Representable and Radon-Nikodým polynomials

Abstract

Every k-homogeneous (continuous) polynomial P ∈ P(^kX, Y ) between Banach spaces admits a unique Aron-Berner extension e P ∈ P(^kX^**, Y^**) to the biduals. Our main result states that, for every σ-finite measure μ, every polynomial P ∈ P(^kL₁(μ), Y ) with Y -valued Aron-Berner extension is representable, in other words, it factors through ℓ₁ in the form P = A◦Q where Q is a polynomial and A is an operator. This may be viewed as a polynomial strengthening of the Dunford-Pettis-Phillips theorem stating that every weakly compact operator on L₁(μ) is representable. We introduce the Radon-Nikodým polynomials and show that every polynomial P ∈ P(^kX, Y ) with Y -valued Aron-Berner extension is Radon-Nikodým. Finally, we prove that every Radon-Nikodým polynomial is unconditionally converging.