Closed injective ideals of multilinear operators, related measures and interpolation

This is the peer reviewed version of the following article: Manzano, A, Rueda, P, Sánchez-Pérez, EA. Closed injective ideals of multilinear operators, related measures and interpolation. Mathematische Nachrichten. 2020; 293: 510-532, which has been published in final form at https://doi.org/10.1002/mana.201800415. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[EN] We introduce and discuss several ways of extending the inner measure arisen from the closed injective hull of an ideal of linear operators to the multilinear case. In particular, we consider new measures that allow to characterize the operators that belong to a closed injective ideal of multilinear operators as those having measure equal to zero. Some interpolation formulas for these measures, and consequently interpolation results involving ideals of multilinear operators, are established. Examples and applications related to summing multilinear operators are also shown.Ministerio de Economia, Industria y Competitividad and FEDER, Grant/Award Numbers: MTM2016-77054-C2-1-P, MTM2017-84058-PManzano, A.; Rueda, P.; Sánchez Pérez, EA. (2020). Closed injective ideals of multilinear operators, related measures and interpolation. Mathematische Nachrichten. 293(3):510-532. https://doi.org/10.1002/mana.2018004155105322933Arens, R. (1951). The adjoint of a bilinear operation. Proceedings of the American Mathematical Society, 2(6), 839-848. doi:10.1090/s0002-9939-1951-0045941-1Aron, R. M., & Berner, P. D. (1978). A Hahn-Banach extension theorem for analytic mappings. Bulletin de la Société mathématique de France, 79, 3-24. doi:10.24033/bsmf.1862Aron, R. M., Cole, B. J., & Gamelin, T. W. (1995). Weak-Star Continuous Analytic Functions. Canadian Journal of Mathematics, 47(4), 673-683. doi:10.4153/cjm-1995-035-1Bergh, J., & Löfström, J. (1976). Interpolation Spaces. Grundlehren der mathematischen Wissenschaften. doi:10.1007/978-3-642-66451-9Besoy, B. F., & Cobos, F. (2019). Interpolation of the measure of non-compactness of bilinear operators among quasi-Banach spaces. Journal of Approximation Theory, 243, 25-44. doi:10.1016/j.jat.2019.03.004Beucher, O. J. (1989). On interpolation of strictly (co-)singular linear operators. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 112(3-4), 263-269. doi:10.1017/s0308210500018734Botelho, G. (2002). Weakly Compact and Absolutely Summing Polynomials. Journal of Mathematical Analysis and Applications, 265(2), 458-462. doi:10.1006/jmaa.2001.7674Botelho, G., Pellegrino, D., & Rueda, P. (2007). On Composition Ideals of Multilinear Mappings and Homogeneous Polynomials. Publications of the Research Institute for Mathematical Sciences, 43(4), 1139-1155. doi:10.2977/prims/1201012383Braunss, H.-A., & Junek, H. (2004). Factorization of injective ideals by interpolation. Journal of Mathematical Analysis and Applications, 297(2), 740-750. doi:10.1016/j.jmaa.2004.04.045Cobos, F. (1999). Interpolation theory and measures related to operator ideals. The Quarterly Journal of Mathematics, 50(200), 401-416. doi:10.1093/qjmath/50.200.401Cobos, F., Manzano, A., Martínez, A., & Matos, P. (2000). On interpolation of strictly singular operators, strictly co-singular operators and related operator ideals. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 130(5), 971-989. doi:10.1017/s0308210500000524Cobos, F., & Martínez, A. (1999). Remarks on Interpolation Properties of the Measure of Weak Non -- Compactness and Ideal Variations. Mathematische Nachrichten, 208(1), 93-100. doi:10.1002/mana.3212080104Dahia, E., Achour, D., & Sánchez Pérez, E. A. (2013). Absolutely continuous multilinear operators. Journal of Mathematical Analysis and Applications, 397(1), 205-224. doi:10.1016/j.jmaa.2012.07.034González, M., & Gutiérrez, J. M. (1999). Injective factorization of holomorphic mappings. Proceedings of the American Mathematical Society, 127(6), 1715-1721. doi:10.1090/s0002-9939-99-04917-5González, M., & Gutiérrez, J. (2000). Erratum to «Injective factorization of holomorphic mappings». Proceedings of the American Mathematical Society, 129(4), 1255-1256. doi:10.1090/s0002-9939-00-06055-xGrothendieck, A. (1954). Sur Certains Sous-Espaces Vectoriels De Lp. Canadian Journal of Mathematics, 6, 158-160. doi:10.4153/cjm-1954-017-xHeinrich, S. (1980). Closed operator ideals and interpolation. Journal of Functional Analysis, 35(3), 397-411. doi:10.1016/0022-1236(80)90089-0Jarchow, H. (1981). Locally Convex Spaces. Mathematische Leitfäden. doi:10.1007/978-3-322-90559-8Kato, T. (1958). Perturbation theory for nullity, deficiency and other quantities of linear operators. Journal d’Analyse Mathématique, 6(1), 261-322. doi:10.1007/bf02790238López Molina, J. A., & Sánchez Pérez, E. A. (2000). On operator ideals related to (p,σ)-absolutely continuous operators. Studia Mathematica, 138(1), 25-40. doi:10.4064/sm-138-1-25-40Maligranda, L. (1986). Interpolation between sum and intersection of Banach spaces. Journal of Approximation Theory, 47(1), 42-53. doi:10.1016/0021-9045(86)90045-6Maligranda, L., & Quevedo, A. (1990). Interpolation of weakly compact operators. Archiv der Mathematik, 55(3), 280-284. doi:10.1007/bf01191169Matter, U. (1987). Absolutely Continuous Operators and Super-Reflexivity. Mathematische Nachrichten, 130(1), 193-216. doi:10.1002/mana.19871300118Pellegrino, D., Rueda, P., & Sánchez-Pérez, E. A. (2015). Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 110(1), 285-302. doi:10.1007/s13398-015-0224-8A.Pietsch Ideals of multilinear functionals (designs of a theory) Proceedings of the Second International Conference on Operator Algebras Ideals and Their Applications in Theoretical Physics Teubner‐Texte Math. vol. 67 Teubner Leipzig 1983 pp.185–199.Popa, D. (2016). A note on the concept of factorable strongly p-summing operators. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 111(2), 465-471. doi:10.1007/s13398-016-0307-1Tylli, H.-O. (1995). The essential norm of an operator is not self-dual. Israel Journal of Mathematics, 91(1-3), 93-110. doi:10.1007/bf0276164