Isometric factorization of vector measures and applications to spaces of integrable functions

2020/07/31 by Nygaard, Olav, Rodríguez, José · 1 citation
#46B20 #46G10 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2008.00090

Abstract

Let X be a Banach space, Σ be a σ-algebra, and m:Σ→ X be a (countably additive) vector measure. It is a well known consequence of the Davis-Figiel-Johnson-Pelczýnski factorization procedure that there exist a reflexive Banach space Y, a vector measure m:Σ→ Y and an injective operator J:Y → X such that m factors as m=J∘ m. We elaborate some theory of factoring vector measures and their integration operators with the help of the isometric version of the Davis-Figiel-Johnson-Pelczýnski factorization procedure. Along this way, we sharpen a result of Okada and Ricker that if the integration operator on L₁(m) is weakly compact, then L₁(m) is equal, up to equivalence of norms, to some L₁( m) where Y is reflexive; here we prove that the above equality can be taken to be isometric.

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