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

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.

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