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Complex Interpolation between Hilbert, Banach and Operator Spaces
Gilles Pisier, Texas A&M University, College Station, TX, and Université Paris VI, France
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Memoirs of the American Mathematical Society
2010; 78 pp; softcover
Volume: 208
ISBN-10: 0-8218-4842-9
ISBN-13: 978-0-8218-4842-5
List Price: US$68 Individual Members: US$40.80
Institutional Members: US\$54.40
Order Code: MEMO/208/978

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).

• Introduction
• Preliminaries. Regular operators
• Regular and fully contractive operators
• Remarks on expanding graphs
• A duality operators/classes of Banach spaces
• Complex interpolation of families of Banach spaces
• $$\pmb{\theta}$$-Hilbertian spaces
• Arcwise versus not arcwise
• Fourier and Schur multipliers
• A characterization of uniformly curved spaces
• Extension property of regular operators
• Generalizations
• Operator space case
• Generalizations (Operator space case)
• Examples with the Haagerup tensor product
• References