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 property: there is a function $varepsilon o Delta_X(varepsilon)$ tending to zero with $varepsilon>0$ such that every operator $Tcolon L_2 o L_2$ 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 $alpha>0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $ heta$-Hilbertian spaces for some $ heta>0$ (see Corollary 6.7), where $ heta$-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).