Abstract
We provide a tensor product representation of Köthe-Bochner function spaces of vector valued integrable functions. As an application, we show that the dual space of a Köthe-Bochner function space can be understood as a space of operators satisfying a certain extension property. We apply our results in order to give an alternate representation of the dual of the Bochner spaces of p-integrable functions and to analyze some properties of the natural norms \(\Delta _p\) that are defined on the associated tensor products.
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References
Bochner, S.: Integration von Funktionen, deren Werte die Elemente eines Vectorraumes sind. Fundamenta Mathematicae 20, 262–276 (1933)
Calabuig, J.M., Delgado, O., Juan, M.A., Sánchez, E.A.: Pérez, On the Banach lattice structure of \(L^1_w\) of a vector measure on a \(\delta \)-ring. Collect. Math. 65, 6567–85 (2014)
Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364(1), 88–103 (2010)
Calabuig, J.M., Gregori, P., Sánchez, E.A.: Pérez, Radon-Nikodým derivatives for vector measures belonging to Köthe function spaces. J. Math. Anal. Appl. 348, 469–479 (2008)
Cerdà, J., Hudzik, H., Mastyło, M.: Geometric properties of Köthe-Bochner spaces. Math. Proc. Cambridge Philos. Soc. 120(3), 521–533 (1996)
Chakraborty, N.D., Basu, S.: Spaces of p-tensor integrable functions and related Banach space properties. Real Anal. Exchange 34, 87–104 (2008)
Chakraborty, N.D., Basu, S.: Integration of vector-valued functions with respect to vector measures defined on \(\delta \)-rings. Ill. J. Math. 55(2), 495–508 (2011)
Defant, A., Floret, K.: Tensor norms and operator ideals. North-Holland, Amsterdam (1993)
Delgado, O., Juan, M.A.: Representation of Banach lattices as \(L^{1}_{w}\) spaces of a vector measure defined on a \(\delta -\)ring. Bull. Belgian Math. Soc. 19, 239–256 (2012)
Diestel, J., Uhl, J.J.: Vector measures. Am. Math. Soc, Providence (1977)
Dobrakov, I.: On integration in Banach spaces, VII. Czechoslovak Math. J. 38, 434–449 (1988)
García-Raffi, L.M., Jefferies, B.: An application of bilinear integration to quantum scattering. J. Math. Anal. Appl. 415, 394–421 (2014)
Gregori Huerta, P.: Espacios de medidas vectoriales. Thesis, Universidad de Valencia, ISBN:8437060591 (2005)
Jefferies, B., Okada, S.: Bilinear integration in tensor products. Rocky Mt. J. Math. 28, 517–545 (1998)
Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)
Lin, P.-K.: Köthe-Bochner function spaces. Birkhauser, Boston (2004)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)
Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domains and integral extensions of operators acting in function spaces. Operator Theory Advances and Applications, vol. 180. Birkhäuser, Basel (2008)
Pallu de La Barriére, R.: Integration of vector functions with respect to vector measures. Studia Univ. Babes-Bolyai Math. 43, 55–93 (1998)
Rodríguez, J.: On integration of vector functions with respect to vector measures. Czechoslovak Math. J. 56, 805–825 (2006)
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First and third authors are supported by grant MTM201453009-P of the Ministerio de Economía y Competitividad (Spain). Second and fourth authors are supported by grant MTM2012-36740-C02-02 of the Ministerio de Economía y Competitividad (Spain).
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Calabuig, J.M., Jiménez Férnandez, E., Juan, M.A. et al. Tensor product representation of Köthe-Bochner spaces and their dual spaces. Positivity 20, 155–169 (2016). https://doi.org/10.1007/s11117-015-0347-3
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DOI: https://doi.org/10.1007/s11117-015-0347-3