Representable and Radon-Nikodým polynomials

Abstract

Every k-homogeneous (continuous) polynomial between Banach spaces admits a unique Aron-Berner extension to the biduals. Our main result states that, for every σ-finite measure µ, every polynomial P ∈ with Y -valued Aron-Berner extension is representable, in other words, it factors through ℓ 1 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 L 1(µ) is representable. We introduce the Radon-Nikodým polynomials and show that every polynomial with Y -valued Aron-Berner extension is Radon-Nikodým. Finally, we prove that every Radon-Nikodým polynomial is unconditionally converging.