Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces X satisfying the following property: there is a function varepsilonto Delta_X(varepsilon) tending to zero with varepsilon>0 such that every operator Tcolon L_2to L_2 with |T|le varepsilon that is simultaneously contractive (i.e., of norm le 1) on L_1 and on L_infty must be of norm le Delta_X(varepsilon) on L_2(X). The author shows that Delta_X(varepsilon) in O(varepsilon^alpha) for some alpha>0 if X is isomorphic to a quotient of a subspace of an ultraproduct of theta-Hilbertian spaces for some theta>0 (see Corollary 6.7), where theta-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).