On the norm of products of polynomials on ultraproduct of Banach spaces

TL;DR

This paper investigates the lower bounds of polynomial product norms in ultraproducts of Banach spaces, establishing links to the original spaces under specific conditions.

Contribution

It provides new insights into the behavior of polynomial norms in ultraproduct Banach spaces and relates these to the norms in the component spaces.

Findings

Established sharp lower bounds for polynomial product norms in ultraproducts

Identified conditions linking ultraproducts to original Banach spaces

Extended understanding of polynomial behavior in complex Banach space constructions

Abstract

The purpose of this article is to study the problem of finding sharp lower bounds for the norm of the product of polynomials in the ultraproducts of Banach spaces $(X_i)_{\mathfrak U}$. We show that, under certain hypotheses, there is a strong relation between this problem and the same problem for the spaces $X_i$.