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A note on Pisier's method in interpolation of abstract Hardy spaces

Hugues Moyart

Abstract

In his approach to Jones theorem on the interpolation of Hardy spaces on the torus, Pisier introduced an original method allowing the computation of complex interpolation spaces by means of real interpolation techniques. This approach has been successfully extended to noncommutative analytic Hardy spaces arising from subdiagonal algebras. In this paper, we formulate and prove an abstract version of Pisier s method in a more general setting. The method is then applied in the study of noncommutative martingale transforms.

A note on Pisier's method in interpolation of abstract Hardy spaces

Abstract

In his approach to Jones theorem on the interpolation of Hardy spaces on the torus, Pisier introduced an original method allowing the computation of complex interpolation spaces by means of real interpolation techniques. This approach has been successfully extended to noncommutative analytic Hardy spaces arising from subdiagonal algebras. In this paper, we formulate and prove an abstract version of Pisier s method in a more general setting. The method is then applied in the study of noncommutative martingale transforms.
Paper Structure (25 sections, 38 theorems, 102 equations)

This paper contains 25 sections, 38 theorems, 102 equations.

Table of Contents

  1. Preliminaries
  2. Abstract interpolation theory
  3. Compatible couples
  4. Compatible bounded operators
  5. Interpolation functors
  6. Subcouples
  7. Quotient couples
  8. Complementation
  9. Duality
  10. The complex method
  11. The real method
  12. -functionals
  13. -functionals
  14. The real interpolation spaces
  15. Quasi-complementation
  16. ...and 10 more sections

Key Result

Proposition I.1

Let $(A_0,A_1)$ be a normal subcouple of a compatible couple $(E_0,E_1)$. Then the projection $E_0\cap E_1\to(E_0/A_0)\cap(E_1/A_1)$ is surjective.

Theorems & Definitions (54)

  • Remark
  • Remark
  • Proposition I.1
  • proof
  • Proposition I.2: JansonRealInterpolation(Proposition 6.2)
  • Proposition I.3: BerghInterpolation(Theorem 4.2.2)
  • Theorem I.4: Duality Theorem
  • Corollary I.5
  • proof
  • Theorem I.6: Reiteration theorem
  • ...and 44 more