Complex Interpolation between Hilbert, Banach and Operator Spaces

Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces X satisfying the following there is a function varepsilonto DeltaX(varepsilon) tending to zero with varepsilon>0 such that every operator Tcolon L2to L2 with Tle varepsilon that is simultaneously contractive (i.e., of norm le 1) on L1 and on Linfty must be of norm le DeltaX(varepsilon) on L2(X). The author shows that DeltaX(varepsilon) in O(varepsilonalpha) 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).