Orthogonally additive polynomials on spaces of continuous functions
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)].
scientific article; zbMATH DE number 2173088
scientific article