Wavelet approach to operator-valued Hardy spaces

TL;DR

This paper develops a wavelet-based framework for operator-valued Hardy spaces, demonstrating their equivalence to existing definitions and providing an explicit basis for H1(R) with an operator space structure.

Contribution

It introduces a wavelet approach to operator-valued Hardy spaces, establishing their equivalence with Tao Mei's definitions and constructing an explicit basis for H1(R).

Findings

Hardy spaces via wavelet coincide with Tao Mei's definitions.

Provided an explicit unconditional basis for H1(R).

Extended wavelet methods to operator-valued Hardy spaces.

Abstract

This paper is devoted to the study of operator-valued Hardy spaces via wavelet method. This approach is parallel to that in noncommutative martingale case. We show that our Hardy spaces defined by wavelet coincide with those introduced by Tao Mei via the usual Lusin and Littlewood-Paley square functions. As a consequence, we give an explicit complete unconditional basis of the Hardy space H1(R) when H1(R) is equipped with an appropriate operator space structure.