On operator valued Hardy spaces

TL;DR

This paper proves that the operator-valued Hardy space , defined through Littlewood-Paley theory, is equivalent to a space of Bochner integrable functions with analytic parts, using noncommutative maximal inequalities.

Contribution

It establishes the equivalence of two definitions of operator-valued Hardy spaces, advancing the understanding of noncommutative harmonic analysis.

Findings

Operator-valued Hardy space  coincides with a space of Bochner integrable functions.

The proof utilizes noncommutative maximal inequalities for Poisson groups.

The result bridges Littlewood-Paley theory and analytic function spaces in the noncommutative setting.

Abstract

The note shows that the operator-valued Hardy space $\sH^1$ introduced via Littlewood-Paley $g$-function coincides with the space of $H^1_R(\T, \sL^1)$ of all Bochner integrable operator-valued functions with integrable analytic part. The proof is based on the noncommutative maximal inequality for Poisson group.