Convergent martingales of operators and the Radon Nikodým property in Banach spaces
HTML articles powered by AMS MathViewer
- by Stuart F. Cullender and Coenraad C. A. Labuschagne PDF
- Proc. Amer. Math. Soc. 136 (2008), 3883-3893 Request permission
Abstract:
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 Nikodým property in terms of convergence of norm bounded martingales defined on the Chaney-Schaefer $l$-tensor product $E\widetilde {\otimes }_l Y$. This extends a classical martingale characterization of the Radon Nikodým property, formulated in the Lebesgue-Bochner spaces $L^p(\mu ,Y)$ ($1< p <\infty$).References
- John Chaney, Banach lattices of compact maps, Math. Z. 129 (1972), 1–19. MR 312329, DOI 10.1007/BF01229536
- Stuart F. Cullender and Coenraad C. A. Labuschagne, A description of norm-convergent martingales on vector-valued $L^p$-spaces, J. Math. Anal. Appl. 323 (2006), no. 1, 119–130. MR 2261155, DOI 10.1016/j.jmaa.2005.10.032
- Stuart F. Cullender and Coenraad C. A. Labuschagne, A note on the $M$-norm of Chaney-Schaefer, Quaest. Math. 30 (2007), no. 2, 151–158. MR 2337359, DOI 10.2989/16073600709486190
- Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
- N. Dinculeanu, Integral representation of linear operators. I, Stud. Cerc. Mat. 18 (1966), 349–385 (Romanian). MR 213504
- N. Dinculeanu, Integral representation of linear operators. I, Stud. Cerc. Mat. 18 (1966), 349–385 (Romanian). MR 213504
- Nicolae Dinculeanu, Linear operations on $L^{p}$-spaces, Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972) Academic Press, New York, 1973, pp. 109–124. MR 0341066
- L. Egghe, Stopping time techniques for analysts and probabilists, London Mathematical Society Lecture Note Series, vol. 100, Cambridge University Press, Cambridge, 1984. MR 808582, DOI 10.1017/CBO9780511526176
- H. Jacobs, Ordered topological tensor products, Ph.D. thesis, University of Illinois, 1969.
- G.A.M. Jeurnink, Integration of functions with values in a Banach lattice, Ph.D. thesis, University of Nijmegen, The Netherlands, 1982.
- Shizuo Kakutani, Concrete representation of abstract $(L)$-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523–537. MR 4095, DOI 10.2307/1968915
- J. L. Krivine, Théorèmes de factorisation dans les espaces réticulés, Séminaire Maurey-Schwartz 1973–1974: Espaces $L^{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 22 et 23, Centre de Math., École Polytech., Paris, 1974, pp. 22 (French). MR 0440334
- C. C. A. Labuschagne, Characterizing the one-sided tensor norms $\Delta _p$ and $^t\Delta _p$, Quaest. Math. 27 (2004), no. 4, 339–363. MR 2125036, DOI 10.2989/16073600409486104
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
- Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093, DOI 10.1007/978-3-642-76724-1
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6
- Vladimir G. Troitsky, Martingales in Banach lattices, Positivity 9 (2005), no. 3, 437–456. MR 2188530, DOI 10.1007/s11117-004-2769-1
Additional Information
- Stuart F. Cullender
- Affiliation: School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, P.O. WITS 2050, South Africa
- Email: scullender@gmail.com
- Coenraad C. A. Labuschagne
- Affiliation: School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, P.O. WITS 2050, South Africa
- Email: Coenraad.Labuschagne@wits.ac.za
- Received by editor(s): August 1, 2007
- Published electronically: June 24, 2008
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 3883-3893
- MSC (2000): Primary 46B28, 47B60, 60G48
- DOI: https://doi.org/10.1090/S0002-9939-08-09537-3
- MathSciNet review: 2425728