Unconditionality for m-homogeneous polynomials on l(infinity)n

[EN] Let x(m, n) be the unconditional basis constant of the monomial basis Z alpha, alpha is an element of N(0)n with vertical bar alpha vertical bar = m, of the Banach space of all m-homogeneous polynomials inn complex variables, endowed with the supremum norm on the n-dimensional unit polydisc Dn. We prove that the quotient of sup(m) m root sup(m)chi(m, n) and root n/log n tends to 1 as n -> infinity. This reflects a quite precise dependence of chi(m, n) on the degree m of the polynomials and their number n of variables. Moreover, we give an analogous formula for m-linear forms, a reformulation of our results in terms of tensor products, and as an application a solution for a problem on Bohr radii.This research was supported by MCINN project MTM2014-57838-C2-2-P.Defant, A.; Sevilla Peris, P. (2016). Unconditionality for m-homogeneous polynomials on l(infinity)n. Studia Mathematica. 232(1):45-55. https://doi.org/10.4064/sm8386-2-2016S4555232

Convergence of monomial expansions in banach spaces

[EN] If E is a Banach sequence space, then each holomorphic function defines a formal power series ¿ ¿ c ¿(f) z ¿. The problem of when such an expansion converges absolutely and actually represents the function goes back to the very beginning of the theory of holomorphic functions on infinite-dimensional spaces. Several very deep results have been given for scalar-valued functions by Ryan, Lempert and Defant, Maestre and Prengel. We go on with this study, looking at monomial expansions of vector-valued holomorphic functions on Banach spaces. Some situations are very different from the scalar-valued case. © 2011 Published by Oxford University Press. All rights reserved.Both authors were supported by the MEC Project MTM2008-03211. The second cited author was partially supported by grants PR2007-0384 (MEC) and UPV-PAID-00-07.Defant, A.; Sevilla Peris, P. (2012). Convergence of monomial expansions in banach spaces. Quarterly Journal of Mathematics. 63(3):569-584. https://doi.org/10.1093/qmath/haq053S56958463

Convergence of Dirichlet polynomials in Banach spaces

[EN] Recent results on Dirichlet series Sigma(n) a(n) 1/n(s), s is an element of C, with coefficients a(n) in an infinite dimensional Banach space X show that the maximal width of uniform but not absolute convergence coincides for Dirichlet series and for m-homogeneous Dirichlet polynomials. But a classical non-trivial fact fue to Bohnenblust and Hille shows that if X is one dimensional, this maximal width heavily depends on the degree m of the Dirichlet polynomials. We carefully analyze this phenomenon, in particular in the setting of l(p)-spaces.Both authors were supported by the MEC Project MTM2008-03211 The second author was partially supported by grants PR2007 0384 (MEC) and UPV PAID 00-07Defant, A.; Sevilla Peris, P. (2011). Convergence of Dirichlet polynomials in Banach spaces. Transactions of the American Mathematical Society. 363(2):691-697. https://doi.org/10.1090/S0002-9947-2010-05146-3S691697363