Spaces of operator-valued functions measurable with respect to the strong operator topology

2008/11/14 by Óscar Blasco, Oscar Blasco, Blasco, Oscar +2
Mathematics · #28B05 #46G10 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:28B05 #msc:46G10

paper · pdf · doi:10.48550/arxiv.0811.2284

Minor revisions; to appear in the proceedings of 3rd Meeting on Vector Measures, Integration and Applications (Eichstaett, 2008)

openalex publication_date 2008/11/14 · arxiv created 2009/04/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X and Y be Banach spaces and (Ω,Σ,μ) a finite measure space. In this note we introduce the space L^p[μ;L(X,Y)] consisting of all (equivalence classes of) functions Φ:Ω↦ L(X,Y) such that ω↦ Φ(ω)x is strongly μ-measurable for all x∈ X and ω↦ Φ(ω)f(ω) belongs to L¹(μ;Y) for all f∈ L^p'(μ;X), 1/p+1/p'=1. We show that functions in L^p[μ;Ł(X,Y)] define operator-valued measures with bounded p-variation and use these spaces to obtain an isometric characterization of the space of all L(X,Y)-valued multipliers acting boundedly from L^p(μ;X) into L^q(μ;Y), 1≤ q< p<∞.

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