Factorization through Lorentz spaces for operators acting in Banach function spaces

Abstract

[EN] We show a factorization through Lorentz spaces for Banach-space-valued operators defined in Banach function spaces. Although our results are inspired in the classical factorization theorem for operators from Ls-spaces through Lorentz spaces Lq,1 due to Pisier, our arguments are different and essentially connected with Maurey's theorem for operators that factor through Lp-spaces. As a consequence, we obtain a new characterization of Lorentz Lq,1-spaces in terms of lattice geometric properties, in the line of the (isomorphic) description of Lp-spaces as the unique ones that are p-convex and p-concave.Funding was provided by Secretaria de Estado de Investigacion, Desarrollo e Innovacion and FEDER (Grant No. MTM2016-77054-c2-1-P).Sánchez Pérez, EA. (2019). Factorization through Lorentz spaces for operators acting in Banach function spaces. Positivity. 23(1):75-88. https://doi.org/10.1007/s11117-018-0593-2S7588231Achour, D., Mezrag, L.: Factorisation des opèrateurs sous-linéaires par $$L^{p,\infty }(\varOmega , \nu )$$ L p , ∞ ( Ω , ν ) et $$L^{q,1} (\varOmega ,\nu )$$ L q , 1 ( Ω , ν ) . Ann. Sci. Math. Québec. 29, 109–121 (2002)Berg, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Heidelberg (1976)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi-Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Sánchez Pérez, E.A.: Domination of operators on function spaces. Math. Proc. Camb. Philos. Soc. 146, 57–66 (2009)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Kalton, N.J., Montgomery-Smith, S.J.: Set-functions and factorization. Arch. Math. 61, 183–200 (1993)Krivine, J.L.: Théorèmes de factorisation dans les espaces réticulés. Séminaire d’analyse fonctionelle Maurey-Schwartz 1973–1974. Exposés XXII et XXIII. p.1–22. École Polytechnique, Paris (1974)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Mastyło, M., Sánchez Pérez, E.A.: Factorization of operators through Orlicz spaces. Bull. Malays. Math. Sci. Soc. 40, 1653–1675 (2017)Mastyło, M., Szwedek, R.: Interpolative construction and factorization of operators. J. Math. Anal. Appl. 401, 198–208 (2013)Maurey, B.: Theorémes de factorisation pour les opèrateurs linéaires à valeurs dans les spaces Lp. Séminaire d’analyse fonctionelle Maurey-Schwartz. 1972–1973. Exposés XVII, p.1–5. École Polytechnique, Paris (1973)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Birkhäuser, Basel (2008)Pisier, G.: Factorization of operators through $$L_{p\infty }$$ L p ∞ or $$L_{p1}$$ L p 1 and noncommutative generalizations. Math. Ann. 276, 105–136 (1986)Rosenthal, H.P.: On subspaces of $$L^{p}$$ L p . Ann. Math. 97, 344–373 (1973