By Gilles Pisier
Encouraged via a query of Vincent Lafforgue, the writer experiences the Banach areas X enjoyable 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 has 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 marginally extra normal experience than within the author's prior paper (1979)
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Ii) For some index set I, there are a metric surjection Q : 1 (I) → X and an isometric embedding J : X → ∞ (I) such that JQ belongs to the closure of ΓH ( 1 (I), ∞ (I)) in B( 1 (I), ∞ (I)). (iii) For any δ > 0 there is a constant C(δ) satisfying the following: for any measure spaces (Ω, μ), (Ω , μ ), for any maps v : X → L∞ (μ) and v : L1 (μ ) → X and for any δ > 0 there is u in ΓH (L1 (μ ), L∞ (μ)) with γH (u) ≤ C(δ) v v and u − vv < δ v v . (iv) For each δ > 0, there is C(δ) satisfying: there are a Hilbert space H and functions ϕ : X → H, ψ : X ∗ → H such that for all (ξ, x) in X ∗ × X we have |ξ(x) − ψ(ξ), ϕ(x) | ≤ δ ξ ϕ(x) ≤ C(δ) 1/2 x x ψ(ξ) ≤ C(δ)1/2 ξ .
Assume for simplicity γB (u) < C. Then, for some integer n, we N n have a factorization u = αβ with α : np (X ) → N ∞ and β : 1 → p (X ) such that n α β < C. Let S ⊂ p (X ) be the range of β and let S0 ⊂ S be the kernel of α|S . Let q : S → S/S0 denote the quotient map and let α ˜ : S/S0 → N ∞ be the N ˜ ˜ map defined by α|S = αq. ˜ Also let β : 1 → S/S0 be defined by β(·) = qβ(·). ˜ But now α ˜ β. ˜ is injective and β˜ surjective. Consider We have then u = J1 J2∗ = α ˜ (·). 1) we find the mapping w : S/S0 → X defined by w(·) = J1−1 α w ≤ (1 + ε) α ˜ ≤ (1 + ε) α .
Consider an operator u : E/F → E /F . Assume that u ⊗ IX : (E/F ) ⊗ X → (E /F ) ⊗ X is bounded for the norms of E(X)/F (X) and E (X)/F (X). In that case, we denote by uX : (E/F )[X] → (E /F )[X] the resulting operator and we simply say that uX is bounded. 2. Let F ⊂ E ⊂ Lp (μ) and F ⊂ E ⊂ Lp (μ ) be as above. Let C ≥ 1. The following properties of a linear map u : E/F → E /F as equivalent: (i) There is a regular operator u ˜ : Lp (μ) → Lp (μ ) dilating u (in particular such that u ˜(E) ⊂ E and u ˜(F ) ⊂ F ) with u ˜ reg ≤ C.