Operator martingale decomposition and the Radon-Nikodym property in Banach spaces

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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 ⊗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 Lp(μ, Y )-space analogues of some of the well-known results on uniform amarts in L1(μ, Y )-spaces.
Lingua originaleEnglish
pagine (da-a)357-365
Numero di pagine9
RivistaJournal of Mathematical Analysis and Applications
Volume363
Stato di pubblicazionePublished - 2010

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

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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title = "Operator martingale decomposition 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 ⊗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– 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 Lp(μ, Y )-space analogues of some of the well-known results on uniform amarts in L1(μ, Y )-spaces.",
keywords = "Banach lattice, Banach space, Bochner norm, Cone absolutely summing operator, Convergent martingale, Convergent submartingale, Dinculeanu operator, Radon–Nikod{\'y}m property, Uniform amart",
author = "Valeria Marraffa",
year = "2010",
language = "English",
volume = "363",
pages = "357--365",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
publisher = "Academic Press Inc.",

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TY - JOUR

T1 - Operator martingale decomposition and the Radon-Nikodym property in Banach spaces

AU - Marraffa, Valeria

PY - 2010

Y1 - 2010

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 ⊗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 Lp(μ, Y )-space analogues of some of the well-known results on uniform amarts in L1(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 ⊗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 Lp(μ, Y )-space analogues of some of the well-known results on uniform amarts in L1(μ, Y )-spaces.

KW - Banach lattice, Banach space, Bochner norm, Cone absolutely summing operator, Convergent martingale, Convergent submartingale, Dinculeanu operator, Radon–Nikodým property, Uniform amart

UR - http://hdl.handle.net/10447/44620

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

ER -