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On set-valued cone absolutely summing maps

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Labuschagne C., Marraffa V.

Open Mathematics

2010

tom 8

nr 1

148-157

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L p(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $$ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $$ of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L 1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $$ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $$ , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.

Ograniczanie wyników

1 Open Mathematics

1 Labuschagne C.

1 Marraffa V.

1 2010

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On set-valued cone absolutely summing maps