The Two-Parameter Free Unitary Segal-Bargmann Transform and its Biane-Gross-Malliavin Identification

TL;DR

This paper introduces a two-parameter free unitary Segal-Bargmann transform, providing an integral representation, exploring its large-N limit, and establishing a Biane-Gross-Malliavin type theorem in free probability.

Contribution

It derives an integral transform representation for the two-parameter free unitary Segal-Bargmann transform and extends the Biane-Gross-Malliavin theorem to this setting.

Findings

Integral transform representation derived

Limit behavior analyzed as N approaches infinity

Biane-Gross-Malliavin theorem extended to two-parameter free setting

Abstract

Motivated by the two-parameter free unitary Segal-Bargmann transform in the form of conditional expectation, we derive the integral transform representation of the two-parameter free unitary Segal-Bargmann transform which coincides to the large-$N$ limit of the two-parameter Segal-Bargmann transform on the unitary group $\mathbb{U}(N)$ and explore its limiting behavior. We also extend the notion of circular systems in order to define a two-parameter free Segal-Bargmann transform and prove a version of Biane-Gross-Malliavin Theorem of the two-parameter free unitary Segal-Bargmann transform.