Principal functions for bi-free central limit distributions

TL;DR

This paper determines the principal function and essential spectrum of a specific non-normal operator associated with bi-free central limit distributions, advancing understanding in free probability and operator theory.

Contribution

It introduces the principal function for a class of non-normal operators linked to bi-free central limit distributions, providing new spectral analysis tools.

Findings

Principal function explicitly computed for the operator.

Essential spectrum characterized for the operator.

Advances spectral analysis in bi-free probability.

Abstract

We find the principal function of the completely non-normal operator l(v_1)+l(v_1)^*+i(r(v_2)+r(v_2)^*) on a subspace of the full Fock space F(H) which arises from a bi-free central limit distribution. As an application, we find the essential spectrum of this operator.