  We prove the little Grothendieck theorem for any 2-convex noncommutative
symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal
faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0, \8)$. Let
$E(\M)$ be the associated symmetric space of measurable operators. Then to any
bounded linear map $T$ from $E(\M)$ into a Hilbert space $\mathcal H$
corresponds a positive norm one functional $f\in E_{(2)}(\M)^*$ such that
$$\forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*),$$ where
$E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant.
As a consequence we obtain the noncommutative Khintchine inequalities for
$E(\M)$ when $E$ is either 2-concave or 2-convex and $q$-concave for some
$q<\8$. We apply these results to the study of Schur multipliers from a
2-convex unitary ideal into a 2-concave one.
