2019/08/21 by Verónica Dimant, Dimant, Verónica, Daniel Galicer +3
Mathematics · #32A05 #32A22 (primary) #46B15 #46B20 #46E50 (secondary) #46G25 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #math.CV #math.FA #msc:32A05 #msc:32A22 #msc:46B15 #msc:46B20 #msc:46E50 #msc:46G25
paper · pdf · doi:10.48550/arxiv.1908.08107
Math. Proc. Cambridge Philos. Soc., accepted
arxiv created 2020/11/11 · arxiv updated 2020/11/12
The polarization constant of a Banach space X is defined as \mathbf c(X):= \limsupk→ ∞ \mathbf c(k, X)^(1)/(k), where \mathbf c(k, X) stands for the best constant C>0 such that \Vert \overset\veeP \Vert ≤ C \Vert P \Vert for every k-homogeneous polynomial P ∈ \mathcal P(kX). We show that if X is a finite dimensional complex space then \mathbf c(X)=1. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak's complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related with the geometry of X for \mathbf c(2,X)=1 to hold. Additionally we link polarization's constants with certain estimates of the nuclear norm of the product of polynomials.