Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$

Keywords: polynomial, $*$-polynomial, symmetric polynomial, symmetric $*$-polynomial, algebraic basis
Published online: 2020-06-12

Abstract


This work is devoted to the study of algebras of continuous symmetric polynomials, that is, invariant with respect to permutations of coordinates of its argument, and of $*$-polynomials on Banach spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ of $p$-power summable sequences of $n$-dimensional vectors of real and complex numbers respectively, where $1\leq p < +\infty.$

We construct the subset of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$ such that every continuous symmetric polynomial on the space $\ell_p(\mathbb{R}^n)$ can be uniquely represented as a linear combination of products of elements of this set. In other words, we construct an algebraic basis of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$. Using this result, we construct an algebraic basis of the algebra of all continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n)$.

Results of the paper can be used for investigations of algebras, generated by continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$, and algebras, generated by continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n)$.

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How to Cite
(1)
Vasylyshyn T. Symmetric Functions on Spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$. Carpathian Math. Publ. 2020, 12 (1), 5-16.

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