Mappings between Banach spaces that preserve convergence of series

Abstract

We know that a numerical series is absolutely convergent, if, and only if, it is unconditionally convergent. Dvoretzky and Rogers proved in 1950 that in any infinite dimensional Banach space there are unconditionally convergent series not absolutely convergent. This result was the origin of the development of the study of the linear mappings between Banach spaces sending unconditionally summable sequences into absolutely summable sequences (the Theory of absolutely Summing Mappings). The Nonlinear Theory started with Pietsch in 1983, when he presented a few results for scalar multilinear mappings and homogeneous polynomials defined on Banach spaces. In 1989 this author started the study of absolutely summing holomorphic mappings between Banach spaces. In a minicourse, thaught in 1997 at a Seminario Brasileiro de Analise, this author presented several results dated of 1996 and 1997 on nonlinear absolutely summing mappings (not necessarily holomorphic mappings) between Banach spaces. This course included an interesting characterization of regularly summing mappings f between Banach spaces, that is those mappings such that is absolutely summable whenever is absolutely summable. This result, applied in a convenient setting, implied a nice characterization result for nonlinear absolutely summing mappings, bearing striking similarities to the correspond result for linear absolutely summing mappings. These and other results, as well as historical references, can be found in M. C. Matos, Math. Nachr. 258, 71-89 (2003). In this work we introduce and develop the concept of uniformly regularly summing mappings between Banach spaces, allowing an interesting characterization of the uniformly absolutely summing mappings between Banach spaces, thus extending previous results on nolinear absolutely summing mappings. We also state a relation between uniform regularity and the Lipschitz property for mappings between Banach spaces.