H${}^2$ Spaces of Non-Commutative Functions

TL;DR

This paper introduces Hardy spaces for free noncommutative functions on noncommutative domains, combining noncommutative function theory with asymptotic integration techniques to advance the understanding of free noncommutative analytic spaces.

Contribution

It defines and studies properties of Hardy spaces in noncommutative settings, pioneering the development of free noncommutative bounded symmetric domains and their analytic function analogues.

Findings

First definitions of Hardy spaces for free noncommutative functions

Establishment of basic properties of these spaces

Foundation for future development of noncommutative bounded symmetric domains

Abstract

We define the Hardy spaces of free noncommutative functions on the noncommutative polydisc and the noncommutative ball and study their basic properties. Our technique combines the general methods of noncommutative function theory and asymptotic formulae for integration over the unitary group. The results are the first step in developing the general theory of free noncommutative bounded symmetric domains on the one hand and in studying the asymptotic free noncommutative analogues of classical spaces of analytic functions on the other.