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The Plurisubharmonic Dentability and the Analytic Radon-Nikodym Property for Bounded Subsets in Complex Banach Spaces

  • Shangquan Bu
Part of the Mathematics and Its Applications book series (MAIA, volume 356)

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: DX has radial limits almost everywhere on the torus T = {e : θ ∈ [0, 2π[} in X, which means that for almost all θ ∈ [0, 2π[, limr↑1 f(re ) exists. An upper semi- continuous function ø : XR 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, gPSH 0(X) define d(f, g) as the Lipschitz constant ‖fgLip 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 0x 0X, f nL 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)e iθn . Among other known characterizations of the analytic Radon-Nikodym property, we have the following

Keywords

Initial Segment Bounded Subset Plurisubharmonic Function Open Unit Disk Complex Banach Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Shangquan Bu
    • 1
  1. 1.Department of Applied MathematicsTsinghua UniversityBeijingChina

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