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Orthogonally additive polynomials on spaces of continuous functions. (English) Zbl 1076.46035
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)], 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, Y. Benyamini, S. Lassalle and 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)].

46G25 (Spaces of) multilinear mappings, polynomials
46G20 Infinite-dimensional holomorphy
46B42 Banach lattices
Full Text: DOI
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