Orthogonally additive polynomials on spaces of continuous functions
type: Item
instance of: scholarly article
title: Orthogonally additive polynomials on spaces of continuous functions
zbMATH Open document ID: 1076.46035
author: David Pérez-García
author: Ignacio Villanueva
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-06-01T00:00:00Z
full work available at URL: 2005Orthogonallyadditive.pdf
review text: Given a Banach lattice \(X\) and a Banach space \(Y,\) a function \(\varphi:X \to Y\) is said to be {orthogonally additive} if for every disjointly supported pair \(f, g \in X,\) we have \(\varphi(f + g) = \varphi(f) + \varphi(g).\) In [DIMACS, Ser. Discret. Math. Theor. Comput. Sci. 4, 571--586 (1991; Zbl 0745.46028)], \textit{K. Sundaresan} showed that for finite \(p\) and \(n \in \mathbb{N}, 1 \leq n < p,\) every scalar-valued \(n\)-homogeneous orthogonally additive polynomial \(P:L_p(\mu) \to \mathbb{K}\) is determined by some \(g \in L_{\frac{p}{p-n}}\) via the formula \(P(f) = \int f^ng d\mu.\) In their main result, the authors prove the following analogous result for \(\mathcal{C}(K)\) spaces. Theorem: Let \(P:\mathcal{C}(K) \to Y\) be an orthogonally additive \(n\)-homogeneous polynomial and let \(T:\mathcal{C}(K) \times \cdots \times \mathcal{C}(K) \to Y\) be the unique associated symmetric multilinear operator. Then there exists a linear operator \(S:\mathcal{C}(K) \to Y\) such that \(\| S \| = \| T \| \) and there exists a finitely additive measure \(\nu:\Sigma \to Y^{\ast\ast}\) such that for every \(f \in \mathcal{C}(K),\) we have \(P(f) = S(f^n) = \int_K f^n d\nu.\) Here, \(\Sigma\) is the Borel \(\sigma\)-algebra of \(K.\) The proof consists of a characterization of orthogonally additive homogeneous polynomials on \(\mathcal{C}(K).\) The paper concludes with the remark that if \(P\) is an orthogonally additive \(n\)-homogeneous polynomial on \(\mathcal{C}(K),\) then the associated linearization, defined on \(\mathcal{C}(K) \otimes_\epsilon \cdots \otimes_\epsilon \mathcal{C}(K)\) is continuous; as the authors observe, the converse is false. Finally, the authors comment that using different techniques, \textit{Y. Benyamini, S. Lassalle} and \textit{J. G. Llavona} have proven the analogue of the Sundaresan result for general Banach lattices [``Homogeneous orthogonally-additive polynomials on Banach lattices,'' Bull. Lond. Math. Soc. (to appear)].
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 2173088
zbMATH Keywords: orthogonally additive polynomials
zbMATH Keywords: C(K)
reviewed by: Richard M. Aron
MaRDI profile type: Publication
OpenAlex ID: W2152695407
Wikidata QID: Q59474396
cites work: Q5748432
cites work: Multilinear operators on spaces of continuous functions
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cites work: Q4257163
cites work: Q5596558
cites work: Additive Functionals on Lp Spaces
cites work: A representation theorem for additive functionals
cites work: Q3974993
cites work: Completely continuous multilinear operators on $C(K)$ spaces
cites work: Remarks on a theorem of Taskinen on spaces of continuous functions
DOI: 10.1016/J.JMAA.2004.12.036
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description: scientific article; zbMATH DE number 2173088