A generalization of p-convexity and q-concavity on Banach lattices

In this paper, considering a real Banach sequence lattice Y instead of a Lebesgue sequence space $l_p$ we generalize p-convexity of a linear operator $T:E\to X$, where E is a Banach space and X is a Banach lattice. Then we prove that basic properties of p-convexity remain valid for Y-convex linear operators. Analogous generalizations are given for q-concavity and p-summability and composition properties between these operators are analyzed.Comment: 25 pages, currently in peer review phase at Positivity Journa

Equivalent norms in a banach function space and the subsequence property

[EN] Consider a finite measure space (Omega, Sigma, mu) and a Banach space X(mu) consisting of (equivalence classes of) real measurable functions defined on Omega such that f chi(A) is an element of X(mu) and parallel to f chi(A)parallel to <= parallel to f parallel to, for all f is an element of X(mu), A is an element of Sigma. We prove that if it satisfies the subsequence property, then it is an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.All the authors were supported by Ministerio de Ciencia, Innovacion y Universidades (Spain), Agencia Estatal de Investigaciones, and FEDER. J.M. Calabuig and M. Fernandez-Unzueta under project MTM2014-53009-P. F. Galaz-Fontes under project MTM2009-14483-C02-01 and E. A. Sanchez Perez under project MTM2016-77054-C2-1-P. M. FernandezUnzueta was also supported by CONACYT 284110.Calabuig, JM.; Fernández-Unzueta, M.; Galaz-Fontes, F.; Sánchez Pérez, EA. (2019). Equivalent norms in a banach function space and the subsequence property. Journal of the Korean Mathematical Society. 56(5):1387-1401. https://doi.org/10.4134/JKMS.j180682S1387140156