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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, Paris, France
Complex Interpolation between Hilbert, Banach and Operator Spaces
eBook ISBN:  978-1-4704-0592-2
Product Code:  MEMO/208/978.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
Complex Interpolation between Hilbert, Banach and Operator Spaces
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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, Paris, France
eBook ISBN:  978-1-4704-0592-2
Product Code:  MEMO/208/978.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2082010; 78 pp
    MSC: 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2082010; 78 pp
MSC: 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).

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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