Factorization of operators through subspaces of L-1-spaces

[EN] We analyze domination properties and factorization of operators in Banach spaces through subspaces of L1-spaces. Using vector measure integration and extending classical arguments based on scalar integral bounds, we provide characterizations of operators factoring through subspaces of L1-spaces of finite measures. Some special cases involving positivity and compactness of the operators are considered.Research supported by MINECO/FEDER under projects MTM2014-53009-P (J.M Calabuig), MTM2014-54182-P (J. Rodriguez) and MTM2012-36740-C02-02 (E. A. Sanchez-Perez).Calabuig, JM.; Rodríguez, J.; Sánchez Pérez, EA. (2017). Factorization of operators through subspaces of L-1-spaces. Journal of the Australian Mathematical Society. 103(3):313-328. https://doi.org/10.1017/S14467887160005133133281033Lindenstrauss, J., & Tzafriri, L. (1979). Classical Banach Spaces II. doi:10.1007/978-3-662-35347-9Pisier, G. (1986). Factorization of Linear Operators and Geometry of Banach Spaces. CBMS Regional Conference Series in Mathematics. doi:10.1090/cbms/060Okada, S., Ricker, W. J., & Sánchez Pérez, E. A. (2008). Optimal Domain and Integral Extension of Operators. doi:10.1007/978-3-7643-8648-1Lacey, H. E. (1974). The Isometric Theory of Classical Banach Spaces. doi:10.1007/978-3-642-65762-7Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., & Sánchez-Pérez, E. A. (2005). Vector measure Maurey–Rosenthal-type factorizations and ℓ-sums of L1-spaces. Journal of Functional Analysis, 220(2), 460-485. doi:10.1016/j.jfa.2004.06.010Juan, M. A., & Sánchez Pérez, E. A. (2013). Maurey-Rosenthal domination for abstract Banach lattices. Journal of Inequalities and Applications, 2013(1). doi:10.1186/1029-242x-2013-213Avilés, A., Cabello Sánchez, F., Castillo, J. M. F., González, M., & Moreno, Y. (2013). On separably injective Banach spaces. Advances in Mathematics, 234, 192-216. doi:10.1016/j.aim.2012.10.013Defant, A., & Sánchez Pérez, E. A. (2004). Maurey–Rosenthal factorization of positive operators and convexity. Journal of Mathematical Analysis and Applications, 297(2), 771-790. doi:10.1016/j.jmaa.2004.04.047DEFANT, A., & PÉREZ, E. A. S. (2009). Domination of operators on function spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 146(1), 57-66. doi:10.1017/s0305004108001734Bartle, R. G., Dunford, N., & Schwartz, J. (1955). Weak Compactness and Vector Measures. Canadian Journal of Mathematics, 7, 289-305. doi:10.4153/cjm-1955-032-1Rosenthal, H. P. (1974). A Characterization of Banach Spaces Containing l1. Proceedings of the National Academy of Sciences, 71(6), 2411-2413. doi:10.1073/pnas.71.6.2411Diestel, J., Jarchow, H., & Tonge, A. (1995). Absolutely Summing Operators. doi:10.1017/cbo9780511526138Rueda, P., & Sánchez-Pérez, E. A. (2015). Compactness in spaces of p-integrable functions with respect to a vector measure. Topological Methods in Nonlinear Analysis, 45(2), 641. doi:10.12775/tmna.2015.030Rosenthal, H. P. (1973). On Subspaces of L p. The Annals of Mathematics, 97(2), 344. doi:10.2307/1970850Diestel, J., & Uhl, J. (1977). Vector Measures. Mathematical Surveys and Monographs. doi:10.1090/surv/015[16] M. Mastyło and E. A. Sánchez-Pérez , ‘Factorization of operators through Orlicz spaces’, Bull. Malays. Math. Sci. Soc. doi:10.1007/s40840-015-0158-5, to appear.Calabuig, J. M., Lajara, S., Rodríguez, J., & Sánchez-Pérez, E. A. (2014). Compactness in L1of a vector measure. Studia Mathematica, 225(3), 259-282. doi:10.4064/sm225-3-6Defant, A. (2001). Positivity, 5(2), 153-175. doi:10.1023/a:1011466509838Fabian, M., Habala, P., Hájek, P., Montesinos, V., & Zizler, V. (2011). Banach Space Theory. CMS Books in Mathematics. doi:10.1007/978-1-4419-7515-