The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let MM be a von Neumann algebra equipped with a normal faithful semifinite trace τ, and let E   be an r.i. space on (0,∞)(0,∞). Let E(M)E(M) be the associated symmetric space of measurable operators. Then to any bounded linear map T   from E(M)E(M) into a Hilbert space HH corresponds a positive norm one functional f∈E(2)(M)∗f∈E(2)(M)∗ such that∀x∈E(M)‖T(x)‖2⩽K2‖T‖2f(x∗x+xx∗), where E(2)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)E(M) when E is either 2-concave or 2-convex and q  -concave for some q<∞q<∞. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one.