Completability and optimal factorization norms in tensor products of Banach function spaces

[EN] Given s-finite measure spaces ( 1, 1, mu 1) and ( 2, 2, mu 2), we consider Banach spaces X1(mu 1) and X2(mu 2), consisting of L0(mu 1) and L0(mu 2) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product X1(mu 1). p X2(mu 2) is continuously included in the metric space of measurable functions L0(mu 1. mu 2). In particular, we prove that the elements of the completion of the projective tensor product of L p-spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally showthat given a bounded linear operator T : X1(mu 1). p X2(mu 2). E (where E is a Banach space), a norm can be found for T to be bounded, which is ` minimal' with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces.J. M. Calabuig and M. Fernandez-Unzueta were supported by Ministerio de Economia, Industria y Competitividad (Spain) under project MTM2014-53009-P. M. Fernandez-Unzueta was also suported by CONACyT 284110. F. Galaz-Fontes was supported by Ministerio de Ciencia e Innovacion (Spain) and FEDER under project MTM2009-14483-C02-01. E. A. Sanchez Perez was supported by Ministerio de Economia, Industria y Competitividad (Spain) and FEDER under project MTM2016-77054-C2-1-P.Calabuig, JM.; Fernández-Unzueta, M.; Galaz-Fontes, F.; Sánchez Pérez, EA. (2019). Completability and optimal factorization norms in tensor products of Banach function spaces. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):3513-3530. https://doi.org/10.1007/s13398-019-00711-7S351335301134Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics, Vol 50, AMS (2002)Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)Bu, Q., Buskes, G., Kusraev, A.G.: Bilinear maps on products of vector lattices: a survey. In: Boulabiar, K., Buskes, G., Triki, A. (eds.) Positivity-Trends in Mathematics. Birkhäser Verlag AG, Basel, pp. 97–26 (2007)Buskes, G., Van Rooij, A.: Bounded variation and tensor products of Banach lattices. Positivity 7, 47–59 (2003)Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F., Sánchez-Pérez, E.A.: Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. RACSAM 108(2), 353–367 (2014)Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F., Sánchez-Pérez, E.A.: Equivalent norms in a Banach function space and the subsequence property. J. Korean Math. Soc. https://doi.org/10.4134/JKMS.j180682Curbera, G.P., Ricker, W.J.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47–63 (2002)Curbera, G.P., Ricker, W.J.: Vector measures, integration and applications. In: Positivity. Birkhäuser Basel, pp. 127–160 (2007)Gil de Lamadrid, J.: Uniform cross norms and tensor products. J. Duke Math. 32, 797–803 (1965)Dunford, N., Schwartz, J.: Linear Operators, Part I: General Theory. Interscience Publishers Inc., New York (1958)Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94(3), 777–798 (1972)Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211(2), 87–106 (1974)Yew, K.L.: Completely $$p$$-summing maps on the operator Hilbert space OH. J. Funct. Anal. 255, 1362–1402 (2008)Kwapien, S., Pelczynski, A.: The main triangle projection in matrix spaces and its applications. Stud. Math. 34(1), 43–68 (1970)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland Publishing Company, Amsterdam (1971)Milman, M.: Some new function spaces and their tensor products. Depto. de Matemática, Facultad de Ciencias, U. de los Andes, Mérida, Venezuela (1978)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. Oper. Theory Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Schep, A.R.: Factorization of positive multilinear maps. Illinois J. Math. 579–591 (1984)Zaanen, A.C.: Integration. North-Holland Publishing Company, Amsterdam-New York (1967)Zaanen, A.C.: Riesz Spaces II. North-Holland Publishing Company, Amsterdam (1983

