Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces
For $n\geq 2$ and a real Banach space $E,$ ${\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself. Let $$\Pi(E)=\Big\{[x^*, (x_1, \ldots, x_n)]: x^{*}(x_j)=\|x^{*}\|=\|x_j\|=1~\mbox{for}~{j=1, \ldots, n}\Big\}.$$ For $T\in {\mathcal L}(^n E:E),$ we define $...
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Ivan Franko National University of Lviv
2022-03-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/270 |
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| author | S. G. Kim |
| author_facet | S. G. Kim |
| author_sort | S. G. Kim |
| collection | DOAJ |
| description | For $n\geq 2$ and a real Banach space $E,$ ${\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.
Let $$\Pi(E)=\Big\{[x^*, (x_1, \ldots, x_n)]: x^{*}(x_j)=\|x^{*}\|=\|x_j\|=1~\mbox{for}~{j=1, \ldots, n}\Big\}.$$
For $T\in {\mathcal L}(^n E:E),$ we define $$\qopname\relax o{Nr}({T})=\Big\{[x^*, (x_1, \ldots, x_n)]\in \Pi(E): |x^{*}(T(x_1, \ldots, x_n))|=v(T)\Big\},$$
where $v(T)$ denotes the numerical radius of $T$.
$T$ is called {\em numerical radius peak mapping} if there is $[x^{*}, (x_1, \ldots, x_n)]\in \Pi(E)$ such that $\qopname\relax o{Nr}({T})=\{\pm [x^{*}, (x_1, \ldots, x_n)]\}.$
In this paper, we investigate some class of numerical radius peak mappings in ${\mathcal
L}(^n l_p:l_p)$ for $1\leq p<\infty.$ Let $(a_{j})_{j\in \mathbb{N}}$ be a bounded sequence in $\mathbb{R}$ such that $\sup_{j\in \mathbb{N}}|a_j|>0.$
Define $T\in {\mathcal L}(^n l_p:l_p)$ by
$$T\Big(\sum_{i\in \mathbb{N}}x_i^{(1)}e_i, \cdots, \sum_{i\in \mathbb{N}}x_i^{(n)}e_i \Big)=
\sum_{j\in \mathbb{N}}a_{j}~x_{j}^{(1)}\cdots x_{j}^{(n)}~e_j.\qquad\eqno(*)$$
In particular is proved the following statements:\
$1.$\ If $1< p<+\infty$ then $T$ is a numerical radius peak mapping if and only if there is $j_0\in \mathbb{N}$ such that
$$|a_{j_0}|>|a_{j}|~\mbox{for every}~j\in \mathbb{N}\backslash\{j_0\}.$$
$2.$\ If $p=1$ then $T$ is not a numerical radius peak mapping in ${\mathcal L}(^n l_1:l_1).$ |
| format | Article |
| id | doaj-art-d56ee0c04adb4139976f3fc0fcb0a7d9 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2022-03-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-d56ee0c04adb4139976f3fc0fcb0a7d92025-08-20T03:28:41ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202022-03-01571101510.30970/ms.57.1.10-15270Some class of numerical radius peak $n$-linear mappings on $l_p$-spacesS. G. Kim0Kyungpook National UniversityFor $n\geq 2$ and a real Banach space $E,$ ${\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself. Let $$\Pi(E)=\Big\{[x^*, (x_1, \ldots, x_n)]: x^{*}(x_j)=\|x^{*}\|=\|x_j\|=1~\mbox{for}~{j=1, \ldots, n}\Big\}.$$ For $T\in {\mathcal L}(^n E:E),$ we define $$\qopname\relax o{Nr}({T})=\Big\{[x^*, (x_1, \ldots, x_n)]\in \Pi(E): |x^{*}(T(x_1, \ldots, x_n))|=v(T)\Big\},$$ where $v(T)$ denotes the numerical radius of $T$. $T$ is called {\em numerical radius peak mapping} if there is $[x^{*}, (x_1, \ldots, x_n)]\in \Pi(E)$ such that $\qopname\relax o{Nr}({T})=\{\pm [x^{*}, (x_1, \ldots, x_n)]\}.$ In this paper, we investigate some class of numerical radius peak mappings in ${\mathcal L}(^n l_p:l_p)$ for $1\leq p<\infty.$ Let $(a_{j})_{j\in \mathbb{N}}$ be a bounded sequence in $\mathbb{R}$ such that $\sup_{j\in \mathbb{N}}|a_j|>0.$ Define $T\in {\mathcal L}(^n l_p:l_p)$ by $$T\Big(\sum_{i\in \mathbb{N}}x_i^{(1)}e_i, \cdots, \sum_{i\in \mathbb{N}}x_i^{(n)}e_i \Big)= \sum_{j\in \mathbb{N}}a_{j}~x_{j}^{(1)}\cdots x_{j}^{(n)}~e_j.\qquad\eqno(*)$$ In particular is proved the following statements:\ $1.$\ If $1< p<+\infty$ then $T$ is a numerical radius peak mapping if and only if there is $j_0\in \mathbb{N}$ such that $$|a_{j_0}|>|a_{j}|~\mbox{for every}~j\in \mathbb{N}\backslash\{j_0\}.$$ $2.$\ If $p=1$ then $T$ is not a numerical radius peak mapping in ${\mathcal L}(^n l_1:l_1).$http://matstud.org.ua/ojs/index.php/matstud/article/view/270numerical radius points; numerical radius peak mappings. |
| spellingShingle | S. G. Kim Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces Математичні Студії numerical radius points; numerical radius peak mappings. |
| title | Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces |
| title_full | Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces |
| title_fullStr | Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces |
| title_full_unstemmed | Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces |
| title_short | Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces |
| title_sort | some class of numerical radius peak n linear mappings on l p spaces |
| topic | numerical radius points; numerical radius peak mappings. |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/270 |
| work_keys_str_mv | AT sgkim someclassofnumericalradiuspeaknlinearmappingsonlpspaces |