Orthogonally additive polynomials on Banach function algebras

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:448

Article

Villena, A. R.¹ 

[1] Univ Granada, Fac Ciencias, Dept Anal Matemat, E-18071 Granada, Spain

关键词: Orthogonally additive polynomial;    Algebra of Lipschitz functions;    Algebra of absolutely continuous functions;    Algebra of differentiable functions;    Fourier algebra;    Figa-Talamanca-Herz algebra;   

DOI  :  10.1016/j.jmaa.2016.11.010

来源: Elsevier

【 摘 要 】

For a Banach function algebra A, we consider the problem of representing a continuous d-homogeneous polynomial P: A -> X, where X is an arbitrary Banach space, that satisfies the property P(f + g) = P(f) + P(g) whenever f, g is an element of A are such that supp(f) boolean AND supp(g) = theta. We show that such a polynomial can be represented as P(f) = T(f(d)) (f is an element of A) for some continuous linear map T: A -> X for a variety of Banach function algebras such as the algebra of continuous functions C-0(Omega) for any locally compact Hausdorff space Omega, the algebra of Lipschitz functions lip(alpha)(K) for any compact metric space K and alpha is an element of]0, 1[, the Figa-Talamanca-Herz algebra A(p)(G) for some locally compact groups G and p is an element of]1, + infinity[, the algebras AC([a, b]) and BVC([a,b) of absolutely continuous functions and of continuous functions of bounded variation on the interval [a,b]. In the case where A = C-n ([a, b]), P can be represented as P(f) = Sigma T-(n1,T- ...,T- nd) (f((n1)) ... f((nd))) where the sum is taken over (n(1), ..., n(d)) is an element of Z(d) with 0 <= n(1) <= ... <= n(d) <= n, for appropriate continuous linear maps T-(n1,T- ...,T- nd) : Cn-nd ([a, b]) -> X. (C) 2016 Elsevier Inc. All rights reserved.

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