STRUCTURE OF CONTINUOUS SYMMETRIC POLYNOMIALS ON THE SPACE OF ABSOLUTELY SUMMABLE SEQUENCES - Vasylyshyn, T. V.; Kotselovska, M.R. (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2025)
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 $τ$ on $\mathbb{N}$ generates some isometrical isomorphism $J_τ: \ell_1→\ell_1$ that rearranges elements of the standard Schauder basis according to $τ.$ 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.