On the Banach lattice structure of L-w(1) of a vector measure on a delta-ring

We study some Banach lattice properties of the space L-w(1)(v) of weakly integrable functions with respect to a vector measure v defined on a delta-ring. Namely, we analyze order continuity, order density and Fatou type properties. We will see that the behavior of L-w(1)(v) differs from the case in which is defined on a sigma-algebra whenever does not satisfy certain local sigma-finiteness property.J. M. Calabuig and M. A. Juan were supported by the Ministerio de Economia y Competitividad (project MTM2008-04594). O. Delgado was supported by the Ministerio de Economia y Competitividad (project MTM2009-12740-C03-02). E. A. Sanchez Perez was supported by the Ministerio de Economia y Competitividad (project MTM2009-14483-C02-02).Calabuig Rodriguez, JM.; Delgado Garrido, O.; Juan Blanco, MA.; Sánchez Pérez, EA. (2014). On the Banach lattice structure of L-w(1) of a vector measure on a delta-ring. Collectanea Mathematica. 65(1):67-85. doi:10.1007/s13348-013-0081-8S6785651Brooks, J.K., Dinculeanu, N.: Strong additivity, absolute continuity and compactness in spaces of measures. J. Math. Anal. Appl. 45, 156–175 (1974)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, 88–103 (2010)Calabuig, J.M., Juan, M.A., Sánchez Pérez, E.A.: Spaces of $$p$$ -integrable functions with respect to a vector measure defined on a $$\delta$$ -ring. Oper. Matrices 6, 241–262 (2012)Curbera, G.P.: El espacio de funciones integrables respecto de una medida vectorial. Ph. D. thesis, University of Sevilla, Sevilla (1992)Curbera, G.P.: Operators into $$L^1$$ of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. (N.S.) 17, 187–204 (2006)G. P. Curbera and W. J. Ricker, Vector measures, integration and applications. In: Positivity (in Trends Math.), Birkhäuser, Basel, pp. 127–160 (2007)Curbera, G.P., Ricker, W.J.: The Fatou property in $$p$$ -convex Banach lattices. J. Math. Anal. Appl. 328, 287–294 (2007)Delgado, O.: $$L^1$$ -spaces of vector measures defined on $$\delta$$ -rings. Arch. Math. 84, 432–443 (2005)Delgado, O.: Optimal domains for kernel operators on $$[0,\infty )\times [0,\infty )$$ . Studia Math. 174, 131–145 (2006)Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244, 119–133 (2007)Delgado, O., Juan, M.A.: Representation of Banach lattices as $$L_w^1$$ spaces of a vector measure defined on a $$\delta$$ -ring. Bull. Belg. Math. Soc. Simon Stevin 19(2), 239–256 (2012)Diestel, J., Uhl, J.J.: Vector measures (Am. Math. Soc. surveys 15). American Mathematical Society, Providence (1997)Dinculeanu, N.: Vector measures, Hochschulbcher fr Mathematik, vol. 64. VEB Deutscher Verlag der Wissenschaften, Berlin (1966)Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., Sánchez Pérez, E.A.: Spaces of $$p$$ -integrable functions with respect to a vector measure. Positivity 10, 1–16 (2006)Fremlin, D.H.: Measure theory, broad foundations, vol. 2. Torres Fremlin, Colchester (2001)Jiménez Fernández, E., Juan, M.A., Sánchez Pérez, E.A.: A Komlós theorem for abstract Banach lattices of measurable functions. J. Math. Anal. Appl. 383, 130–136 (2011)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Luxemburg, W.A.J., Zaanen, A.C.: Riesz spaces I. North-Holland, Amsterdam (1971)Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on $$\delta$$ -rings. Adv. Math. 73, 204–241 (1989)Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 75, 121–167 (1989)Thomas, E.G.F.: Vector integration (unpublished) (2013)Turpin, Ph.: Intégration par rapport à une mesure à valeurs dans un espace vectoriel topologique non supposé localement convexe, Intègration vectorielle et multivoque, (Colloq., University Caen, Caen, 1975), experiment no. 8, Dèp. Math., UER Sci., University Caen, Caen (1975)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces (Oper. Theory Adv. Appl.), vol. 180. Birkhäuser, Basel (2008)Zaanen, A.C.: Riesz spaces II. North-Holland, Amsterdam (1983