Integral representation of linear operators on Orlicz-Bochner spaces

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.