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Integral representation of linear operators on Orlicz-Bochner spaces

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Abstract

Let (Ω, Σ, μ) be a σ-finite measure space and let\(\mathcal{L}(X,Y)\) stand for the space of all bounded linear operators between Banach spaces (X; ‖ • ‖ X ) and (Y; ‖ • ‖ Y ). We study the problem of integral representation of linear operators from an Orlicz-Bochner spaceL ϕ(μ,X) toY with respect to operator measures\(m : \sum \to \mathcal{L}(X,Y) \). It is shown that a linear operatorT:L ϕ (μ,X) →Y has the integral representationT(f = ∫Ω f(ω)dm with respect to a ϕ*-variationally μ-continuous operator measurem if and only ifT is (γϕ ‖ • ‖ Y )-continuous, where γϕ stands for a natural mixed topology onL ϕ (μ,X). As an application, we derive Vitali-Hahn-Saks type theorems for families of operator measures.

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Authors and Affiliations

  1. Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65-516, Zielona Góra, Poland

    Krzysztof Feledziak & Marian Nowak

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  1. Krzysztof Feledziak
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  2. Marian Nowak
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Correspondence to Krzysztof Feledziak.

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Feledziak, K., Nowak, M. Integral representation of linear operators on Orlicz-Bochner spaces. Collect. Math. 61, 277–290 (2010). https://doi.org/10.1007/BF03191233

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  • DOI: https://doi.org/10.1007/BF03191233

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