Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces $X$ satisfying the following property: there is a function $/varepsilon/to /Delta_X(/varepsilon)$ tending to zero with $/varepsilon>0$ such that every operator $T/colon / L_2/to L_2$ 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 $/theta$-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).
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