Absolutely continuous operators on function spaces and vector measures

Absolutely continuous operators on function spaces and vector measures Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L 0(μ) such that $${L^\infty(\mu) \subset E \subset L^1(\mu)}$$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space $${(X, \xi)}$$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T m : L ∞(μ) → X. In particular, we characterize relatively compact sets $${\mathcal{M}}$$ in ca μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology $${\mathcal{T}_s}$$ of simple convergence in terms of the topological properties of the corresponding set $${\{T_m : m \in \mathcal{M}\}}$$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ∞(μ) → X. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Absolutely continuous operators on function spaces and vector measures

Positivity , Volume 17 (3) – Jun 26, 2012

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Copyright © 2012 by The Author(s)
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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