Isomorphic copies of l(1) for m-homogeneous non-analytic Bohnenblust-Hille polynomials

Abstract

[EN] We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m-homogeneous non-analytic polynomials on c(o) contains an isomorphic copy of l(1). Moreover, we can have this copy of l(1) in such a way that every non-zero element of it fails to be analytic at precisely the same point.This work was partially supported by Ministerio de Educacion, Cultura y Deporte, projects MTM201347093-P, MTM2014-57838-C2-2-P, and MTM2015-65825-PConejero, JA.; Seoane-Sepulveda, JB.; Sevilla Peris, P. (2017). Isomorphic copies of l(1) for m-homogeneous non-analytic Bohnenblust-Hille polynomials. Mathematische Nachrichten. 290(2-3):218-225. https://doi.org/10.1002/mana.201600082S2182252902-3Aron, R. M., Bernal-Gonzalez, L., Pellegrino, D. M., & Sepulveda, J. B. S. (2015). Lineability. doi:10.1201/b19277F. Bayart A. Defant L. Frerick M. Maestre P. Sevilla-Peris Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables, arXiv:1405.7205Bayart, F., Pellegrino, D., & Seoane-Sepúlveda, J. B. (2014). The Bohr radius of the n-dimensional polydisk is equivalent to(log⁡n)/n. Advances in Mathematics, 264, 726-746. doi:10.1016/j.aim.2014.07.029Bernal-González, L., Pellegrino, D., & Seoane-Sepúlveda, J. B. (2013). Linear subsets of nonlinear sets in topological vector spaces. Bulletin of the American Mathematical Society, 51(1), 71-130. doi:10.1090/s0273-0979-2013-01421-6Bohnenblust, H. F., & Hille, E. (1931). On the Absolute Convergence of Dirichlet Series. The Annals of Mathematics, 32(3), 600. doi:10.2307/1968255Defant, A., Frerick, L., Ortega-Cerdà, J., Ounaïes, M., & Seip, K. (2011). The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Annals of Mathematics, 174(1), 485-497. doi:10.4007/annals.2011.174.1.13Defant, A., & Sevilla-Peris, P. (2014). The Bohnenblust-Hille cycle of ideas from a modern point of view. Functiones et Approximatio Commentarii Mathematici, 50(1), 55-127. doi:10.7169/facm/2014.50.1.2Dineen, S. (1999). Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics. doi:10.1007/978-1-4471-0869-6Enflo, P. H., Gurariy, V. I., & Seoane-Sepúlveda, J. B. (2014). On Montgomery’s conjecture and the distribution of Dirichlet sums. Journal of Functional Analysis, 267(4), 1241-1255. doi:10.1016/j.jfa.2014.04.001Enflo, P. H., Gurariy, V. I., & Seoane-Sepúlveda, J. B. (2013). Some results and open questions on spaceability in function spaces. Transactions of the American Mathematical Society, 366(2), 611-625. doi:10.1090/s0002-9947-2013-05747-9O. Toeplitz Über eine bei den Dirichletschen Reihen auftretende Aufgabe aus der Theorie der Potenzreihen von unendlichvielen Veränderlichen, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen 417 432 191