A Partial Solution To An Open Problem

Abstract

Let $\left( {X,\left\| {.,...,.} \right\|} \right)$ be a real$n$-normed space, as introduced by S. Gahler [1] in 1969. The setof all bounded multilinear $n$-functionals on $\left( {X,\left\|{.,...,.} \right\|} \right)$ forms a vector space. A boundedmultilinear $n$-functional $F$ is defined by $\left\| F\right\|: = {\rm{sup}}\left\{ {\left| {F\left( {{x_1},...,{x_n}}\right)} \right|:\left\| {{x_1},...,{x_n}} \right\| \le 1}\right\}$. \textbf{\bigskip }This formula defines a norm on $X'$ (the space of all bounded multilinear $n$-functionals on $X$). \textbf{\bigskip }Let $Y: = \left\{{{y_1},...,{y_n}} \right\}$ in $\ell^{q}$, where $q$ is the dualexponent of $p$. \textbf{\bigskip }Batkunde et al. [2] defined the following multilinear$n$-functional on $\ell^{p}$ where $1 \le p < \infty$:\begin{equation*}{F_Y}\left( {{x_1},...,{x_n}} \right): =\frac{1}{{n!}}\sum\limits_{{j_1}} {...} \sum\limits_{{j_n}}{\left| {\begin{array}{*{20}{c}} {{x_{1{j_1}}}} & \cdots & {{x_{1{j_n}}}} \\ \vdots & \ddots & \vdots \\ {{x_{n{j_1}}}} & \ldots & {{x_{n{j_n}}}} \\\end{array}} \right|} \left| {\begin{array}{*{20}{c}} {{y_{1{j_1}}}} & \cdots & {{y_{1{j_n}}}} \\ \vdots & \ddots & \vdots \\ {{y_{n{j_1}}}} & \ldots & {{y_{n{j_n}}}} \\\end{array}} \right|\end{equation*}for ${x_1},...,{x_n} \in \ell^{p}$.\textbf{\bigskip }Regarding the $n$-functional on $\left( {\ell^{p},\left\| {.,...,.}\right\|_p^{}} \right)$, an open problem was given by Batkunde et al. [2]. Theywant to compute the exact norm of ${F_Y}$, especially for $p \ne2$. In this paper, we deal with a partial solution to this openproblem given in their paper.