STRUCTURE OF CONTINUOUS SYMMETRIC POLYNOMIALS ON THE SPACE OF ABSOLUTELY SUMMABLE SEQUENCES

Abstract

The work is devoted to the study of symmetric continuous homogeneous polynomials on the Banach space $\ell_1$ over the field $\mathbb{K}\in\{\mathbb{R}, \mathbb{C}\}$ of all absolutely summable sequences of elements of $\mathbb{K}.$
We consider the notion of symmetry in the following sense.
Every bijection $\tau$ on $\mathbb{N}$ generates some isometrical isomorphism $J_\tau: \ell_1\to\ell_1$ that rearranges elements of the standard Schauder basis according to $\tau.$ As a result, every subgroup $S$ of the group of all bijections on $\mathbb{N}$ generates the group $G_S$ of corresponding isometrical isomorphisms on $\ell_1.$ The structure of $G_S$-symmetric homogeneous continuous $\mathbb{K}$-valued polynomials on $\ell_1$ is investigated in the present work.