eBook ISBN:  9781470405922 
Product Code:  MEMO/208/978.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 
eBook ISBN:  9781470405922 
Product Code:  MEMO/208/978.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 208; 2010; 78 ppMSC: Primary 46; 47;
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

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


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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).

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