Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces

Abstract

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 ${\mathcalL}(^n l_p:l_p)$ 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 ${\mathcal L}(^n l_1:l_1).$