# 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
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## References

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