Extreme polynomials and multilinear forms on \(\ell_1\)
type: Item
instance of: scholarly article
title: Extreme polynomials and multilinear forms on \(\ell_1\)
zbMATH Open document ID: 0922.46042
author: Yun Sung Choi
author: Sung Guen Kim
author: Haseo Ki
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-10-14T00:00:00Z
review text: The authors' main interest is in the geometry of the Banach space \({\mathcal P}({^2\ell_1})\) of 2-homogeneous polynomials \(P\) on \(\ell_1\), endowed with the norm \(\| P\|= \sup\{| P(x)| | x\in \ell_1, \| x\|\leq 1\}\). They first examine the interesting special case of 2-homogeneous polynomials \(P(x,y)= ax^2+ by^2+ cxy\), where \(a,b,c\in\mathbb{R}\) and where \(x,y\in\mathbb{K}= \mathbb{R}\) or \(\mathbb{C}\). In this case, they show that \(\| P\|_{\mathbb{R}}=\| P\|_{\mathbb{C}}\) and they find necessary and sufficient conditions on the coefficients \(a\), \(b\), and \(c\) in order for \(P\) to have norm \(1\). Using this, they characterize the extreme points of the closed unit ball of \({\mathcal P}({^2\ell^2_1})\) in the real case, thereby complementing work of \textit{R. A. Ryan} and \textit{B. Turett} [J. Math. Anal. Appl. 221, No. 2, 698-711 (1998)]. The authors then use this to study extreme and strongly extreme points of the ball \({\mathcal P}({^2\ell_1})\). Among other things, they provide an example of a 2-homogeneous polynomial \(P\) which is not extreme, with the property that for the uniquely associated symmetric bilinear form, \(\check P\), \(\|\check P\|= 2\| P\|\). It is worth noting that this is an area of continued interesting open problems. For example, there does not appear to be a characterization of the extreme points of the ball of \({\mathcal P}({^2\ell^2_1})\) in the complex case. See also recent work by [\textit{C. Boyd} and \textit{R. A. Ryan}, ``Geometric theory of the space of integral polynomials and symmetric tensor products'' (to appear); \textit{Y. S. Choi} and \textit{S. G. Kim}, Arch. Math. 71, 472-480 (1998); and \textit{B. Grecu}, ``Extreme polynomials on Hilbert spaces'' (to appear)].
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 1246330
zbMATH Keywords: polynomials on Banach spaces
zbMATH Keywords: strongly extreme points
reviewed by: Richard M. Aron
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6161
OpenAlex ID: W2090113024
cites work: Polynomials in many variables: Real vs complex norms
cites work: Supremum norms for quadratic polynomials
cites work: Q3688914
cites work: Norm attaining bilinear forms on \(L^ 1[0,1]\)
cites work: Q3496777
cites work: Multilinear forms, subsymmetric polynomials, and spreading models on Banach spaces
cites work: Holomorphic Mappings on l 1
DOI: 10.1006/JMAA.1998.6161
Recommended article: On \(L_p\) extremal polynomials
Recommended article: Extreme polynomials on \(c_0\)
Recommended article: Q3606013
Recommended article: Q5436544
Recommended article: Q3317517
Recommended article: Polynomials, symmetric multilinear forms and weak compactness
Recommended article: On an extremal problem for polynomials in a multi-dimensional case
Recommended article: Extremal problems in the theory of polynomial operators
Recommended article: Q4291320
Recommended article: Extremal Multilinear Forms on Banach Spaces
description: scientific article; zbMATH DE number 1246330