The norming sets of $${{\mathcal {L}}}(^2 {\mathbb {R}}^2_{h(w)})$$
2023; Birkhäuser; Volume: 89; Issue: 1-2 Linguagem: Inglês
10.1007/s44146-023-00078-7
ISSN2064-8316
Autores Tópico(s)Mathematical Analysis and Transform Methods
ResumoAn element $$(x_1, \ldots , x_n)\in E^n$$ is called a norming point of $$T\in {{\mathcal {L}}}(^n E)$$ if $$\Vert x_1\Vert =\cdots =\Vert x_n\Vert =1$$ and $$|T(x_1, \ldots , x_n)|=\Vert T\Vert ,$$ 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 $$\begin{aligned} \text {Norm}(T)=\{(x_1, \ldots , x_n)\in E^n: (x_1, \ldots , x_n)~\text{ is } \text{ a } \text{ norming } \text{ point } \text{ of }~T\}. \end{aligned}$$ Let $${\mathbb {R}}^2_{h(w)}$$ denote the plane with the hexagonal norm with weight $$0<w<1$$ $$\begin{aligned} \Vert (x, y)\Vert _{h(w)}=\max \Big \{|y|, |x|+(1-w)|y|\Big \}. \end{aligned}$$ We classify $$\text {Norm}(T)$$ for every $$T\in {{\mathcal {L}}}(^2 {\mathbb {R}}_{h(w)}^2)$$ .
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