A description of Banach space-valued Orlicz hearts

• Autorzy: Coenraad Labuschagne , Theresa Offwood
• Języki publikacji: EN

Abstrakty

EN

Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $$ H_\varphi \left( \mu \right)\tilde \otimes _l Y $$ of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $$ \left( {H_\varphi \left( \mu \right)\tilde \otimes _l Y} \right)* $$ and $$ H_\varphi \left( \mu \right)*\tilde \otimes _l Y* $$ in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým property on Y. Using the associativity of the l-norm, an alternative proof is given of the known fact that for any separable Banach lattice E and any Banach space Y, E and Y have the Radon-Nikodým property if and only if $$ E\tilde \otimes _l Y $$ has the Radon-Nikodým property. As a corollary, the Radon-Nikodým property in H φ(μ, Y) is described in terms of the Radon-Nikodým property on H φ(μ) and Y.

Słowa kluczowe

EN

Bochner norm; l-norm; Banach lattice; Banach space; Martingale; Orlicz space; Orlicz heart; Tensor product; Radon-Nikodým property

Kategorie tematyczne

• 46A45: Sequence spaces (including K\"othe sequence spaces)
• 46B45: Banach sequence spaces
• 46B42: Banach lattices
• 46B28: Spaces of operators; tensor products; approximation properties

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 6
• Strony: 1109-1119

Daty

wydano

2010-12-01

online

2010-10-30

Twórcy

autor

Coenraad Labuschagne

• University of the Witwatersrand

autor

Theresa Offwood

• University of the Witwatersrand

Bibliografia

• [1] Birnbaum Z., Orlicz W., Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen, Studia Math., 1931, 3, 1–67
• [2] Bombal F., On l 1 subspaces of Orlicz vector-valued function spaces, Math. Proc. Cambridge Philos. Soc., 1987, 101, 107–112 http://dx.doi.org/10.1017/S0305004100066445
• [3] Chaney J., Banach lattices of compact maps, Math. Z., 1972, 129(1), 1–19 http://dx.doi.org/10.1007/BF01229536
• [4] Cullender S.F., Labuschagne C.C.A., A description of norm-convergent martingales on vector-valued L p-spaces, J. Math. Anal. Appl., 2006, 323(1), 119–130 http://dx.doi.org/10.1016/j.jmaa.2005.10.032
• [5] Cullender S.F., Labuschagne C.C.A., Convergent martingales of operators and the Radon Nikodým property in Banach spaces, Proc. Amer. Math. Soc., 2008, 136(11), 3883–3893 http://dx.doi.org/10.1090/S0002-9939-08-09537-3
• [6] Dinculeanu N., Integral representation of linear operators. I, II, Stud. Cerc. Mat., 1966, 18, 349–385, 483–536
• [7] Diestel J., An approach to the theory of Orlicz spaces of Lesbesgue-Bochner measurable functions, Math. Ann., 1970, 186(1), 20–33 http://dx.doi.org/10.1007/BF01350637
• [8] Diestel J., On the representation of bounded, linear operators from Orlicz spaces of Lebesgue-Bochner measurable functions to any Banach space, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., 1970, 18, 375–378
• [9] Diestel J., Some remarks on subspaces of Orlicz spaces of vector-valued finitely additive functions, Studia Math., 1971, 39, 161–164
• [10] Diestel J., Uhl J.J., Vector Measures, Math. Surveys Monogr., 15, AMS, Providence, 1977
• [11] Edgar G.A., Sucheston L., Stopping Times and Directed Processes, Encyclopedia Math. Appl., 47, Cambridge University Press, Cambridge, 1992
• [12] Greub W.H., Multilinear Algebra, Grundlehren Math. Wiss., 136, Springer, Berlin-Heidelberg-New York, 1967
• [13] Jeurnink G.A.M., Integration of functions with values in a Banach lattice, Ph.D. thesis, University of Nijmegen, the Netherlands, 1982
• [14] Krasnosel’skii M.A., Rutitskii Ya.B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961
• [15] Labuschagne C.C.A., A note on the order continuity of the norm of \( E\tilde \otimes _m F \) , Arch. Math., 1994, 62(4), 335–337 http://dx.doi.org/10.1007/BF01201786
• [16] Labuschagne C.C.A., Riesz reasonable cross norms on tensor products of Banach lattices, Quaest. Math., 2004, 27(3), 243–266
• [17] Labuschagne C.C.A., Characterizing the one-sided tensor norms Δp and tΔp, Quaest. Math., 2004, 27(4), 339–363
• [18] Labuschagne C.C.A., A Dodds-Fremlin property for Asplund and Radon-Nikodým operators, Positivity, 2006, 10(2), 391–407 http://dx.doi.org/10.1007/s11117-005-0023-0
• [19] Lindenstrauss Y., Tzafriri L., Classical Banach Spaces, I and II, Ergeb. Math. Grenzgeb., 92 and 97, Springer, Berlin-Heidelberg-New York, 1977 and 1979
• [20] Luxemburg W.A.J., Banach function spaces, Ph.D. thesis, Technische Hogeschool te Delft, 1955
• [21] Meyer-Nieberg P., Banach Lattices, Universitext, Springer, Berlin-Heidelberg-New York, 1991
• [22] Orlicz W., Über eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Polon. Sci. A, 1932, 8–9, 207–220
• [23] Popa N., Die Permanenzeigenschaften der Tensorprodukte von Banachverbänden, In: Romanian-Finnish Seminar on Complex Analysis, Bucharest, 1976, Lecture Notes in Math., 743, Springer, Berlin, 1979, 627–647
• [24] Rao M.M., Ren Z.D., Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 250, Marcel Dekker, New York, 2002
• [25] Schaefer H.H., Banach Lattices and Positive Operators, Grundlehren Math. Wiss., 215, Springer, Berlin-Heidelberg-New York, 1974
• [26] Sundaresan K., The Radon-Nikodym theorem for Lebesgue-Bochner function spaces, J. Funct. Anal., 1977, 24(3), 276–279 http://dx.doi.org/10.1016/0022-1236(77)90058-1
• [27] Talagrand M., La structure des espaces de Banach réticulés ayant la propriété de Radon-Nikodym, Israel J. Math., 1983, 44(3), 213–220 http://dx.doi.org/10.1007/BF02760972
• [28] Talagrand M., Pettis Integral and Measure Theory, Mem. Amer. Math. Soc., 307, AMS, Providence, 1984
• [29] Turett B., Uhl J.J., L p(μ, X) (1 < p < ∞) has the Radon-Nikodým property if X does by martingales, Proc. Amer. Math. Soc., 1976, 61(2), 347–350
• [30] Uhl J.J., Applications of Radon-Nikodým theorems to martingale convergence, Trans. Amer. Math. Soc., 1969, 145, 271–285
• [31] Uhl J.J., The Radon-Nikodým theorem and the mean convergence of Banach space valued martingales, Proc. Amer. Math. Soc., 1969, 21(1), 139–144
• [32] Zaanen A.C., Linear Analysis, Interscience, New York & North-Holland, Amsterdam & Noordhoff, Groningen, 1953
• [33] Zaanen A.C., Riesz Spaces II, North-Holland Math. Library, 30, North-Holland, Amsterdam-New York-Oxford, 1983

Identyfikatory

DOI

10.2478/s11533-010-0065-9