SYMMETRIC POLYNOMIALS ON CARTESIAN PRODUCTS OF REAL BANACH SPACES OF ABSOLUTELY SUMMABLE SEQUENCES

Abstract

Let $\ell_p^{(\mathbb{R})}$  be the real Banach space of all absolutely summable in the power $p\in [1,+\infty)$ sequences of real numbers. For $n\in\N,$ let $\ell_{p_1}^{(\mathbb{R})}\times\ldots\times \ell_{p_n}^{(\mathbb{R})}$ be the Cartesian product of Banach spaces $\ell_{p_1}^{(\mathbb{R})},\ldots, \ell_{p_n}^{(\mathbb{R})},$ where $p_1,\ldots,p_n\in [1,+\infty).$ We consider the space $\ell_{p_1}^{(\mathbb{R})}\times\ldots\times \ell_{p_n}^{(\mathbb{R})}$ as the space of all sequences $x= (x_1,x_2,\ldots)$ of $n$-dimensional vectors $x_j = \bigl( x_j^{(1)},\ldots,x_j^{(n)} \bigr),$ where $j\in\mathbb{N},$ such that the sequence $\bigl( x_1^{(s)}, x_2^{(s)},\ldots \bigr)$ belongs to the space $\ell_{p_s}^{(\mathbb{R})}$ for every $s\in \{1,\ldots,n\}.$ A function $f$ defined on the space $\ell_{p_1}^{(\mathbb{R})}\times\ldots\times \ell_{p_n}^{(\mathbb{R})}$ is called symmetric if   $\displaystyle f\bigl( \bigl( x_{\sigma(1)},x_{\sigma(2)},\ldots \bigr)         \bigr) = f((x_1,x_2,\ldots)) $   for every $(x_1,x_2,\ldots) \in \ell_{p_1}^{(\mathbb{R})}\times\ldots\times \ell_{p_n}^{(\mathbb{R})}$ and for every bijection $\sigma:\mathbb{N}\to\mathbb{N}.$ We show that the set $\displaystyle \left\{ H_{\boldsymbol{p},\boldsymbol{k}}^{(\R)}:\; \k\in\mathbb{Z}_+^n\;\textrm{such that}\;k_1/p_1+\ldots+k_n/p_n \geq 1 \right\} $