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On set-valued cone absolutely summing maps

100%
Labuschagne C., Marraffa V.
Open Mathematics
|
2010
|
tom 8
|
nr 1
148-157
EN
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.
2
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On stability and robust stability of positive linear Volterra equations in Banach lattices

100%
Murakami S., Ngoc P.
Open Mathematics
|
2010
|
tom 8
|
nr 5
966-984
EN
We study positive linear Volterra integro-differential equations in Banach lattices. A characterization of positive equations is given. Furthermore, an explicit spectral criterion for uniformly asymptotic stability of positive equations is presented. Finally, we deal with problems of robust stability of positive systems under structured perturbations. Some explicit stability bounds with respect to these perturbations are given.
3
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Properties of modulus of monotonicity and Opial property in direct sums

100%
Markowicz J., Prus S.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2017
|
tom 71
|
nr 2
EN
We give an example of a Banach lattice with a non-convex modulus of monotonicity, which disproves a claim made in the literature. Results on preservation of the non-strict Opial property and Opial property under passing to general direct sums of Banach spaces are established.
4
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A description of Banach space-valued Orlicz hearts

72%
Labuschagne C., Offwood T.
Open Mathematics
|
2010
|
tom 8
|
nr 6
1109-1119
EN
Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $$ H_\varphi \left( \mu \right)\tilde \otimes _l Y $$ of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $$ \left( {H_\varphi \left( \mu \right)\tilde \otimes _l Y} \right)* $$ and $$ H_\varphi \left( \mu \right)*\tilde \otimes _l Y* $$ in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým property on Y. Using the associativity of the l-norm, an alternative proof is given of the known fact that for any separable Banach lattice E and any Banach space Y, E and Y have the Radon-Nikodým property if and only if $$ E\tilde \otimes _l Y $$ has the Radon-Nikodým property. As a corollary, the Radon-Nikodým property in H φ(μ, Y) is described in terms of the Radon-Nikodým property on H φ(μ) and Y.
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