Abstract
Let X be a complex Banach space. Following [2], X is said to have the analytic Radon- Nikodym property if every uniformly bounded analytic function from the open unit disk of C with values in X, f: D → X has radial limits almost everywhere on the torus T = {eiθ : θ ∈ [0, 2π[} in X, which means that for almost all θ ∈ [0, 2π[, limr↑1 f(reiθ) exists. An upper semi- continuous function ø : X → R is plurisubharmonic ( see [3] ) if, for every \( x,y \in X,f(x) \leqslant \int_0^{2\pi } {f(x + e^{i\theta } y)} \tfrac{{d\theta }} {{2\pi }}. \) We shall denote by PSH 0(X) the the set of all Lipschitz plurisubharmonic functions f on X satisfying f(0) = 0. For f, g ∈ PSH 0(X) define d(f, g) as the Lipschitz constant ‖f−g ‖Lip of f−g. PSH 0 X equipped with the metric d becomes a complete metric space. A sequence of functions (f n)n≥0 in L 1([0, 2π[N, X) is called an X-valued analytic martingale if f 0 ≡ x 0 ∈ X, f n ∈ L 1([0, 2π[n, X) and for every (θ 1, θ 2, … θ n-1) ∈ [0, 2π[n-1, f n(θ 1, θ 2, … θ n) = f n-1(θ 1, θ 2, … θ n-1) + d n(θ 1, θ 2, … θ n-1)eiθn. Among other known characterizations of the analytic Radon-Nikodym property, we have the following
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© 1996 Kluwer Academic Publishers
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Bu, S. (1996). The Plurisubharmonic Dentability and the Analytic Radon-Nikodym Property for Bounded Subsets in Complex Banach Spaces. In: Li, B., Wang, S., Yan, S., Yang, CC. (eds) Functional Analysis in China. Mathematics and Its Applications, vol 356. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0185-8_23
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DOI: https://doi.org/10.1007/978-94-009-0185-8_23
Publisher Name: Springer, Dordrecht
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