درون‌یابی پیچیده بین فضاهای هیلبرت، باناک و عملگر (یادداشت‌های انجمن ریاضی آمریکا)

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). Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function ε→ΔX(ε) tending to zero with ε > 0 such that every operator T: L2→L2 with ∥T∥≤ε that is simultaneously contractive (i.e. of norm ≤1) on L1 and on L∞ must be of norm ≤ΔX(ε) on L2(X). We show that ΔX(ε)∈O(εα) for some α > 0 iff XX is isomorphic to a quotient of a subspace of an ultraproduct of θ-Hilbertian spaces for some θ > 0 (see Corollary 6.7), where θ-Hilbertian is meant in a slightly more general sense than in our previous paper (1979)