Numerical radius points of ${\mathcal L}(^m l_{\infty}^n:l_{\infty}^n)$

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 $${\rm Nrad}({T})=\Big\{~[x^*, (x_1, \ldots, x_n)]\in \Pi(E): |x^{*}(T(x_1, \ldots, x_n))|=v(T)~\Big\},$$where $v(T)$ denotes the numerical radius of $T$.$T$ is called {\em numerical radius peak mapping} if there is $[x^{*}, (x_1, \ldots, x_n)]\in \Pi(E)$ that satisfies ${\rm Nrad}({T})=\Big\{~\pm [x^{*}, (x_1, \ldots, x_n)]~\Big\}.$ In this paper we classify ${\rm Nrad}({T})$ for every $T\in {\mathcal L}(^2 l_{\infty}^2: l_{\infty}^2)$ in connection with the set of the norm attaining points of $T$.We also characterize all numerical radius peak mappings in ${\mathcalL}(^m l_{\infty}^n:l_{\infty}^n)$ for $n, m\geq 2,$ where $l_{\infty}^n=\mathbb{R}^n$ with the supremum norm.