A Partial Solution To An Open Problem

Authors : Şükran KONCA

Pages : 0-0

Doi:10.17678/beujst.41573

View : 14 | Download : 7

Publication Date : 2015-12-28

Article Type : Research Paper

Abstract :Let $\leftinsert ignore into journalissuearticles values( {X,\left\| {.,...,.} \right\|} \right);$ be a real $n$-normed space, as introduced by S. Gahler [1] in 1969. The set of all bounded multilinear  $n$-functionals on $\leftinsert ignore into journalissuearticles values( {X,\left\| {.,...,.} \right\|} \right);$ forms a vector space. A bounded multilinear  $n$-functional $F$ is defined by  $\left\| F \right\|: = {\rm{sup}}\left\{ {\left| {F\leftinsert ignore into journalissuearticles values( {{x_1},...,{x_n}} \right);} \right|:\left\| {{x_1},...,{x_n}} \right\| \le 1} \right\}$. \textbf{\bigskip } This formula defines a norm on $X`$ insert ignore into journalissuearticles values(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 dual exponent 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}\leftinsert ignore into journalissuearticles values( {{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 $\leftinsert ignore into journalissuearticles values( {\ell^{p},\left\| {.,...,.} \right\|_p^{}} \right);$, an open problem was given by Batkunde et al. [2]. They want to compute the exact norm of ${F_Y}$, especially for $p \ne 2$. In this paper, we deal with a partial solution to this open problem given in their paper.
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