This is an announcement for the paper "Optimal domain of $q$-concave operators and vector measure representation of $q$-concave Banach lattices" by O. Delgado and E.A. Sanchez Perez.
Abstract: Given a Banach space valued $q$-concave linear operator $T$ defined on a $\sigma$-order continuous quasi-Banach function space, we provide a description of the optimal domain of $T$ preserving $q$-concavity, that is, the largest $\sigma$-order continuous quasi-Banach function space to which $T$ can be extended as a $q$-concave operator. We show in this way the existence of maximal extensions for $q$-concave operators. As an application, we show a representation theorem for $q$-concave Banach lattices through spaces of integrable functions with respect to a vector measure. This result culminates a series of representation theorems for Banach lattices using vector measures that have been obtained in the last twenty years.
Archive classification: math.FA
Mathematics Subject Classification: 47B38, 46G10, 46E30, 46B42
Submitted from: easancpe@mat.upv.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1511.02337
or