Orthogonally additive polynomials on Dedekind \(\sigma\)-complete vector lattices (Q2837105)

In [DIMACS, Ser. Discret. Math. Theor. Comput. Sci. 4, 571--586 (1991; Zbl 0745.46028)], \textit{K. Sundaresan} presented a characterization of the space of continuous \(n\)-homogeneous orthogonally additive polynomials on the classical Banach lattices \(L^p\) and \(\ell^p\), \(1\leq p<\infty\). Since then, many researchers have generalized this result to other classical Banach lattices.NEWLINENEWLINEIn the present paper, the author proves a similar result for the case of Dedekind \(\sigma\)-complete vector lattices, that is, if \(A\) is a Dedekind \(\sigma\)-complete vector lattice, \(B\) is a topological vector space and \(P :A \rightarrow B\) is a continuous \(n\)-homogeneous orthogonally additive polynomial, then for every \(x \in A\) there exists a linear map \(S\, : \Pi^n(A) \rightarrow B\) such that \(P(x)=S(x^n)\). It is also shown, as a corollary, that this characterization generalizes the result of \textit{K. Sundaresan} [loc.\,cit.].