Given a Banach space valued q-concave linear operator T defined on a
σ-order continuous quasi-Banach function space, we provide a description
of the optimal domain of T preserving q-concavity, that is, the largest
σ-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
