Hardy spaces of operator-valued analytic functions

TL;DR

This paper extends classical Hardy and BMO space theories to operator-valued analytic functions, establishing atomic decompositions, duality theorems, and characterizations analogous to scalar cases.

Contribution

It introduces atomic decomposition for operator-valued Hardy spaces and proves duality and characterization theorems extending scalar results.

Findings

Atomic decomposition for operator-valued Hardy spaces

Extension of BMOA properties to operator-valued functions

Duality between operator-valued H^1 and BMOA spaces

Abstract

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of $\mathbb{C}.$ In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued $\mathrm{H}^1$-BMOA duality theorem is proved. Finally, by the $\mathrm{H}^1$-BMOA duality we present the Lusin area integral and Littlewood-Paley $g$-function characterizations of the operator-valued analytic Hardy space.