A characterization of operators with p-summing adjoint via p-limited sets

A CHARACTERIZATION OF OPERATORS WITH p-SUMMING ADJOINT VIA p-LIMITED SETS JUAN MANUEL DELGADO AND C´ ANDIDO PI˜ NEIRO A subset Aof a Banach space Xis limited if and only if for every weak∗null sequence (x∗ n) in X∗there exists a sequence (αn)∈c0such that |hx∗ n, xi| ≤ αnfor all x∈Aand n∈N. As an extension of this notion, a subset Aof a Banach space Xis said to be p-limited (p∈[1,∞)) if for every weakly p-summable sequence (x∗ n) in X∗there exists (αn)∈`psuch that |hx∗ n, xi| ≤ αnfor all x∈Aand n∈N[2]. Some basic properties related to this notion are showed as well as its connections with (different forms of) compactness. As an application, we give a characterization of operators with p-summing adjoint as those mapping relatively compact sets to p-limited sets. The research was supported by MTM2009-14483-C02-01 and MTM201236740-C02-01 projects (Spain). References [1] J. M. Delgado and C. Pi˜ neiro,A note on p-limited sets. preprint. [2] A. K. Karn and D. P. Sinha,An operator summability in sequences in Banach spaces, arXiv:1207.3620 [math.FA] (2012). Higher Technical School of Architecture, University of Seville, 41012 Seville, Av. Reina Mercedes, 2, Spain E-mail address:[email protected] Faculty of Experimental Sciences, University of Huelva, 21071 Huelva, Campus Universitario de El Carmen, Spain E-mail address:[email protected] 1