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# memo_has_moved_text(); Complex interpolation between Hilbert, Banach and operator spaces

Gilles Pisier, Texas A&M University, College Station, Texas 77843 and Université Paris VI, Equipe d’Analyse, Case 186, 75252, Paris Cedex 05, France

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: Volume 208, Number 978
ISBNs: 978-0-8218-4842-5 (print); 978-1-4704-0592-2 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00601-0
Posted: May 11, 2010
MathSciNet review: 2732331
MSC (2000): Primary 46B70; Secondary 43A15, 46B20, 46B28, 46L07, 46M35

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Chapters

• Introduction
• Chapter 1. Preliminaries. Regular operators
• Chapter 2. Regular and fully contractive operators
• Chapter 3. Remarks on expanding graphs
• Chapter 4. A duality operators/classes of Banach spaces
• Chapter 5. Complex interpolation of families of Banach spaces
• Chapter 6. $\theta$-Hilbertian spaces
• Chapter 7. Arcwise versus not arcwise
• Chapter 8. Fourier and Schur multipliers
• Chapter 9. A characterization of uniformly curved spaces
• Chapter 10. Extension property of regular operators
• Chapter 11. Generalizations
• Chapter 12. Operator space case
• Chapter 13. Generalizations (operator space case)
• Chapter 14. Examples with the Haagerup tensor product

### Abstract

Motivated by a question of Vincent Lafforgue, we study 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 $L_2(X)$. We show 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 our previous paper (1979).