MULTIPLICATIVE CONVOLUTION FOR THE BLOCK-SYMMETRIC POLYNOMIALS
DOI:
https://doi.org/10.31471/2304-7399-2025-20(76)-48-58Keywords:
block-symmetric polynomial, block-symmetric analytic function, spectrum, multiplicative convolutionAbstract
This paper investigates the concept of multiplicative convolution for block-symmetric polynomials defined on infinite-dimensional spaces of the form $\ell_p(C^s)$. A multiplicative shift operation and the corresponding convolution operator are developed for the algebra of block-symmetric analytic functions of bounded type. It is shown that the defined convolution is both commutative and associative, forming a commutative semiring with unity on the spectrum of the algebra. Estimates for the spectral radius function of homomorphisms are provided, and essential properties such as linearity, continuity, and compatibility with the algebraic structure are established. The construction of spectral elements with prescribed values on elementary polynomials is proposed. This work generalizes earlier results for the $\ell_1(C^s)$ case to the broader setting of arbitrary $1 < p < \infty$, contributing to the theory of symmetric analytic functions in infinite-dimensional analysis.
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