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

TL;DR

This paper generalizes Pisier's interpolation method for Hardy spaces, extending it to noncommutative settings and applying it to noncommutative martingale transforms, thus broadening the theoretical framework.

Contribution

It formulates an abstract version of Pisier's method applicable to a wider class of spaces and demonstrates its use in noncommutative martingale analysis.

Findings

Extended Pisier's method to abstract noncommutative spaces

Applied the method to noncommutative martingale transforms

Provided new tools for interpolation in noncommutative analysis

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.