integral operators, and succeeds then in a second step (in a real tour de force) to show that each $\alpha^C_{\mathbf{r}}$-integral operator in fact factorizes in a natural way through one of these protypes.
scientific article; zbMATH DE number 6946893
scientific article