Integral representation of linear operators on Orlicz-Bochner spaces
Tóm tắt
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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