Uniqueness of extensions of homogeneous polynomials on \(c_{0}\)-sum of Hilbert spaces
type: Item
instance of: scholarly article
title: Uniqueness of extensions of homogeneous polynomials on \(c_{0}\)-sum of Hilbert spaces
zbMATH Open document ID: 1158.46033
DOI: 10.1016/S0019-3577(07)80064-4
author: B. E. Eshmatov
published in: Indagationes Mathematicae. New Series
publication date: 2008-08-12T00:00:00Z
review text: The authors study homogeneous polynomials on the \(c_0\)-sum \(X\) of complex Hilbert spaces \(\{H_i\}_{i\in I}\). They show that if \(P\) is a 2-homogeneous polynomial on \(X''\) which attains its norm at a point \(x=(x_j)_j\) of \(X\), then the set of \(j\) in \(I\) where \(\| x_j\| _{H_j}=1\) is finite. From this it follows that every 2-homogeneous polynomial on \(X\) which attains its norm has a unique norm-preserving extension to \(X''\). When \(n\geq 3\), the authors provide examples of \(n\)-homogeneous polynomials on \(X\) which have two distinct norm-preserving extensions to \(X''\). \textit{P.--K.\thinspace Lin} [Acta Math.\ Sin.\ 24, No.\,5, 877--880 (2008; Zbl 1158.46032), see the preceding review] has recently shown that every \(2\)-homogeneous polynomial on complex \(c_0\) has a unique norm-preserving extension to \(\ell_\infty\). Thus it is reasonable to conjecture that every \(2\)-homogeneous polynomial on \(X\) has a unique norm-preserving extension to \(X''\)
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 5311051
zbMATH Keywords: homogeneous polynomials
zbMATH Keywords: extension of polynomials
zbMATH Keywords: unique Hahn-Banach extension
MaRDI profile type: Publication
cites work: A Hahn-Banach extension theorem for analytic mappings
cites work: Unique Hahn-Banach theorems for spaces of homogeneous polynomials
cites work: EXTENSIONS OF POLYNOMIALS ON PREDUALS OF LORENTZ SEQUENCE SPACES
cites work: A Theorem on Polynomial-Star Approximation
cites work: Q4257163
cites work: Q5479994
cites work: Q3711155
full work available at URL: s0019-3577(07)80064-4
OpenAlex ID: W2060533876
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description: scientific article; zbMATH DE number 5311051