Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces

Abstract

We consider the problem of extending or factorizing a bounded bilinear map defined on a couple of order continuous Banach function spaces to its optimal domain, i.e. the biggest couple of Banach function spaces to which the bilinear map can be extended. As in the case of linear operators, we use vector measure techniques to find this space, and we show that this procedure cannot be always successfully used for bilinear maps. We also present some applications to find optimal factorizations of linear operators between Banach function spaces.J. M. Calabuig was supported by Ministerio de Economia y Competitividad (Spain) (project MTM2011-23164) and by "Jose Castillejo 2009" (MEC). E. A. Sanchez-Perez was supported by MEC and FEDER (project MTM2009-14483-C02-02). J. M. Calabuig and E. A. Sanchez-Perez were also supported by Ayuda para Estancias de PDI de la UPV en Centros de Investigacion de Prestigio (PAID-00-11).Calabuig Rodriguez, JM.; Fernandez Unzueta, M.; Galaz Fontes, F.; Sánchez Pérez, EA. (2014). Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas (RACSAM). 108(2):353-367. https://doi.org/10.1007/s13398-012-0101-7S3533671082Calabuig, J.M., Galaz-Fontes, F., Jiménez Fernández, E., Sánchez Pérez, E.A.: Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series. Math. Z. 257, 381–402 (2007)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)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.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47–63 (2002)Curbera, G.P. , Ricker, W.J.: Optimal domains for the kernel operator associated with Sobolev’s inequality. Studia Math. 158(2), 131–152 (2003) [see also Corrigenda in the same journal, 170 (2005) 217–218)]Delgado, O.: Banach function subspaces of $$L^1$$ of a vector measure and related Orlicz spaces. Indag. Math. (N. S.) 15, 485–495 (2004)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)Diestel, J., Uhl, J.J.: Vector measures. In: Math. Surveys, vol. 15. Amer. Math. Soc., Providence (1977)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Galdames, O., Sánchez Pérez, E.A.: Optimal range theorems for operators with $$p$$ -th power factorable adjoints. Banach J. Math. Anal. 6(1), 61–73 (2012)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. In: Oper. Theory Adv. Math. Appl., vol. 180. Birkäuser, Basel (2008