Resumen de Integral representation of linear operators on Orlicz-Bochner spaces

Krzystof Feledziak, Marian Nowak

• 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 (γ_ϕ ‖ • ‖ Y )-continuous, where γ_ϕ 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.