By Gilles Pisier

Stimulated through a question of Vincent Lafforgue, the writer reviews the Banach areas X pleasing the next estate: there's a functionality \varepsilon\to \Delta_X(\varepsilon) tending to 0 with \varepsilon>0 such that each operator T\colon \ L_2\to L_2 with \|T\|\le \varepsilon that's concurrently contractive (i.e., of norm \le 1) on L_1 and on L_\infty needs to be of norm \le \Delta_X(\varepsilon) on L_2(X). the writer exhibits that \Delta_X(\varepsilon) \in O(\varepsilon^\alpha) for a few \alpha>0 if X is isomorphic to a quotient of a subspace of an ultraproduct of \theta-Hilbertian areas for a few \theta>0 (see Corollary 6.7), the place \theta-Hilbertian is intended in a touch extra normal experience than within the author's previous paper (1979)

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**Example text**

Here again, one can replace 1 by a ﬁxed constant to treat the “isomorphic” (as opposed to isometric) variant of this. 10. More generally, let 1 ≤ p ≤ ∞. e. X, Y ∈ B ⇒ X ⊕p Y ∈ B) then the “norm” of factorization through a space belonging to B is indeed a norm. More precisely, let us deﬁne for any operator u : E → F acting between Banach spaces the norm γB (u) = inf{ u1 u2 } u u 2 1 X −→ F. where the inﬁmum runs over all X in B and all factorizations E −→ Let ΓB (E, F ) denote the space of those u that admit such a factorization.

13) B(θ,n) . Our assumption implies by homogeneity that for any such T TX ≤ C T α T 1−α reg . By a classical real interpolation result (see [2, p. 58]), this can be rewritten as TX ≤ C T (Br ( n ),B( n )) α,1 2 2 . Therefore, the conclusion follows from the general fact (see [2, p. 102]) that whenever we have an inclusion B0 ⊂ B1 , there is a constant C (depending only on α and θ) such that for any θ < α and any T in B0 T (B0 ,B1 )α,1 ≤C T (B0 ,B1 )θ . 13). 4. 8. Let C ≥ 1 be a constant. The following are equivalent: (i) X is C-isomorphic to a quotient of a subspace of an arcwise θ-Hilbertian space.

4) are preserved in this change) assume that λi (z) = Fi (z) and Gj (z) = μj (z). 4) we have (aij (θ)) (Br ( n ),B( n )) θ 2 2 ≤ 1. Thus we obtain vij = λi aij μj with λi = Fi (θ), μj = Gj (θ), aij = aij (θ). Note that λ, μ are in the unit ball of n2 by the maximum principle applied to (Fi ) and (Gj ). This proves the “only if part” except that we replaced the closed unit ball by the open one. An elementary compactness argument completes the proof. The main result of this section is perhaps the following.