Representable and Radon-Nikodým polynomials

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...

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Bibliographic Details
Published in:Quaestiones mathematicae 2023-07, Vol.46 (7), p.1293-1313
Main Authors: Cilia, Raffaella, Gutiérrez, Joaquín M.
Format: Article
Language:English
Subjects:
Aron-Berner extension of polynomials
Banach spaces
Linear operators
Operators (mathematics)
Polynomials
Primary: 46G25
Radon
Radon-Nikodým polynomials
Riesz-representable polynomials
Secondary: 47B10
unconditionally converging polynomials
µ
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Summary: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.
ISSN:1607-3606
1727-933X
DOI:10.2989/16073606.2022.2073481