### Abstract

Lingua originale | English |
---|---|

pagine (da-a) | 357-365 |

Numero di pagine | 9 |

Rivista | Journal of Mathematical Analysis and Applications |

Volume | 363 |

Stato di pubblicazione | Published - 2009 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cita questo

*Journal of Mathematical Analysis and Applications*,

*363*, 357-365.

**Operator martingale decompositions and the Radon Nikodym property in Banach spaces.** / Marraffa, Valeria; Labuschagne, Coenraad C.A.

Risultato della ricerca: Article

*Journal of Mathematical Analysis and Applications*, vol. 363, pagg. 357-365.

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TY - JOUR

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

AU - Marraffa, Valeria

AU - Labuschagne, Coenraad C.A.

PY - 2009

Y1 - 2009

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

ER -