Operator martingale decompositions and the Radon Nikodym property in Banach spaces

Valeria Marraffa, Coenraad C.A. Labuschagne

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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 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.
Lingua originaleEnglish
pagine (da-a)357-365
Numero di pagine9
RivistaJournal of Mathematical Analysis and Applications
Volume363
Stato di pubblicazionePublished - 2009

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Radon-Nikodym Property
Banach Lattice
Radon
Banach spaces
Martingale
Banach space
Decomposition
Decompose
Submartingale
Norm
Operator
Riesz Decomposition
Supermartingale
Analogue
Tensors
Tensor Product
If and only if

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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title = "Operator martingale decompositions and the Radon Nikodym property in Banach spaces",
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 Nikod{\'y}m property if and only if every uniformly norm bounded martingale defined on the Chaney–Schaefer l-tensor product E 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– Nikod{\'y}m 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–Nikod{\'y}m 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.",
author = "Valeria Marraffa and Labuschagne, {Coenraad C.A.}",
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language = "English",
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pages = "357--365",
journal = "Journal of Mathematical Analysis and Applications",
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T1 - Operator martingale decompositions and the Radon Nikodym property in Banach spaces

AU - Marraffa, Valeria

AU - Labuschagne, Coenraad C.A.

PY - 2009

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N2 - 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 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.

AB - 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 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.

UR - http://hdl.handle.net/10447/47510

M3 - Article

VL - 363

SP - 357

EP - 365

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

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