A representation theorem for orthogonally additive polynomials on Riesz spaces

Abstract

Abstract. The aim of this article is to prove a representation theorem for orthogonally additive polynomials in the spirit of the recent theorem on representation of orthogonally additive polynomials on Banach lattices but for the setting of Riesz spaces. To this purpose the notion of p-orthosymmetric multilinear form is introduced and it is shown to be equivalent to the orthogonally additive property of the corresponding polynomial. Then the space of positive orthogonally additive polynomials on an Archimedean Riesz space taking values on an uniformly complete Archimedean Riesz space is shown to be isomorphic to the space of positive linear forms on the n-power in the sense of Boulabiar and Buskes of the original Riesz space.

Factorization of (q,p)-summing polynomials through Lorentz spaces

“…Actually, the identification that we establish implicitly in the proof of Theorem 8 between polynomials from X to Y and linear maps from X m to Y has an abstract counterpart in the representation theorem for positive orthogonally additive polynomials on Riesz spaces given in [15,Th.3.4]; see also the references therein for more results regarding polynomials on Banach lattices.…”

Product factorable multilinear operators defined on sequence spaces

“…The Banach space of n-homogeneous orthogonally additive polynomials is closely related to the zero product preserving n-linear operators and several papers can be found in this direction in the literature (see [3,5,12,15,16] and references therein). Now we will give a generalization of the isomorphisms between orthogonally additive n-homogeneous polynomial forms and sequences given in the papers [15] and [16].…”

Orthogonally Additive Polynomials and Orthosymmetric Maps in Banach Algebras with Properties 𝔸 and 𝔹

“…, y n ∈ A. By Cohen's factorization theorem (see [6,§ 11,Corollary 11]), each y ∈ A can be written in the form y = y 1 • • • y n with y 1 , . .…”

“…. , x), x ∈ A, for some continuous n-linear map ϕ : A n → C. Orthogonally additive polynomials have been widely discussed in the context of Banach lattices (see [11] and the references therein). However, we were primarily motivated by the results stating that every continuous orthogonally additive n-homogeneous polynomial P from A into C can be represented in the form P (x) = ω(x n ), x ∈ A, for some ω ∈ A * in the case in which A is a commutative C * -algebra [5,9,13] or A is the Fourier algebra A(G) of a locally compact group G having an abelian subgroup of finite index [4] (some restriction on G is inevitable).…”