Norm attaining bilinear forms of ${\mathcal L}(^2 d_{*}(1, w)^2)$ at given vectors

Abstract

For given unit vectors $x_1, \cdots, x_n$ of a real Banach space $E,$ we define $$NA({\mathcal L}(^nE))(x_1, \cdots, x_n)=\{T\in {\mathcal L}(^nE): |T(x_1, \cdots, x_n)|=\|T\|=1\},$$ where ${\mathcal L}(^nE)$ denotes the Banach space of all continuous $n$-linear forms on $E$ endowed with the norm $\|T\|=\sup_{\|x_k\|=1, 1\leq k\leq n}{|T(x_1, \ldots, x_n)|}$.In this paper, we classify $NA({\mathcal L}(^2 d_{*}(1, w)^2))(Z_1, Z_2)$ for unit vectors $Z_1, Z_2\in d_{*}(1, w)^2,$ where $d_{*}(1, w)^2=\mathbb{R}^2$ with the norm of weight $0<w<1$ endowed with $\|(x, y)\|_{d_*(1, w)}=\max\Big\{|x|, |y|, \frac{|x|+|y|}{1+w}\Big\}$.