Operator Valued Hardy Spaces

TL;DR

This paper systematically studies operator-valued Hardy spaces within non-commutative L^p spaces linked to semifinite von Neumann algebras, motivated by matrix harmonic analysis and non-commutative martingale inequalities.

Contribution

It introduces a new framework for non-commutative Hardy spaces defined via the non-commutative Lusin integral, expanding the theoretical understanding of operator-valued harmonic analysis.

Findings

Established properties of non-commutative Hardy spaces

Connected non-commutative Hardy spaces with martingale inequalities

Provided foundational results for operator-valued harmonic analysis

Abstract

We give a systematic study on the Hardy spaces of functions with values in the non-commutative $L^p$-spaces associated with a semifinite von Neumann algebra ${\cal}M.$ This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities. Our non-commutative Hardy spaces are defined by the non-commutative Lusin integral function.