Remarks on the norming sets of ${\mathcal L}(^ml_{1}^n)$ and description of the norming sets of ${\mathcal L}(^3l_{1}^2)$

Abstract

Let $n\in \mathbb{N}, n\geq 2.$ An element $x=(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and
$|T(x)|=\|T\|,$ where ${\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$
For $T\in {\mathcal L}(^n E)$ we define the {\em norming set} of $T$

$\mathop{Norm}(T)=\Big\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\Big\}.$

By $i=(i_1,i_2,\ldots,i_m)$ we denote the multi-index. In this paper we show the following:

\noi (a) Let $n, m\geq 2$ and let $l_1^n=\mathbb{R}^n$ with the $l_1$-norm. Let $T=\big(a_{i}\big)_{1\leq i_k\leq n}\in {\mathcal L}(^ml_{1}^n)$ with $\|T\|=1.$
Define $S=\big(b_{i}\big)_{1\leq i_k\leq n}\in {\mathcal L}(^n l_1^m)$ be such that $b_{i}=a_{i}$ if
$|a_{i}|=1$ and $b_{i}=1$ if
$|a_{i}|<1.$

Let $A=\{1, \ldots, n\}\times \cdots\times\{1, \ldots, n\}$ and $M=\{i\in A: |a_{i}|<1\}.$
Then,

$\mathop{Norm}(T)=\bigcup_{(i_1, \ldots, i_m)\in M}\Big\{\Big(\big(t_1^{(1)}, \ldots, t_{{i_1}-1}^{(1)}, 0, t_{{i_1}+1}^{(1)}, \ldots, t_{n}^{(1)}\big), \big(t_1^{(2)}, \ldots, t_{n}^{(2)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big),$

$\Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \big(t_1^{(2)}, \ldots, t_{{i_2}-1}^{(2)}, 0, t_{{i_2}+1}^{(2)}, \ldots, t_{n}^{(2)}\big), \big(t_1^{(3)}, \ldots, t_{n}^{(3)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big),\ldots$

$\ldots, \Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \ldots, \big(t_1^{(m-1)}, \ldots, t_{n}^{(m-1)}\big), \big(t_1^{(m)}, \ldots, t_{{i_m}-1}^{(m)}, 0, t_{{i_m}+1}^{(m)}, \ldots, t_{n}^{(m)}\big)\Big)\colon$

$\Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big)\in \mathop{Norm}(S)\Big\}.$

This statement extend the results of [9].

\noi (b) Using the result (a), we describe the norming sets of every $T\in {\mathcal L}(^3l_{1}^2).$