{"title":"Remarks on the norming sets of ${\\mathcal L}(^ml_{1}^n)$ and description of the norming sets of ${\\mathcal L}(^3l_{1}^2)$","authors":"Sung Guen Kim","doi":"10.30970/ms.58.2.201-211","DOIUrl":null,"url":null,"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 $T$ \n\\centerline{$\\qopname\\relax o{Norm}(T)=\\Big\\{(x_1, \\ldots, x_n)\\in E^n: (x_1, \\ldots, x_n)~\\mbox{is a norming point of}~T\\Big\\}.$} \nBy $i=(i_1,i_2,\\ldots,i_m)$ we denote the multi-index. In this paper we show the following: \n\\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$|a_{i}|<1.$ \nLet $A=\\{1, \\ldots, n\\}\\times \\cdots\\times\\{1, \\ldots, n\\}$ and $M=\\{i\\in A: |a_{i}|<1\\}.$Then, \n\\centerline{$\\qopname\\relax o{Norm}(T)=\\bigcup_{(i_1, \\ldots, i_m)\\in M}\\Big\\{\\Big(\\big(t_1^{(1)}, \\ldots, t_{{i_1}-1}^{(1)}, 0, t_{{i_1}+1}^{(1)}, \\ldots, t_{n}^{(1)}\\big), \\big(t_1^{(2)}, \\ldots, t_{n}^{(2)}\\big), \\ldots, \\big(t_1^{(m)}, \\ldots, t_{n}^{(m)}\\big)\\Big),$} \n\\centerline{$\\Big(\\big(t_1^{(1)}, \\ldots, t_{n}^{(1)}\\big), \\big(t_1^{(2)}, \\ldots, t_{{i_2}-1}^{(2)}, 0, t_{{i_2}+1}^{(2)}, \\ldots, t_{n}^{(2)}\\big), \\big(t_1^{(3)}, \\ldots, t_{n}^{(3)}\\big), \\ldots, \\big(t_1^{(m)}, \\ldots, t_{n}^{(m)}\\big)\\Big),\\ldots$} \n\\centerline{$\\ldots, \\Big(\\big(t_1^{(1)}, \\ldots, t_{n}^{(1)}\\big), \\ldots, \\big(t_1^{(m-1)}, \\ldots, t_{n}^{(m-1)}\\big), \\big(t_1^{(m)}, \\ldots, t_{{i_m}-1}^{(m)}, 0, t_{{i_m}+1}^{(m)}, \\ldots, t_{n}^{(m)}\\big)\\Big)\\colon$} \n\\centerline{$ \\Big(\\big(t_1^{(1)}, \\ldots, t_{n}^{(1)}\\big), \\ldots, \\big(t_1^{(m)}, \\ldots, t_{n}^{(m)}\\big)\\Big)\\in \\qopname\\relax o{Norm}(S)\\Big\\}.$} \nThis statement extend the results of [9]. \n\\noi (b) Using the result (a), we describe the norming sets of every $T\\in {\\mathcal L}(^3l_{1}^2).$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.58.2.201-211","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
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 $T$
\centerline{$\qopname\relax o{Norm}(T)=\Big\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\Big\}.$}
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,
\centerline{$\qopname\relax o{Norm}(T)=\bigcup_{(i_1, \ldots, i_m)\in M}\Big\{\Big(\big(t_1^{(1)}, \ldots, t_{{i_1}-1}^{(1)}, 0, t_{{i_1}+1}^{(1)}, \ldots, t_{n}^{(1)}\big), \big(t_1^{(2)}, \ldots, t_{n}^{(2)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big),$}
\centerline{$\Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \big(t_1^{(2)}, \ldots, t_{{i_2}-1}^{(2)}, 0, t_{{i_2}+1}^{(2)}, \ldots, t_{n}^{(2)}\big), \big(t_1^{(3)}, \ldots, t_{n}^{(3)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big),\ldots$}
\centerline{$\ldots, \Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \ldots, \big(t_1^{(m-1)}, \ldots, t_{n}^{(m-1)}\big), \big(t_1^{(m)}, \ldots, t_{{i_m}-1}^{(m)}, 0, t_{{i_m}+1}^{(m)}, \ldots, t_{n}^{(m)}\big)\Big)\colon$}
\centerline{$ \Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big)\in \qopname\relax o{Norm}(S)\Big\}.$}
This statement extend the results of [9].
\noi (b) Using the result (a), we describe the norming sets of every $T\in {\mathcal L}(^3l_{1}^2).$