# Complex Interpolation between Hilbert, Banach and Operator Spaces

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*Gilles Pisier*

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).

#### Table of Contents

# Table of Contents

## Complex Interpolation between Hilbert, Banach and Operator Spaces

- Introduction 19 free
- Chapter 1. Preliminaries. Regular operators 715 free
- Chapter 2. Regular and fully contractive operators 1321
- Chapter 3. Remarks on expanding graphs 1725
- Chapter 4. A duality operators/classes of Banach spaces 2129
- Chapter 5. Complex interpolation of families of Banach spaces 2735
- Chapter 6. - .4 -Hilbertian spaces 3341
- Chapter 7. Arcwise versus not arcwise 4149
- Chapter 8. Fourier and Schur multipliers 4351
- Chapter 9. A characterization of uniformly curved spaces 4755
- Chapter 10. Extension property of regular operators 5159
- Chapter 11. Generalizations 5563
- Chapter 12. Operator space case 6169
- Chapter 13. Generalizations (Operator space case) 6977
- Chapter 14. Examples with the Haagerup tensor product 7381
- References 7583