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Complex interpolation between Hilbert, Banach and operator spaces
About this Title
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: https://doi.org/10.1090/S0065-9266-10-00601-0
Published electronically: May 11, 2010
MSC: Primary 46B70, 47B10, 46M05, 47A80
Table of Contents
Chapters
- Introduction
- 1. Preliminaries. Regular operators
- 2. Regular and fully contractive operators
- 3. Remarks on expanding graphs
- 4. A duality operators/classes of Banach spaces
- 5. Complex interpolation of families of Banach spaces
- 6. $\pmb {\theta }$-Hilbertian spaces
- 7. Arcwise versus not arcwise
- 8. Fourier and Schur multipliers
- 9. A characterization of uniformly curved spaces
- 10. Extension property of regular operators
- 11. Generalizations
- 12. Operator space case
- 13. Generalizations (Operator space case)
- 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). Let $B_{{r}}(L_2(\mu ))$ be the space of all regular operators on $L_2(\mu )$. We are able to describe the complex interpolation space \[ (B_{{r}}(L_2(\mu )), B(L_2(\mu )))^\theta . \] We show that $T\colon \ L_2(\mu )\to L_2(\mu )$ belongs to this space iff $T\otimes id_X$ is bounded on $L_2(X)$ for any $\theta$-Hilbertian space $X$.
More generally, we are able to describe the spaces \[ (B(\ell _{p_0}), B(\ell _{p_1}))^\theta \ \mathrm {or}\ (B(L_{p_0}), B(L_{p_1}))^\theta \] for any pair $1\le p_0,p_1\le \infty$ and $0<\theta <1$. In the same vein, given a locally compact Abelian group $G$, let $M(G)$ (resp. $PM(G)$) be the space of complex measures (resp. pseudo-measures) on $G$ equipped with the usual norm $\|\mu \|_{M(G)} = |\mu |(G)$ (resp. \[ \|\mu \|_{PM(G)} = \sup \{|\hat \mu (\gamma )| \ \big | \ \gamma \in \widehat G\}). \] We describe similarly the interpolation space $(M(G), PM(G))^\theta$. Various extensions and variants of this result will be given, e.g. to Schur multipliers on $B(\ell _2)$ and to operator spaces.
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