Norm or numerical radius attaining polynomials on \(C(K\))
type: Item
instance of: scholarly article
title: Norm or numerical radius attaining polynomials on \(C(K\))
zbMATH Open document ID: 1059.46026
author: Yun Sung Choi
author: Domingo García
author: Sung Guen Kim
author: Manuel Maestre
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-20T00:00:00Z
review text: A real Banach space X has the Badé property if the set of extreme points \(\operatorname{ext} B_X\) of the closed ball \(B_X\) is a norming subset of \(X^\ast.\) Also, \(X\) has the \(E\)-property if every norm attaining linear form \(x^\ast\) attains its norm at a point in \(\text{ext} B_X.\) Finally, \(X\) is said to have the \(\lambda\)-property if \(B_X\) coincides with the sequential convex hull of \(\operatorname{ext} B_X.\) \textit{A. Aizpuru} [Arch. Math. 69, 333--337 (1997; Zbl 0917.46010)] proved the following two results. Theorem 1. The following are equivalent for \(X = {\mathcal C}(K,\mathbb C): \) \(X\) has the E-property; \(X\) has the \(\lambda\)-property; \(K\) is \(0\) or \(1\) dimensional. Theorem 2. The following are equivalent for \(X = {\mathcal{C}}(K,\mathbb R):\) \(\text{ext} B_X\) is a norming set for \(X^\ast;\) \(X\) has the Badé property; \(X\) has the \(\lambda\)-property; \(K\) is \(0\)-dimensional. The two main questions posed here are the following. Question 1. Given \(K,\) a compact Hausdorff space such that \(X = {\mathcal C}(K,\mathbb K)\) has the E-property, \(n \in \mathbb N\), \(n \geq 2,\) and a norm attaining \(n\)-homogeneous polynomial \(P:{\mathcal C}(K,\mathbb K) \to \mathbb K,\) does it follow that \(\| P\| \) is attained at some point of \(\operatorname{ext} B_X?\) Question 2. Suppose now that \(X = {\mathcal C}(K,\mathbb K)\) has the Badé property. Is it true that \(\operatorname{ext} B_X\) is a norming set for the space of \(n\)-homogeneous polynomials \(P: X \to \mathbb K\), \(n\geq 2?\) The authors show that Question 1 has an affirmative answer if \(\mathbb K = \mathbb C\) and \(K\) is compact of dimension at most 1. If \(K\) is also metrizable, the authors show that the conditions in Theorem 1 are also equivalent to the statement that every norm attaining \(n\)-homogeneous polynomial \(P:{\mathcal C}(K,\mathbb C) \to \mathbb C\) attains its norm at an extreme point. Also, if \(K\) is scattered (dispersed) [\textit{A…
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 2093794
zbMATH Keywords: norm attaining
zbMATH Keywords: numerical radius attaining
zbMATH Keywords: polynomials
zbMATH Keywords: \(C(K)\)
zbMATH Keywords: dimension
reviewed by: Richard M. Aron
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.03.005
OpenAlex ID: W2039985375
cites work: Every real Banach space can be renormed to satisfy the denseness of numerical radius attaining operators
cites work: On multilinear mappings attaining their norms.
cites work: There is no bilinear Bishop-Phelps theorem
cites work: Denseness of Operators Whose Second Adjoints Attain Their Numerical Radii
cites work: Numerical Radius Attaining Operators and the Radon-Nikodym Property
cites work: On operators which attain their norm at extreme points
cites work: Norm Attaining Bilinear Forms on Spaces of Continuous Functions
cites work: Q4283362
cites work: Q4869336
cites work: A geometric function determined by extreme points of the unit ball of a normed space
cites work: Rotundity, the C.S.R.P., and the λ-Property in Banach Spaces
cites work: Q5630192
cites work: Q5186189
cites work: A proof that every Banach space is subreflexive
cites work: Extreme Points in Spaces of Continuous Functions
cites work: Numerical Radius-Attaining Operators on C(K)
cites work: Norm attaining bilinear forms on \(L^ 1[0,1]\)
cites work: Norm or Numerical Radius Attaining Multilinear Mappings and Polynomials
cites work: Q4257163
cites work: Stone-Čech compactification of a product
cites work: Boundaries for polydisc algebras in infinite dimensions
cites work: Q4772256
cites work: On operators which attain their norm
cites work: A counterexample on numerical radius attaining operators
cites work: A property of multilinear operations
cites work: Spaces of continuous functions (III) (Spaces C(Ω) for Ω without perfect subsets)
DOI: 10.1016/J.JMAA.2004.03.005
Recommended article: Norm or Numerical Radius Attaining Multilinear Mappings and Polynomials
Recommended article: Numerical-Radius-Attaining Polynomials
Recommended article: Norm attaining polynomials
Recommended article: On norm attaining polynomials.
Recommended article: Norm-attaining polynomials and differentiability
Recommended article: Q3209666
Recommended article: Norm and Numerical Radius of 2-homogeneous Polynomials on the Real Space lp2, (1 < p > ∞)
Recommended article: Numerical Radius-Attaining Operators on C(K)
Recommended article: Explicit norm attaining polynomials
Recommended article: Q4424437
description: scientific article; zbMATH DE number 2093794