  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.
