The Experts below are selected from a list of 4413 Experts worldwide ranked by ideXlab platform
Xueying Zhang - One of the best experts on this subject based on the ideXlab platform.
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atomic decompositions of Banach Lattice valued martingales
Statistics & Probability Letters, 2012Co-Authors: Xueying Zhang, Chuanzhou ZhangAbstract:In this paper, atomic decompositions of Banach Lattice-valued martingales are given. We discuss the relation between the LERMT property and atomic decompositions. With the help of atomic decompositions, the relation of the martingale spaces is investigated.
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atomic decompositions of Banach Lattice valued martingales with weighted measure
Scientia Sinica Mathematica, 2012Co-Authors: Chuanzhou Zhang, Xueying ZhangAbstract:In this paper the atomic decompositions of Banach Lattice-valued martingales with weighted measure are given. With the help of the atomic decompositions, the interpolation theorems on martingale spaces are investigated.
Anthony Wickstead - One of the best experts on this subject based on the ideXlab platform.
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Banach Lattice algebras some questions but very few answers
Positivity, 2017Co-Authors: Anthony WicksteadAbstract:We pose a number of questions and problems about Banach Lattice algebras. These concern: What should the definition be? How to add an identity. Order theoretic properties of the multiplication. Order theoretic properties of the left regular representation.
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Banach Lattices of L-weakly and M-weakly compact operators
Archiv der Mathematik, 2017Co-Authors: E. Bayram, Anthony WicksteadAbstract:We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach Lattice under the regular norm, for that Banach Lattice to have an order continuous norm, to be an AL-space or an AM-space.
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Free and projective Banach Lattices
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2015Co-Authors: Ben De Pagter, Anthony WicksteadAbstract:We define and prove the existence of free Banach Lattices in the category of Banach Lattices and contractive Lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach Lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach Lattice P to be projective if, whenever X is a Banach Lattice, J is a closed ideal in X, Q : X → X / J is the quotient map, T : P → X / J is a linear Lattice homomorphism and e > 0, there exists a linear Lattice homomorphism : P → X such that T = Q o and ∥ ∥ ≤ (1 + e )∥T∥. We establish the connection between projective Banach Lattices and free Banach Lattices, describe several families of Banach Lattices that are projective and prove that some are not.
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Free and Projective Banach Lattices
arXiv: Functional Analysis, 2012Co-Authors: B De Pagter, Anthony WicksteadAbstract:We define and prove the existence of free Banach Lattices in the category of Banach Lattices and contractive Lattice homomorphisms and establish some of their fundamental properties. We give much more detailed results about their structure in the case that there are only a finite number of generators and give several Banach Lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach Lattice $P$ to be projective if whenever $X$ is a Banach Lattice, $J$ a closed ideal in $X$, $Q:X\to X/J$ the quotient map, $T:P\to X/J$ a linear Lattice homomorphism and $\epsilon>0$ there is a linear Lattice homomorphism $\hat{T}:P\to X$ such that (i) $T=Q\circ \hat{T}$ and (ii) $\|\hat{T}\|\le (1+\epsilon)\|T\|$. We establish the connection between projective Banach Lattices and free Banach Lattices and describe several families of Banach Lattices that are projective as well as proving that some are not.
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relative weak compactness of solid hulls in Banach Lattices
Indagationes Mathematicae, 1998Co-Authors: Z.l. Chen, Anthony WicksteadAbstract:We show that the solid hull of every relatively weakly compact set in a Banach Lattice is again relatively weakly compact if and only if the Banach Lattice is an order direct sum of a KB-space and an atomic Banach Lattice with an order continuous norm. If we assume order continuity of the norm then this is equivalent to requiring that the image of every relatively weakly compact set under the modulus map is again relatively weakly compact. We also show that amongst Banach Lattices with an order continuous norm those that have the property that the Lattice operations are weakly sequentially continuous are precisely the atomic ones. The final section of the paper is devoted to applications of our earlier results to questions concerning the factorization of compact and weakly compact operators through reflexive Banach Lattices.
Chuanzhou Zhang - One of the best experts on this subject based on the ideXlab platform.
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atomic decompositions of Banach Lattice valued martingales
Statistics & Probability Letters, 2012Co-Authors: Xueying Zhang, Chuanzhou ZhangAbstract:In this paper, atomic decompositions of Banach Lattice-valued martingales are given. We discuss the relation between the LERMT property and atomic decompositions. With the help of atomic decompositions, the relation of the martingale spaces is investigated.
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atomic decompositions of Banach Lattice valued martingales with weighted measure
Scientia Sinica Mathematica, 2012Co-Authors: Chuanzhou Zhang, Xueying ZhangAbstract:In this paper the atomic decompositions of Banach Lattice-valued martingales with weighted measure are given. With the help of the atomic decompositions, the interpolation theorems on martingale spaces are investigated.
Coenraad C A Labuschagne - One of the best experts on this subject based on the ideXlab platform.
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operator martingale decompositions and the radon nikodým property in Banach spaces
Journal of Mathematical Analysis and Applications, 2010Co-Authors: Coenraad C A Labuschagne, Valeria MarraffaAbstract:Abstract We consider submartingales and uniform amarts of maps acting between a Banach Lattice and a Banach Lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon–Nikodým property if and only if every uniformly norm bounded martingale defined on the Chaney–Schaefer l-tensor product E ⊗ ˜ l Y , where E is a suitable Banach Lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach Lattices Y with the Radon–Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on E ⊗ ˜ l Y . Secondly, we derive a Riesz decomposition for uniform amarts of maps acting between a Banach Lattice and a Banach space. This result is used to characterize Banach spaces with the Radon–Nikodým property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1 p ∞ , our results yield L p ( μ , Y ) -space analogues of some of the well-known results on uniform amarts in L 1 ( μ , Y ) -spaces.
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Operator martingale decompositions and the Radon–Nikodým property in Banach spaces
Journal of Mathematical Analysis and Applications, 2010Co-Authors: Coenraad C A Labuschagne, Valeria MarraffaAbstract:Abstract We consider submartingales and uniform amarts of maps acting between a Banach Lattice and a Banach Lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon–Nikodým property if and only if every uniformly norm bounded martingale defined on the Chaney–Schaefer l-tensor product E ⊗ ˜ l Y , where E is a suitable Banach Lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach Lattices Y with the Radon–Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on E ⊗ ˜ l Y . Secondly, we derive a Riesz decomposition for uniform amarts of maps acting between a Banach Lattice and a Banach space. This result is used to characterize Banach spaces with the Radon–Nikodým property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1 p ∞ , our results yield L p ( μ , Y ) -space analogues of some of the well-known results on uniform amarts in L 1 ( μ , Y ) -spaces.
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Convergent martingales of operators and the Radon Nikodým property in Banach spaces
Proceedings of the American Mathematical Society, 2008Co-Authors: Stuart F. Cullender, Coenraad C A LabuschagneAbstract:We extend Troitsky's ideas on measure-free martingales on Banach Lattices to martingales of operators acting between a Banach Lattice and a Banach space. We prove that each norm bounded martingale of cone absolutely summing (c.a.s.) operators (also known as 1-concave operators), from a Banach Lattice E to a Banach space Y, can be generated by a single c.a.s. operator. As a consequence, we obtain a characterization of Banach spaces with the Radon Nikodym property in terms of convergence of norm bounded martingales defined on the Chaney-Schaefer ∫-tensor product E)⊗iy. This extends a classical martingale characterization of the Radon Nikodym property, formulated in the Lebesgue-Bochner spaces L P (μ, Y) (1 < p < ∞).
Vladimir G. Troitsky - One of the best experts on this subject based on the ideXlab platform.
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Unbounded norm topology in Banach Lattices
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Marko Kandić, Mohammad A. A. Marabeh, Vladimir G. TroitskyAbstract:Abstract A net ( x α ) in a Banach Lattice X is said to un-converge to a vector x if ‖ | x α − x | ∧ u ‖ → 0 for every u ∈ X + . In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quasi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach Lattice X is a KB-space iff its closed unit ball B X is un-complete. For a Banach Lattice X, B X is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.
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Unbounded Norm Topology in Banach Lattices
arXiv: Functional Analysis, 2016Co-Authors: Marko Kandić, Mohammad A. A. Marabeh, Vladimir G. TroitskyAbstract:A net $(x_\alpha)$ in a Banach Lattice $X$ is said to un-converge to a vector $x$ if $\bigl\lVert\lvert x_\alpha-x\rvert\wedge u\bigr\rVert\to 0$ for every $u\in X_+$. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff $X$ has a strong unit. Un-topology is metrizable iff $X$ has a quasi-interior point. Suppose that $X$ is order continuous, then un-topology is locally convex iff $X$ is atomic. An order continuous Banach Lattice $X$ is a KB-space iff its closed unit ball $B_X$ is un-complete. For a Banach Lattice $X$, $B_X$ is un-compact iff $X$ is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.
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the 2 concavification of a Banach Lattice equals the diagonal of the fremlin tensor square
arXiv: Functional Analysis, 2011Co-Authors: Gerard Buskes, Alexey I Popov, Adi Tcaciuc, Vladimir G. TroitskyAbstract:We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach Lattice $E$ with itself and the 2-concavification of $E$.
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Martingales in Banach Lattices, II
Positivity, 2011Co-Authors: Hailegebriel E. Gessesse, Vladimir G. TroitskyAbstract:This note is a follow-up to Troitsky (Positivity 9(3):437–456, 2005). We provide several sufficient conditions for the space M of bounded martingale on a Banach Lattice F to be a Banach Lattice itself. We also present examples in which M is not a Banach Lattice. It is shown that if F is a KB-space and the filtration is dense then F is a projection band in M .
